philosophy of physics

Turn and Face The Strange … Ch-Ch-Changes not prec lude the possibility that the phenomenon c an be modeled within the lower-level theory in a different way. There may be aspec ts of the phenomenon (suc h as, say, its mac rosc opic similarity to other phenomena) that c annot be c aptured by the desc riptive resourc es of the theory, but the phenomenon itself c an be desc ribed by the theory. Consider, for instanc e, the relationship between neurosc ienc e and folk psyc hology. It might be argued that the latter is explanatorily irreduc ible to the former. Perhaps there is no viable neurosc ientific ac c ount of why the reasons explanations c ommon in folk psyc hology are suc c essful, but a materialist about the mind c ould maintain that this is merely bec ause the neurosc ientific theory operates at too fine a sc ale to disc ern the patterns that ground this sort of explanation. In every token instanc e c overed by the folk psyc hologic al explanation, there is nothing relevant going on that is not c aptured by neurosc ienc e. It is just that the way neurosc ienc e desc ribes what is going on is not c onduc ive to the c onstruc tion or justific ation of reasons explanations. The patterns that the neurosc ientific desc ription fails to see are nonetheless wholly generated by proc esses desc ribable using neuroscience. A substanc e dualist, however, would argue that there is an even deeper failure of reduc tion going on here. The phenomena and proc esses desc ribed by neurosc ienc e are by themselves inadequate to even generate the kinds of patterns that c harac teriz e reasons explanations. This is bec ause the lower-level theory does not have the resourc es to desc ribe a c ruc ial element of the ontologic al furniture of the situation, the mind or the soul. Here we have more than a mere c ase of explanatory irre-duc ibility. We may c all c ases like this, where the lower-level theory c annot even fully desc ribe a phenomenon that c an be modeled by the higher-level theory, examples of ontological irreducibility. This is probably the sense in whic h the British emergentists c onc eived of emergenc e (see Mc Laughlin (1992) for an illuminating analysis of this sc hool of thought). With referenc e to phase transitions, this view is perhaps most starkly expressed in Batterman (2005). Batterman argues that the disc ontinuity in the thermodynamic potential at a phase transition is not an artifac t of a partic ular mathematic al representation of the physic al phenomenon but is a feature of the physic al phenomenon itself. He says, “My c ontention is that thermodynamic s is c orrec t to c harac teriz e phase transitions as real physic al disc ontinuities and it is c orrec t to represent them mathematic ally as singularities” (ibid., 234). If there are genuine disc ontinuities in physic al systems, it seems we c ould not represent them ac c urately using only c ontinuous mathematic al func tions. So, sinc e the statistic al mec hanic s of finite systems does not give us disc ontinuities, it is inc apable of fully desc ribing this physic al phenomenon. We c an only approac h an explanation of the phenomenon by working in the infinite limit. The idealiz ation is a manifestation of the inability of the theory to fully desc ribe the phenomenon of phase transitions in finite systems. We disc uss these ideas further in section 3.3. In the remainder of this c hapter, we disc uss the status of these three notions of emergenc e—c onc eptual novelty, explanatory irreduc ibility, and ontologic al irreduc ibility—as they apply to both the standard statistic al mec hanic al notion of phase transitions and the treatment of c ritic al phenomena by the renormaliz a-tion group. These topic s are treated separately bec ause, as disc ussed above, the renormaliz ation group introduc es new issues bearing on the topic of emergenc e and reduc tion that go beyond issues involving infinite idealiz ation in traditional statistic al mechanics. 3. The Infinite Idealizatio n in Statistical Mechanics In the previous sec tion, we disc ussed three ways in whic h the relationship between statistic al mec hanic s and thermodynamic s might be nonreduc tive. There is a hierarc hy to these different senses of emergenc e set by the varying strengths of the assumptions about explanation required in order for them to represent a genuine failure of the c ore sense of reduc tion. Conc eptual novelty is the weakest notion of emergenc e, explanatory irreduc ibility is stronger, and ontologic al irreduc ibility is stronger still. In this sec tion, we disc uss the c ase that c an be made for phase transitions exemplifying eac h of these notions of emergenc e. We c onc lude that in the domain of ordinary statistic al mec hanic s (exc luding the renormaliz ation group), the c ase for phase transitions being either ontologic ally or explanatorily irreduc ible is weak. The c ase for phase transitions being c onc eptually novel is stronger, but even here there are questions that c an be raised. 3.1 Conceptual Novelty Page 10 of 24

Turn and Face The Strange … Ch-Ch-Changes A natural kind in a higher-level theory is c onc eptually novel if there is no kind in any potential reduc ing theory that c aptures the same set of phenomena. Are thermodynamic phase transitions c onc eptually novel? That is, does the kind ‘phase transition’ have a natural c ounterpart kind in statistic al mec hanic s? If we restric t ourselves to finite N systems, it is c ommonly believed that there is not a kind in statistic al mec hanic s c orresponding to phase transitions and that one c an only find suc h a kind in infinite N statistic al mec hanic s. We believe, to the contrary, that no theory, infinite or finite, statistical mechanical or mechanical, possesses a natural kind that perfectly overlaps with the thermodynamic natural kind. Yet if one relaxes the demand of perfec t overlap, then there are kinds— even in finite N statistic al mec hanic s—that overlap in interesting and explanatorily powerful ways with thermodynamic phase transitions. Stric tly speaking, thermodynamic phase transitions are c onc eptually novel; more loosely speaking, they are not. To begin, one might wonder in what sense “phase transition” is a kind even in thermodynamic s. After all, there are ambiguities in the way we define phases. Is glass a superc ooled liquid or a solid? It depends on whic h c riteria one uses and no set seems obviously superior. Be that as it may, the notion of a transition is relatively c lear in thermodynamic s, and it is defined, as above, as a disc ontinuity in one of the thermodynamic potentials. Let's stic k with this. Now, is the kind pic ked out by Def 1 the c ounterpart of the thermodynamic definition? Despite many c laims that it is, Def 1's extension is c learly very different than that given by thermodynamic s. To mention the most glaring differenc e—and on whic h, more later—there are many systems that do not have well-defined ther-modynamic limits. Do they not have phase transitions? One c an define words as one likes, but the point is that there are many systems that suffer abrupt mac rosc opic c hanges, c hanges that thermodynamic s would c ount as phase transitions, but whic h do not have thermodynamic limits. Systems with very long-range interac tions are prominent examples. But in fac t the c onditions on the existenc e of a thermody-namic limit are numerous and stringent, so in some sense most systems do not have thermodynamic limits. A strong c ase c an be made that Def 1, as a result, provides at best suffic ient c onditions for a phase transition, and not nec essary c onditions. How does finite N statistic al mec hanic s fare? The c onventional wisdom is that finite N statistic al mec hanic s lac ks the resourc es to have c ounterparts of thermodynamic s phase transitions. However, we believe that people often assent to this c laim too quic kly. One of the more interesting developments in statistic al mec hanic s of late has been c hallenges to ordinary statistic al mec hanic s from the realms of the very large and the very small. These are regimes that test the applic ability of normal Boltz mann-Gibbs equilibrium statistic al mec hanic s. The issues arise from the suc c ess of statistic al mec hanic al tec hniques in new areas. In c osmology, statistic al mec hanic s is used not only to explain the inner workings of stars but also to explain the statistic al distribution of galaxies, c lusters, and more. In these c ases, the forc e of interest is of c ourse the gravitational forc e, one that is not sc reened at short distanc es like the Coulomb forc e. Systems like this do not have a well-defined thermodynamic limit, often are not approximately extensive, suffer negative heat c apac ities, and more (see Callender (2011) for disc ussion). There has also been an extension of statistic al mec hanic al tec hniques to the realm of the small. Sodium c lusters obey a solidlike to liquidlike “phase transition,” Bose-Einstein c ondensation oc c urs, and muc h more. These atomic c lusters have been surprisingly amenable to statistic al mec hanic al treatment, yet they too do not satisfy the c onditions for the applic ation of the thermodynamic limit. Physic ally, one way to think about what is happening here is that in small systems a muc h higher proportion of the partic les reside on the surfac e, so surfac e effec ts play a substantial role in the physic s. As a result, these systems also raise issues about extensivity, negative spec ific heats, and muc h more.4 These systems are relevant to our c onc erns here for a very simple reason: they appear to have phase transitions, yet lac k a well-defined thermodynamic limit, so Def 1 seems inadequate. Orthogonal to our philosophic al worries about reduc tion, there are also purely physic al motivations for better understanding thermodynamic phase transitions from the perspec tive of finite statistic al mec hanic s. Naturally, some physic ists appear motivated by both issues, the c onc eptual and the physic al: Conc eptually, the nec essity of the thermodynamic limit is an objec tionable feature: first, the number of degrees of freedom in real systems, although possibly large, is finite, and, sec ond, for systems with long-range interac tions, the thermodynamic limit may even be not well defined. These observations indic ate that the theoretic al desc ription of phase transitions, although very suc c essful in c ertain aspec ts, may not be c ompletely satisfac tory. (Kastner 2008, 168) Page 11 of 24

Turn and Face The Strange … Ch-Ch-Changes As a result of this motivation, there are already several proposals for finite-partic le ac c ounts of phase transitions. These are sometimes c alled smooth phase transitions. The researc h is ongoing, but what exists already provides evidenc e of the existenc e of thermodynamic phase transitions in finite systems. There are many different sc hemes, but we will concentrate on the two most well known. 3.1.1 Back-Bending Figure 5.1 Back-bending of the caloric curve. Inspired in part by van der Waals theory and its S-shaped bends, this theory has been developed by Wales and Berry (1994), Gross and Votyakov (2000) and Chomaz , Gulminelli, and Duflot (2001). Unlike in the traditional theory of phase transitions, here the authors work with the mic roc anonic al ensemble, not the c anonic al ensemble. The general idea is that the signatures of phase transitions of different orders are read off from the c urvature of the mic roc anonic al entropy, S = kb lnΩ(E), where Ω(E) is the mic roc anonic al partition func tion. In partic ular, if written in terms of the assoc iated c aloric c urve, T(E) = 1/∂ E ln[Ω(E)], we c an understand a first-order transition as a “bac k- bending” c urve, where for a given value of T(E) one c an have more than one set of values for E/N (see figure 5.1). For our illustrative purposes, we will use this as our definition: (Def 2) A first-order phase transition oc c urs when there is “bac k-bending” in the mic roc anonic al c aloric curve. Def 2 is equivalent to the entropy being c onvex or the heat c apac ity being negative for c ertain values. As expec ted, bac k-bending c an be seen in finite-N systems. So with Def 2 we have an alternative c riterion of phase transitions that nic ely c harac teriz es phase transitions even in systems that do not have thermodynamic limits. We hasten to add that the theory is not exhausted by a simple definition. Rather, the hope— whic h has to some extent been realiz ed—is that it and its generaliz ations c an predic t and explain both c ontinuous phase transitions and also phase transitions in systems lac king a thermodynamic limit. Def 2 is rather striking when one realiz es that it is equivalent to a region of negative heat c apac ities appearing. The reader familiar with the van Hove theorem may be alarmed, for that theorem forbids bac k-bending in the thermodynamic limit. Sinc e our c onc erns are about the finite c ase, this in itself is not troubling. But if one hopes that this definition goes over to the infinite N definition in the thermody-namic limit, where ensemble equivalenc e holds for many systems, this might be a problem: the c anonic al ensemble c an never have negative heat c apac ity, whereas the mic roc anonic al one c an, and yet they are equivalent for “normal” short-range systems in the thermodynamic limit. Does “ensemble equivalenc e” in the infinite limit squeez e out these negative heat c apac ities? No, for one must remember that ensemble equivalenc e holds, where it does, only when systems are not undergoing phase transitions. This is a point originally made by Gibbs (1902). And indeed, ensemble inequivalenc e c an be used as a marker of phase transitions. What is happening is that the mic roc anonic al ensemble has struc ture that the c anonic al ensemble c annot see; the regions of bac k-bending (or c onvex entropy, or negative heat c apac ity) are missed by the c anonic al ensemble. Yet sinc e the c anonic al ensemble is equivalent to the mic roc anonic al—if at all—only when no phase transition obtains, there is no opportunity for c onflic t with “equivalence” results. This remark provides a c lue to the relation between Def 1 and Def 2 and a way of thinking about the first as a subspec ies of the sec ond. When there is bac k-bending there is ensemble inequivalenc e. From the perspec tive of the c anonic al ensemble for an infinite system, this is where a nonanalytic ity appears in the thermodynamic limit. It Page 12 of 24

Turn and Face The Strange … Ch-Ch-Changes c an “see” the phase transition in that c ase; but when finite it is blind to this struc ture. Def 2 c an then be seen as more general, sinc e it triggers the nonanalytic ity seen in infinite systems and c aptured by Def 1 but also applies to finite systems. Many more interesting fac ts have rec ently been unearthed about the relationships among bac k-bending, nonc onc ave entropies, negative heat c apac ity, ensemble inequivalenc e, phase transitions, and nonextensivity We refer the reader to Touc hette and Ellis (2005) for disc ussion and referenc es. For rigorous c onnec tions between Def 1 and Def 2, see Touc hette (2006). 3.1.2 Distribution of Zeros This approac h grows direc tly out of the Yang-Lee pic ture. The Yang-Lee theorem is about the distribution of z eros of the grand c anonic al ensemble's partition func tion in the c omplex plane. A c ritic al point is enc ountered when this distribution “pinc hes” the real axis, and this c an only oc c ur when the number of z eros is infinite. Fisher and later Grossmann then provided an elaborate c lassific ation of phase transitions in terms of the distribution of z eros of the c anonic al partition func tion in the c omplex temperature plane. Interested in Bose-Einstein c ondensation, nuc lear fragmentation and other “phase transitions” in small systems, a group of physic ists at the University of Oldenburg sought to extend this approac h to the finite c ase (see Borrmann, Mülken, and Harting 2000). For our purposes, we can define their phase transitions as: (Def 3) A phase transition oc c urs when the z eros of the c anonic al partition func tion align perpendic ularly to the real temperature axis and the density sc ales with the number of partic les. The distribution of z eros of a partition func tion c ontains a lot of information. The idea behind this approac h is to extrac t three parameters (α ,γ,τ1) from the partition func tion that tell us about this distribution: τ1 is a func tion of the number of z eros in the c omplex temperature plane, and it is positive for finite systems; γ is the c rossing angle between the real axis and the line of z eros; and α is determined from the approximate density of z eros on that line. What happens as we approac h a phase transition is that the distribution of z eros in the c omplex temperature plane “line up” and gradually gets denser and straighter as N inc reases.5 Figure 5.2 Distribution of zeros in the complex inverse temperature (β= 1/kT) plane. We stress that, as with the previous group, the physic ists involved do not offer a stray definition but rather a c omprehensive theory of phase transitions in small systems. In partic ular, the Oldenburg group c an use this theory to not only predic t whether there is a phase transition but also to identify the c orrec t order of the transition. Their c lassific ation exc els when treating Bose-Einstein c ondensation, as it reproduc es the spac e dimension and partic le number dependenc e of the transition order. Like the approac h using Def 2, the present approac h works for both finite and infinite systems. For finite systems, τ1 is always positive and we look for c ases where α = γ: these c orrespond to first-order transitions in finite systems. More c omplic ated relations between α and γ c orrespond to higher-order transitions. For infinite systems, phase transitions of the first-order oc c ur when α = γ = τ1 = 0 and for higher-order when α 〉 0. So the sc heme inc ludes the Def 1 c ase as a subspec ies. One c an then view Def 3—or more ac c urately, the whole c lassific ation sc heme assoc iated with (α ,γ, τ1)—as a wider, more general definition of phase transitions, one inc luding small systems, with Def 1 as a spec ial c ase when the thermodynamic limit is legitimate. Page 13 of 24

Turn and Face The Strange … Ch-Ch-Changes What is the relationship between Def 2 and Def 3? It turns out that they are almost equivalent. Indeed, if one ignores a c lass of systems that may turn out to be unphysic al, they are demonstrably equivalent; see Touc hette (2006).6 The ric h sc hemes of whic h these definitions form a part may not be equivalent, but on the question of what counts as a phase transition they will largely agree. As a result of the work on finite-N definitions—and while duly rec ogniz ing that it is very muc h ongoing—it seems to us that statistic al mec hanic s is hardly at a loss to desc ribe phase transitions in finite systems. The situation instead seems to us to be more subtle. No definition in statistic al mec hanic s, infinite or finite, exac tly reproduc es the extension pic ked out by thermodynamic s with the kind “phase transition.” What one judges the best definition then hangs on what extension one wants to preserve. If foc using on thermodynamic systems possessing thermodynamic limits, then Def 1 is fine. Then the kind “phase transition” is c onc eptually emergent relative to finite-N statistic al mec hanic s. But if impressed by long-range systems, small systems, nonextensive systems, and “solidlike-to-liquidlike” mesosc opic transitions, then one of the finite-N definitions is nec essary. Relative to these definitions, the kind “phase transition” is not c onc eptually novel. If one wants a c omprehensive definition, for finite and infinite, then the sc hemes desc ribed provide the best bet. Probably none of the definitions provide nec essary and suffic ient c onditions for a phase transition that overlaps perfec tly with thermodynamic phase transitions. That, however, is okay, for thermodynamic s itself does not neatly c harac teriz e all the ways in whic h mac rostates c an change in an “abrupt” way. In any c ase, we do not believe that c onc eptual novelty by itself poses a major threat to reduc tionism. After all even a (too) stric t Nagelian notion of reduc tion c an ac c ommodate c onc eptual novelty (as long as the novel higher-level kind is expressible as a finite disjunc tion of lower-level kinds). Conc eptual novelty is only a problem when you do not have explanatory reduc ibility of the c onc eptually novel kind, a question to whic h we now turn. 3.2 Explanatory Irreducibility Explanatory irreduc ibility oc c urs, we said, when the explanation of a higher-level phenomenon requires a c onc eptual novelty, yet the reduc ing theory does not have the resourc es to explain why the c onc eptual novelty is warranted.7 Where phase transitions are espec ially interesting, philosophic ally, lies in the fac t that, at first glanc e, they seem to be a real-life and prominent instanc e of explanatory irreduc ibility. To arrive at this c laim, let us suppose that the finite-N definitions surveyed above are theoretic ally inadequate. Assume that Def 1 is employed in the best explanation of the phenomena. Then we have already seen that no finite-N statistic al mec hanic s c an suffer phase transitions so understood. If the “reduc ing theory” is finite-N statistic al mec hanic s, then we potentially have a c ase of explanatory irreduc ibility. But should the reduc ing theory be restric ted to finite-N theory? One quic k way out of diffic ulty would be to inc lude the thermodynamic limit as part of the reduc ing theory. However, this would be a c heat. The thermodynamic limit is, we believe, the produc tion of another phenomenologic al theory, not a piec e of the reduc ing theory. The ontology of the c lassic al reduc ing theory is supposed to be finite-N c lassic al mec hanic s. Suc h a theory has surfac e effec ts, fluc tuations, and more, but the thermodynamic limit squashes these out. More importantly, the ontology of the system in the thermodynamic limit is not the c lassic al mec hanic s of billiard balls and the like. A quic k and interesting way to see this point is to note that the thermodynamic limit is mathematic ally equivalent to the c ontinuum limit (Compagner 1989). The c ontinuum limit is one wherein the siz e and number of partic les is dec reased without bound in a finite-siz ed volume. When thermodynamic s emerges from this limit, it is emerging from a theory desc ribing c ontinuous matter, not atomistic matter. New light is shed on all that is regained in the thermodynamic limit if we see it as regained in the c ontinuum limit. For here we do not see properties emerging from an atomic mic roworld behaving thermodynamic ally, but rather properties emerging from a c ontinuum, a realm well “above” the atomic . For this reason, with respec t to the reduc tion of thermodynamic s to statistic al mec hanic s, we do not see proofs that thermodynamic properties emerge in the thermody-namic limit as c ases whereby thermodynamic properties are reduc ed to mec hanic al properties. If this is right, then we have a potential c ase of explanatory irreduc ibility. The best explanation of the phenomenon of phase transitions c ontains an idealiz ation whose effic ac y c annot be explained from the perspec tive of finite-N theory. So are phase transitions ac tually explanatorily irreduc ible? The answer hangs on whether de-idealiz ation c an be ac hieved within finite-N statistic al mec hanic s. We believe that it c an be. We have already hinted at one possibility. If one c ould show that one or more of the finite-N definitions approximate in a c ontrolled way Def 1, then we c ould view Def 1 as “really” talking about one of the other definitions. Indeed, this seems very muc h a live Page 14 of 24

Turn and Face The Strange … Ch-Ch-Changes possibility with either Def 2 or Def 3 above. However, suppose we believe that this is not possible. Is there any other way of de-idealiz ing the standard treatment of phase transitions? We believe that there is, and both Butterfield (2011) and Kadanoff (2009) point toward the right diagnosis. Before getting to that, however, notic e that the ac tual prac tic e of the sc ienc e more or less guarantees that some finite-N approximation must be available. In rec ent years there has been an effloresc enc e of c omputational models of statistic al mec hanic al phenomena (see Krauth 2006). Sinc e we c annot simulate an infinite system, these models give an inkling of how we might approximate the divergenc es assoc iated with c ritic al behavior in a finite system. Consider, for instanc e, the Monte Carlo implementation of the Ising model (see, for instanc e, Wolff (1989)). The Monte Carlo method involves pic king some probabilistic algorithm for propagating fluc tuations in the lattic e c onfiguration of an Ising system as time evolves. Eac h run of the simulation is a random walk through the spac e of c onfigurations, and we study the statistic al properties of ensembles of these walks. It might be argued that the system siz e in these simulations is effec tively infinite, sinc e the lattic e is usually implemented with periodic boundary c onditions. However, this periodic ity should be interpreted merely as a c omputational tool, not as a simulation of infinite system siz e. The algorithm is supposed to study the manner in whic h fluc tuations propagate through the lattic e, but the model will only work if the c orrelation length is less than the periodic ity of the system. If fluc tuations propagate over sc ales larger than the periodic ity, we will have a c onflic t between the propagation of fluc tuations and the c onstraints set by the periodic ity of boundary c onditions. So the periodic boundary c onditions should be interpreted as setting an effec tive system siz e. The model is only useful as long as the c orrelation length remains below this c harac teristic length sc ale. Unfortunately, the periodic boundary c onditions also mean that the model is not ac c urate at the c ritic al point, only c lose to it. As the c orrelation length approac hes system siz e in a real system, surfac e effec ts bec ome relevant, and the simulation neglec ts these effec ts. Nonetheless, the Monte Carlo method does allow us to see how Ising systems approac h c ritic al behavior near the c ritic al point. For instanc e, models exhibit the inc rease of c orrelation length as the c ritic al point is approac hed and the assoc iated slow-down of equilibriation (due to the inc reased length over whic h fluc tuations propagate). As we c onstruc t larger and larger systems, the model is prec ise c loser and c loser to the c ritic al point, and we c an see the c orrelation length get larger. We c an also model the nonequilibrium phenomenon of avalanc hes, where the order parameter of the system c hanges in a series of sharp jumps as the external parameter in the Hamiltonian is varied. As an example, the magnetiz ation of a magnetic material exhibits avalanc hes as the external field is tuned. The avalanc hes are due to the way in whic h fluc tuations of c lusters of spins trigger further fluc tuations. At the c ritic al point, we get avalanc hes of all siz es. Again, the approac h to this behavior c an be studied by examining how the distribution of avalanc hes c hanges as the system approac hes the c ritic al point. These are just some examples of how finite models c an be c onstruc ted to examine the behavior of a system arbitrarily c lose to the c ritic al point. These models fail suffic iently c lose to c ritic ality bec ause they do not adequately deal with boundary effec ts. However, they do give an indic ation of how the behavior of large finite systems c an be seen as smoothly approximating the behavior of infinite systems. We now turn to a more explic it attempt to understand the idealiz ation. Butter-field (2011, § 3.3 and § 7) thinks the treatment of phase transitions does not oc c asion any great mystery. We agree and reproduc e his mathematic al analogy (with slight modific ations) to illustrate the point. Consider a sequenc e of real func tions gN, where N ranges over the natural numbers. For eac h value of N, the func tion gN (x) is c ontinuous. It is equal to −1 when x is less than or equal to −1/N, inc reases linearly with slope N when x is between −1/N and 1/N, and then stays at 1 when x is greater than or equal to 1/N. The slope of the segment c onnec ting the two c onstant segments of the func tion gets steeper and steeper as N inc reases. While every member of this sequenc e of func tions is c ontinuous, the limit of the sequenc e g∞(x) is disc ontinuous at x = 0. Now c onsider another sequenc e of real func tions of x, fN. These are two-valued func tions, defined as follows: Given these definitions, fN(x) is the c onstant z ero func tion for all N. If we just look at the sequenc e of func tions, we Page 15 of 24

Turn and Face The Strange … Ch-Ch-Changes would expec t the limit of the sequenc e fN as N → ∞ to also be c onstant. However, if we c onstruc t f∞ (x) from g∞ (x) using the above definition, we will not get a c onstant func tion. The func tion will be disc ontinuous; it will take on the value 1 at x = 0. If one foc uses only on fN without paying attention to how it is generated from gN, the behavior in the limit will seem mysterious and inexplic able given the behavior at finite N. Imagine that we represent a physic al property in a model in terms of fN(x) taking on the value 1, where N is a measure of the siz e of the physic al system. This property c an only be exemplified in the infinite-N limit, of c ourse. And if we restric ted ourselves to c onsidering fN when trying to explain the property, we would be at a loss. No matter how big N gets, as long as it is finite there is no notion of being nearer or further away from the property obtaining. We might c onc lude that the property is emergent in the infinite limit, sinc e we c annot “de-idealiz e” as we did in the case of extensivity and show how a finite system approximates this property. However, this is only bec ause we are not paying attention to the gN(x). Realiz ing the relationship between fN and gN allows us to ac c ount for the behavior of fN in the infinite limit from a finite system perspec tive, sinc e there is a c lear sense in whic h the func tions gN approac h disc ontinuity as N approac hes infinity. We might put the point as follows. Suppose we have a theory of some physic al property that utiliz es the predic ates g, N, and x. Suppose further that we are partic ularly interested in the rapid inc rease in gN(x) around x = 0 when N is large. Rather than analyz e gN (x) for partic ular finite values of N, it might make sense from a c omputational perspec tive to work with the infinite idealiz ation g∞(x), where the relevant behavior is stark and loc aliz ed at x = 0. We may introduc e a new “kind” represented by the predic ate f that pic ks out the phenomenon of interest in the infinite limit. This kind is conceptually novel to the g, N, x framework. Indeed, one can imagine a whole theory written in terms off, without referenc e to g. Using suc h a theory it c ould be diffic ult to see how f is approximated by some func tion of finite-N. Bec ause f is two-valued, the property it represents will appear to just pop into existenc e in the infinite limit without being approximated in any way by large finite systems. Restric ted to f (and henc e g∞(x)), one would not have the resourc es present to explain how f emerges from the shape of g when N is finite. This is prec isely what happens in phase transitions. As Butterfield shows, the example of f and g translates nic ely into the treatment of phase transitions. The magnetiz ation in an Ising model behaves like gN(x), where N is the number of particles and x is the applied field. For finite systems, the transition of the system between the two phases of magnetiz ation oc c urs c ontinuously as the applied field goes from negative to positive. In the infinite c ase, the transition is disc ontinuous. The sequenc e of func tions fN isolate one aspec t of the behavior of the func tions gN— whether or not they are c ontinuous. If we just foc us on this property, it might seem like there is entirely novel behavior in the infinite partic le c ase. The shape of f∞(x) around x = 0 is not in any sense approximated or approac hed by fN as N gets large. If it is the c ase that large finite systems c an be suc c essfully modeled as infinite systems, this might seem to be a sign of explanatory irreduc ibility. The suc c ess of the infinite partic le idealiz ation c annot be explained bec ause the infinite partic le func tion is not the limit of the finite partic le func tion sequenc e fN. The illusion of explanatory irreduc ibility is dispelled when we realiz e that any explanation involving f∞ c an be rephrased in terms of g∞, and the latter func tion does not display inexplic ably novel behavior. It is in fac t the limit of the finite partic le func tions gN. As N inc reases, gN approac hes g∞ in a well-defined sense. At a suffic iently large but finite system siz e N0 , the resolution of our measuring instruments will not be fine-grained enough to distinguish between gN0 (x) and g∞(x). We have an explanation, muc h like the one we have for extensivity, of the effic ac y of the infinite idealiz ation. Rec ogniz ing that the predic ate f only pic ks out part of the information c onveyed by the predic ate g dissolves the mystery. The new predic ate is useful when we are working with the idealiz ation, but it makes de-idealiz ation a more involved proc ess. To see the c onnec tion between a phase transition defined via Def 1 and real finite systems, one must first “undo” the c onc eptual innovation and write the theory as a limit of nasc ent func tions. At that point one c an then see that the idealiz ation is an innoc ent simplific ation and extrapolation of what happens to c ertain physic al c urves when N grows large.8 3.3 Ontological Irreducibility Ontologic al irreduc ibility involves a very strong failure of reduc tion, and if any phenomenon deserves to be c alled emergent, it is one whose desc ription is ontolog-ic ally irreduc ible to any theory of its parts. Batterman argues that phase transitions are emergent in this sense (Batterman 2005). It is not just that we do not know of an adequate statistic al mec hanic al ac c ount of them, we c annot c onstruc t suc h an ac c ount. Phase transitions, ac c ording to this Page 16 of 24

Turn and Face The Strange … Ch-Ch-Changes view, are c ases of genuine physic al disc ontinuities. The disc ontinuity is there in nature itself. The thermodynamic representation of these phenomena as mathematic al singularities is quite natural on this view. It is hard to see how else to best represent them. However, c anonic al statistic al mec hanic s does not allow for mathematic al singularities in thermodynamic func tions of finite systems, so it does not have the resourc es to adequately represent these physic al disc ontinuities. If the density of a finite quantity of water does as a matter fac t c hange disc ontinuously at a phase transition, then it seems that statistic al mec hanic s is inc apable of desc ribing this phenomenon, so the thermodynamic s of phase transitions is genuinely ontologic ally irreduc ible. Why think phase transitions are physic ally disc ontinuous? Batterman appeals to the qualitative distinc tion between the phases of fluids and magnets. Yet desc ribing the distinc tion between the phases as “qualitative” is potentially misleading. It is true that the different phases of c ertain systems appear mac rosc opic ally distinc t to us. A liquid c ertainly seems very different from a gas. However, from a thermodynamic perspec tive the differenc e is quantitative. Phases are distinguished based on the magnitudes of c ertain thermodynamic parameters. The mere existenc e of distinc t states of the system exhibiting these different magnitudes does not suggest that there is any disc ontinuity in the transition between the systems. This is a point about the mathematic al representation, but the lesson extends to the physic al phenomenon. While it is true that the phases of a system are mac rosc opic ally distinc t, this is not suffic ient to establish that the physic al transition from one of these phases to the other as gross c onstraints are altered involves a physic al disc ontinuity. In order to see whether there really is a disc ontinuity that is appropriately modeled as a singularity we need to understand the dynamic s of the c hange of phase. So we take a c loser look at what happens at a first-order phase transition. Consider the standard representation of an isotherm on the liquid-gas P-V diagram at a phase transition (figure 5.3). Figure 5.3 P- V diagram for a liquid-gas system at a phase transition. The two blac k dots are c oexistenc e points. At these points the pressure on the system is the same, but the system separates into two distinc t phases: low-volume liquid and high-volume gas. The two c oexistenc e points are c onnec ted by a horiz ontal tie-line or Maxwell plateau. On this plateau, the system exists as a two-phase mixture. It is here that the dynamic s of interest takes plac e. However, the representation above is too c oarse-grained to provide a full desc ription of the behavior of the system at transition. This representation c ertainly involves a mathematic al singularity: as the pressure is reduc ed, the volume of the system c hanges disc ontin-uously. But a c loser look at how the transition takes plac e demonstrates that this is just an artifac t of the representation, and not an ac c urate pic ture of what is going on at the transition. The P-V diagram ignores fluc tuations, but fluc tuations are c ruc ial to the transition between phases. The proc ess by whic h this takes plac e is nuc leation. When we inc rease the pressure of a gas above the c oexistenc e point it does not instantaneously switc h to a liquid phase. It c ontinues in its gaseous phase, but this supersaturated vapor is meta-stable. Thermal fluc tuations c ause droplets of liquid to nuc leate within the gaseous phase. In this regime, the liquid phase is energetic ally favored, and this enc ourages the expansion of the droplet. However, surfac e effec ts at the gas– liquid interfac e impede the expansion. When the droplet is small, surfac e effec ts predominate, preventing the liquid phase from spreading, but if there is a fluc tuation large enough to push the droplet over a c ritic al radius, the free energy advantage dominates and the liquid phase c an spread through the entire system. A full ac c ount of the gas– liquid transition will involve a desc ription of the proc ess of nuc leation, a nonequilibrium phenomenon that is not represented on the equilibrium P-V diagram in figure 5.3. Page 17 of 24

Turn and Face The Strange … Ch-Ch-Changes Perhaps the nuc leation of droplets from z ero radius c ould be seen as an example of a physic al disc ontinuity. However, an analysis of this proc ess is not beyond the reac h of finite partic le statistic al mec hanic s. We c an study the nuc leation of a new phase using the Ising model. As the external field c rosses z ero, simulations of the model show that initially loc al c lusters of spins flip. Some of these c lusters are too small, so they shrink bac k to z ero, but onc e there is a large enough c luster—a c ritic al droplet—the flipping spreads ac ross the entire system and the new phase takes over. All of this is observable in a simple finite partic le Ising system, so the phenomenon of nuc leation c an be desc ribed by statistic al mec hanic s without having to invoke the thermodynamic limit. If it is the c ase that physic al disc ontinuities c annot be ac c urately desc ribed by statistic al mec hanic s, then we have good reason for believing there are no suc h disc ontinuities in the proc ess of phase transition. Even if we grant that phase transitions involve a physic al disc ontinuity and c an only be ac c urately represented by a mathematic al singularity, the ontologic al irre-duc ibility of the phenomenon does not follow. Very rec ently it has been shown that the mic roc anonic al entropy, unlike the c anonic al free energy, c an be nonanalytic for finite systems. And indeed, a researc h program has sprung up based on this disc overy that tries to link singularities of the mic roc anonic al entropy to thermodynamic phase transitions (Franz osi, Pettini, and Spinelli 2000, Kastner 2008). That program demonstrates that nonanalytic ities in the entropy are assoc iated with a c hange in the topology of c onfiguration spac e. Consider the subset of c onfiguration spac e Mv that c ontains all points for whic h the potential energy per partic le is lower than v. As v is varied, this subset c hanges, and at some c ritic al values of v the topologic al properties of the subset c hange. This topology c hange is marked by a c hange in the Euler c harac teristic . For finite systems, there is a nonanalytic ity in the entropy wherever there is a topology c hange. For infinite systems there is a c ontinuum of points at whic h the topology c hanges, so a straightforward identific ation of phase transitions with topology c hange is inappropriate.9 Nevertheless, it is widely believed that there is some c onnec tion between these finite nonanalytic ities and thermodynamic phase transitions. This is a fledgling researc h program and there are still a number of open questions. It is unc lear what topologic al c riteria will be nec essary and suffic ient to define phase transitions, if any suc h c riteria c an be found. What is important for our purposes is that it is c lear that the mic roc anonic al ensemble does exhibit singularities even in the finite partic le c ase and that there is a plausible researc h program attempting to understand phase transitions in terms of these singularities. As suc h, it is c ertainly premature to dec lare that phase transitions are ontologic ally irreduc ible even if they involve genuine physic al disc ontinuities. Statistic al mec hanic s might well have the resourc es to adequately represent these disc ontinuities without having to advert to the thermodynamic limit. 4. The Infinite Idealizatio n in the Reno rmalizatio n Gro up We have argued that there is good reason to think the use of the infinite limit in the statistic al mec hanic al desc ription of phase transitions does not show that the phenomenon is either ontologic ally or explanatorily irreduc ible. Here we examine whether similar c laims c an be made about the way the infinite idealiz ation is used in renormaliz ation group theory. While this theory is usually inc luded under the broad rubric of statistic al mec hanic s, there are signific ant differenc es between renormaliz ation group methods and the methods c harac teristic of statistic al mec hanic s. Statistic al mec hanic s allows us to c alc ulate the statistic al properties of a system by analyz ing an ensemble of similar systems. Renormaliz ation group methods enter when c orrelations within a system extend over sc ales long enough to make straightforward ensemble methods imprac tic al (see Kadanoff (2010a) for more on this distinc tion). The properties of the system are c alc ulated not from a single ensemble but from the way in whic h the ensemble c hanges upon resc aling. In statistic al mec hanic s, the infinite idealiz ation is important for the effec t it has on a single ensemble (allowing nonanalytic ities, for instanc e). In renormaliz ation group theory, the infinite idealiz ation is important bec ause it allows unlimited resc aling as we move from ensemble to ensemble. The apparent differenc e in the use of the idealiz ation suggests the possibility of signific ant philosophic al distinc tions. It will not do to blithely extend our c onc lusions about statistic al mec hanic s to c over renormaliz ation group theory. We distinguish two different types of explanation that utiliz e the renormaliz ation group framework. The first is an explanation of the c ritic al behavior of partic ular systems, and the sec ond is the universal behavior of c lasses of systems. The first type of explanation does not raise any fundamentally new issues that we did not already c onsider in our disc ussion of the explanatory reduc ibility of phase transitions in statistic al mec hanic s. The sec ond type of explanation does raise signific ant new issues, sinc e we move from the examination of phenomena in partic ular systems to phenomena c harac teriz ing c lasses of systems. Batterman (2011) argues that the Page 18 of 24

Turn and Face The Strange … Ch-Ch-Changes renormaliz ation group explanation of universality is a c ase of explanatory irreduc ibility. While we might be able to tell a c omplex mic rophysic al story that explains why a partic ular finite system exhibits c ertain c ritic al behavior (the first type of explanation), we c annot ac c ount for the fac t that many mic rosc opic ally distinc t systems exhibit identic al c ritic al behavior (the sec ond type of explanation) without using the infinite idealiz ation. We begin with a brief disc ussion the first type of explanation: the renormaliz ation group applied to the c ritic al behavior of individual systems. We know from theory and experiment that there are large-sc ale c orrelations near the c ritic al point and that mean field theory does not work in these c onditions. We need a method that c an handle systems with long c orrelation lengths, and this is exac tly the purpose that the renormaliz ation group method serves. We idealiz e the c orrelation length of the system as infinite so that it flows to a fixed point under resc aling and then c alc ulate its c ritic al exponent by examining the behavior of the trajec tory near the fixed point. This raises the question of why a system with a large c orrelation length c an be suc c essfully represented as a system with an infinite c orrelation length. If we have no explanation of the suc c ess of this idealiz ation, we have a c ase of explanatory irreduc ibility. However, when we are foc using on the behavior of a partic ular system, any irreduc ibility in the renormaliz ation group theory is inherited from orthodox statistic al mec hanic s. The justific ation of the infinite c orrelation length idealiz ation will c oinc ide with the justific ation for the infinite system siz e idealiz ation. Why does the renormaliz ation group method need the infinite limit? Bec ause it relies on the divergenc e of the c orrelation length at the c ritic al point, whic h is impossible in a finite system. Why does the c orrelation length diverge? Bec ause it is related to the susc eptibility, whic h is a sec ond derivative of the free energy and diverges. Why does the susc eptibility diverge? Bec ause there is a nonanalytic ity in the free energy. Explaining why (or whether) this nonanalytic ity exists takes us bac k to the statistic al mec hanic al definition of phase transitions. If statistic al mec hanic s c an explain phase transitions reduc tively, then the renormaliz ation group does not pose an additional philosophic al problem when we focus on its application to particular systems. It is true that the system must be idealiz ed in order to employ renormaliz ation group theory, but that idealiz ation c an be justified outside renormaliz ation group theory. The more interesting c ase is the sec ond type of explanation, the explanation of universality. Without the renormaliz ation group method, we might examine the behavior of individual finite system and disc over that a number of suc h systems, though mic rosc opic ally distinc t, exhibit strikingly similar mac rosc opic behavior near c ritic ality. However, this would not tell us why we should expect this mac rosc opic similarity, and so it is not really a satisfac tory explanation of universality. The renormaliz ation group method givesusagenuine explanation: when the c orrelation length diverges, there is no c harac teristic length sc ale. If the relevant parameters for the system vanish, as they do at c ritic ality, the system will flow to a fixed point under repeated resc aling. Fixed points c an func tion as attrac tors, leading to similar c ritic al behavior for a number of different systems. If the system siz e is finite, the system will not flow to a fixed point. We might be able to show that a number of distinc t large finite systems flow to points in system spac e that are very c lose to eac h other, but onc e again all that we have done is revealed the universality of c ritic al (or near-c ritic al) behavior. We have not explained it. There is a generic reason to expec t distinc t infinite systems to flow to stable fixed points, but without mentioning fixed points there does not seem to be a generic reason to expec t distinc t finite systems to flow to points that are near eac h other. So it seems that fixed points play an indispensable role in the explanation of universal behavior. We c annot “de-idealiz e” and remove referenc e to fixed points in the explanation, the way we c an for nonanalytic ities in partic ular systems. Think bac k to Butterfield's example desc ribed in sec tion 3.2. In that example, the apparent explanatory irreduc ibility of the behavior of f∞ was resisted by rephrasing our explanations in terms of g∞, a func tion whose behavior in the limit is not novel. In the c ase of the renormaliz ation group, it seems that this move is unavailable to us. Fixed points are a novel feature that only appear in the infinite limit. There does not seem to be a c lear sense in whic h the renormaliz ation flow of finite systems c an approximate a fixed point. A point is either a fixed point for the flow or it is not; it c annot be “almost” a fixed point. And unlike Butterfield's example, there does not seem to be a way of rephrasing the explanation of universality in terms that are approximated by large finite systems. So there is a strong prima fac ie c ase that universality is explanatorily irreduc ible. However, we do not believe that the c ase stands up to sc rutiny. To see how it fails, we begin by showing that we c an explain why finite systems exhibit universal behavior near c ritic ality. However, this explanation does require the full resourc es of the renormaliz ation group method, inc luding fixed points. So it is not an explanation of the sort that we were Page 19 of 24

Turn and Face The Strange … Ch-Ch-Changes c ontemplating above, one that does away with referenc e to fixed points. We will argue that this should not ac tually trouble the reduc tionist, but first we present the explanation. Consider an Ising system extending over a finite length. When the system is resc aled, the separation between the nodes on the lattic e inc reases. Sinc e we are keeping the system siz e fixed, this means the number of nodes will dec rease. So unlike the infinite system c ase, for a finite system the number of nodes is a parameter that is affec ted by resc aling. If the number of nodes is N, we c an now think of 1/N as a relevant parameter (as defined in sec tion 1.3). When we restric t ourselves to the infinite c ase, we are c onsidering a partic ular hypersurfac e of this new parameter spac e where 1/N is set to 0. However, sinc e 1/N is a relevant parameter, perturbing the system off this hypersurfac e (i.e., switc hing from the infinite to a finite system) will take the system away from the c ritic al fixed point. This should be c ause for c onc ern. It seems there is no hope for an explanatory reduc tion. If even a slight perturbation off the 1/N = 0 hypersurfac e c hanges the c ritic al behavior, how c an we think of finite systems as approximating the behavior of infinite systems? As Kadanoff says, “if the bloc k transformation ever reac hes out and sees no more c ouplings in the usual approximation sc hemes … that will signal the system that a weak c oupling situation has been enc ountered and will c asc ade bac k to produc e a weak c oupling phase [a trivial fixed point with K = 0]” (Kadanoff 2010b, 47). However, all is not lost. The differenc e between the behavior of finite and infinite systems depends on the c orrelation length. When the c orrelation length is very small relative to the system siz e, the finite system behaves muc h like the infinite system. The values of thermodynamic observables will not differ substantially from their values for an infinite system. The behavior of the finite system will only exhibit a qualitative distinc tion when the c orrelation length bec omes c omparable to the system siz e. This phenomenon is known as finite siz e c rossover (see Cardy (1996), c h. 4) for a full mathematic al treatment). It is a manifestation of the fac t that the behavior of the system is sensitive to the large-sc ale geometry of the system only when the c orrelation length is large enough to be c omparable to the system siz e. The c rossover is c ontrolled by the reduc ed temperature. As long as this parameter is above a c ertain value (given by an inverse power of the system siz e), the c orrelation length will be small enough that no distinc tion between finite and infinite systems will be measurable. It is only below the c rossover temperature that finite-siz e effec ts bec ome signific ant and the system flows away from the c ritic al point. For a large system, the c rossover temperature will be very small, and its differenc e from the c ritic al temperature t = 0 may be within experimental error. So for a suffic iently large system, it is plausible that the infinite siz e approximation will work all the way to c ritic ality. Renormaliz ation group theory itself predic ts this. A similar point is made in Butterfield (2011). Crossover theory also provides tools for estimating the c hanges to c ritic al behavior that c ome from c hanging the geometry of the system by limiting its siz e. Adding system siz e as a parameter gives us a new sc aling func tion for the susc eptibility, a desc ription of how the susc eptibility c hanges with c hanges in relevant parameters. As desc ribed above, this sc aling func tion gives a behavior for the susc eptibility similar to the infinite limit as long as the ratio of c orrelation length to system siz e is low. It also allows us to predic t the behavior of the susc eptibility when this ratio bec omes c lose to one. The susc eptibility of a finite system will not diverge; it will have a smooth peak. The height of the peak of susc eptibility sc ales as a positive power of siz e of the system. So for a large system, the susc eptibility will be large but not infinite. In addition, the loc ation of the peak shifts, and this shift sc ales as an inverse power of the siz e. This means that for a large system the differenc e between the c ritic al temperature (the temperature at the c ritic al fixed point of the infinite system) and the temperature at whic h it attains maximum susc eptibility is very small. So for a mac rosc opic system, c rossover theory explains why it is a good approximation to treat the susc eptibility as diverging at the c ritic al point. The point of this disc ussion is that we c an tell an explanatory story about the c irc umstanc es under whic h partic ular large finite systems c an be treated like infinite systems. If the c rossover temperature is suffic iently small, then limitations of our measurement proc edures might make it diffic ult or even impossible to distinguish that the system does not flow to the c ritic al point. However, this explanatory story does make referenc e to fixed points in system spac e. So the worry is that it is not a fully reduc tive ac c ount. We may have explained why individual finite systems c an be suc c essfully idealiz ed as flowing to the c ritic al fixed point, but have we ac c ounted for the existenc e of the c ritic al fixed point? We are taking for granted in our explanation the topologic al struc ture of system spac e, a topologic al struc ture that is to a large extent determined by the behavior of infinite systems.10 This is true, but does it lead to explanatory irreduc ibility? Is it illic it to inc lude the topologic al struc ture of system Page 20 of 24

Turn and Face The Strange … Ch-Ch-Changes spac e among the explanatory resourc es of our lower-level theory? It would be if this struc ture involved an idealiz ation whose effic ac y c ould not be ac c ounted for within the lower-level theory. Isn't an irreduc ible infinite idealiz ation involved in the postulation of a renormaliz ation flow with fixed points? It is not. As we have seen, the renormaliz ation flow c an be defined for all systems, finite and infinite alike, sinc e 1/N c an be introduc ed as a relevant parameter. Fixed points will appear on the hypersurfac e where 1/N = 0. There is no infinite idealiz ation involved here. Of c ourse, we are talking about infinite systems and how they behave under the renormaliz ation flow, but this should not be problematic from a reduc tive point of view. The problem would arise if we model finite systems as infinite systems without explanation. But at this stage, when we are setting up the spac e and determining its topologic al c harac teristic s, we are not modeling partic ular systems. Insofar as finite systems are represented in our desc ription of the spac e, they are represented as finite systems, and infinite systems are represented as infinite systems. So the topologic al struc ture of the spac e c an be desc ribed without problematic infinite idealiz ation. When we try to explain the universality of c ritic al behavior in finite systems, we do have to employ the infinite idealiz ation, but as we have seen, this idealiz ation is not irreduc ible if we c an use the topologic al struc ture of system spac e in our reduc tive explanation. We c an de-idealiz e for partic ular systems and see why they c an be treated as if they flow to the c ritic al point. Understanding the behavior of infinite systems is c ruc ial to explaining the behavior of finite systems, since we only get the fixed points by examining the behavior of infinite systems, but this in itself does not imply emergenc e. We agree with Batterman (2011) that mathematic al singularities in the renormaliz ation group method are information sourc es, not information sinks. We disagree with his c ontention that the renormaliz ation group explanation requires the infinite idealiz ation and is thus emergent. It requires c onsideration of the behavior of infinite systems, but it does not require us to idealiz e any finite system as an infinite system. Any ac tual infinite idealiz ations in a renormaliz ation group explanation c an be de-idealiz ed using finite-siz e c rossover theory. Loc ating fixed points does not require an infinite idealiz ation, it just requires that our mic rosc opic theory c an talk about infinite systems, and indeed it c an. 5. Co nclusio n Phase transitions are an important instanc e of putatively emergent behavior. Unlike many things c laimed emergent by philosophers (e.g., tables and c hairs), the alleged emergenc e of phase transitions stems from both philosophic al and sc ientific arguments. Here we have foc used on the c ase for emergenc e built from physic s. We have found that when one c larifies c onc epts and digs into the details, with respec t to standard textbook statistic al mec hanic s, phase transitions are best thought of as c onc eptually novel, but not ontologic ally or explanatorily irreduc ible. And if one goes past textbook statistic al mec hanic s, then an argument c an be made that they are not even c onc eptually novel. In the c ase of renormaliz ation group theory, c onsideration of infinite systems and their singular behavior provides a c entral theoretic al tool, but this is c ompatible with an explanatory reduc tion. Phase transitions may be “emergent” in some sense of this protean term, but not in a sense that is inc ompatible with the reduc tionist projec t broadly construed. References Batterman, R. W. (2005). Critic al phenomena and breaking drops: Infinite idealiz ations in physic s. Studies in History and Philosophy of Modern Physics 36: 225–244. ———. (2011). Emergenc e, singularities, and symmetry breaking. Foundations of Physics 41(6): 1031– 1050. Borrmann, P., Mülken, O., and Harting, J. (2000). Classific ation of phase transitions in small systems. Physical Review Letters 84: 3511–3514. Butterfield, J. (2011). Less is different: emergenc e and reduc tion rec onc iled. Foundations of Physics 41(6): 1065– 1135. Callender, C. (2011). Hot and heavy matters in the foundations of statistic al mec hanic s. Foundations of Physics 41 (6): 960–981 Page 21 of 24

Turn and Face The Strange … Ch-Ch-Changes Cardy, J. (1996). Scaling and renormalization in statistical physics. Cambridge: Cambridge University Press. Cartwright, N. (1983). How the laws of physics lie. Oxford: Clarendon Press. Chomaz , P., Gulminelli, F. and Duflot, V. (2001). Topology of event distributions as a generaliz ed definition of phase transitions in finite systems. Physical Review E 64: 046114. Compagner, A. (1989). Thermodynamic s as the c ontinuum limit of statistic al mec hanic s. American Journal of Physics 57: 106–117. Franz osi, R., Pettini, M. and Spinelli, L. (2000). Topology and phase transitions: Paradigmatic evidenc e. Physical Review Letters 84: 2774–2777. Gibbs, J. W. (1902). Elementary principles in statistical mechanics. New York: Charles Sc ribner's Sons. Griffiths, R. (1972). Phase transitions and critical phenomena, Vol. 1, ed. C. Domb and M. Green. London: Academic Press. Gross, D. H. E. and Votyakov, E. V. (2000). Phase transitions in “small” systems. The European Physical Journal B — Condensed Matter and Complex Systems 15: 115–126. Kadanoff, L. (2009). More is the same; mean field theory and phase transitions. Journal of Statistical Physics 137: 777–797. ———. (2010a). Relating theories via renormaliz ation. Available at http://jfi.uchicago .edu/~leo p/Abo utPapers/Reno rmalizatio nV4.0.pdf. ———. (2010b). Theories of matter: Infinities and renormaliz ation. (arXiv: 1002.2985) Kastner, M. (2008). Phase transitions and c onfiguration spac e topology. Reviews of Modern Physics 80: 167– 187. Krauth, W. (2006). Statistical mechanics: Algorithms and computations. Oxford: Oxford University Press. Lebowitz , J. L. (1999). Statistic al mec hanic s: A selec tive review of two c entral issues. Reviews of Modern Physics 71: S346–S347. Liu, C. (1999). Explaining the emergenc e of c ooperative phenomena. Philosophy of Science 66: S92– S106. Mc Laughlin, B. P. (1992). The rise and fall of British emergentism. In Emergence or Reduction?, ed. A. Bec kermann, H. Flohr, and J. Kim, 49–93. Berlin: Walter de Gruyter. Melnyk, A. (2003). A physicalist manifesto: Thoroughly modern materialism. Cambridge: Cambridge University Press. Nagel, E. (1979). The structure of science: Problems in the logic of scientific explanation. Indianapolis: Hac kett. Strevens, M. (2009). Depth: An account of scientific explanation. Harvard: Harvard University Press. Touchette, H. (2002). When is a quantity additive, and when is it extensive? Physica A: Statistical Mechanics and Its Applications 305: 84–88. ———. (2006). Comment on “First-order phase transition: Equivalenc e between bimodalities and the Yang-Lee theorem.” Physica A: Statistical Mechanics and Its Applications 359: 375–379. Touchette, H. and Ellis, R. S. (2005). Nonequivalent ensembles and metastability. In Complexity, Metastability and Nonextensivity, ed. C. Bec k, G. Benedek, A. Rapisarda, and C. Tsallis. 81– 88. World Sc ientific . Wales, D. J. and Berry, R. S. (1994). Coexistenc e in finite systems. Physical Review Letters, 73: 2875– 2878. U. Wolff (1989). Collec tive monte c arlo updating for spin systems. Physical Review Letters 62: 361– 364. Page 22 of 24

Turn and Face The Strange … Ch-Ch-Changes Notes: (1) As an example, c onsider multiple realiz ation, often presented as a failure of reduc tion. However, it is only a failure if we believe that a lower-level explanation of the higher-level law must be unified (i.e., the explanation must be the same for every instanc e of the higher-level law). If we are willing to allow for disunified explanation, then we may indeed have a genuine lower-level explanation of the higher-level law, preserving the c ore sense of reduction. (2) Stric tly speaking, additivity and extensivity are different properties; see Touc hette (2002). Sinc e they overlap for many real systems, they are commonly run together; however, it is a mistake to do so in general, for some quantities sc ale with partic le number N (and henc e are extensive), yet are not additive. (3) Some textbooks even go in the other direc tion, namely, defining the thermodynamic limit as that state wherein entropy and energy are extensive. (4) For the thermodynamic limit to exist, two c onditions on the potential in the Hamiltonian must be satisfied, one on large distanc es, one on small distanc es. These extensions c an be viewed as c hallenges in either length sc ale. In another sense, however, one c an view both types of systems as unified together as “small” systems. If we define a system as “small” if its spatial extension is less than the range of its dominant interac tion, then even galac tic clusters are small. (5) A small movie of this oc c urring for small magnetic c lusters is available at http: //smallsystems.isn- oldenburg.de/movie.gif (6) This c hapter shows that yet another definition, one based on a bimodality of the energy distribution, is almost equivalent to Def 3. However, the bimodality definition is equivalent to Def 2, so the demonstration links Def 2 and Def 3. (7) There are some potential c onnec tions between “explanatory irreduc ibility” and notions in the literature on idealiz ation. In partic ular, depending upon how one understands Galilean idealiz ation, it is possible that a c onc eptual novelty is explanatorily irreduc ible just in c ase it is not a “harmless” Galilean idealiz ation. Coined by Mc Mullin, a Galilean idealiz ation in a sc ientific model is a deliberate distortion of the target system that simplifies, unifies or generally makes more useful or applic able the model. Cruc ially, a Galilean idealiz ation is also one that allows for c ontrolled “de-idealiz ation.” In other words, it allows for adding realism to the model (at the expense of simplic ity or usefulness, to be sure) so that one c an see that the distortions are justified by c onvenienc e and are not ad hoc . Idealiz ations like this are sometimes dubbed “c ontrollable” idealiz ations and are widely viewed as harmless. What to make of suc h non-Galilean idealiz ations is an ongoing projec t in philosophy of sc ienc e. One prominent idea—see, e.g., Cartwright (1983) or Strevens (2009)—is that the model may faithfully represent the signific ant c ausal relationships involved in the real system. The departure from reality need not then ac c ompany a c orresponding lac k of faith in the deliveranc es of the model. It is possible that we c ould understand the standard explanation of phase transitions as a distortion that nonetheless suc c essfully represents the c ausal relationships of the system. Perhaps the thermodynamic limit is legitimatiz ed by the fac t that surfac e effec ts are not a differenc e- maker (in the sense of Strevens) in the systems of interest. We will leave this line of thought to others to develop. (8) Thanks to Jim Weatherall for kic k-starting our thinking of phase transitions as delta func tions that c an be approximated by analytic func tions and to Jeremy Butterfield for kindly letting us use an advanc e c opy of his 2011 article. (9) The problem with identifying these singularities with phase transitions in thermodynamic s is that as N grows the order of the phase transition also inc reases, roughly as N/2. These transitions are far weaker than the ones enc ountered in thermodynamic s, and in any c ase, unobservable in real noisy data unless N is really small. (10) Our thanks to Robert Batterman for pushing us on this point. Tarun Menon Tarun Menon is a graduate student in Philosophy at the University of California, San Diego. His research interests are in the philosophy of physics and m etaphysics, particularly tim e, probability, and the foundations of statistical m echanics. He is also Page 23 of 24

Turn and Face The Strange … Ch-Ch-Changes interested in form al epistem ology and the cognitive structure of science. C raig C allend er Craig Callender is Professor of Philosophy at the University of California, San Diego. He has written widely in philosophy of science, m etaphysics, and philosophy of physics. He is the editor of Physics m eets philosophy at the Planck length (with Huggett) and the Oxford handbook of the philosophy of tim e. He is currently working on a book m onograph on the relationship between physical tim e and tim e as we experience it.

Effective Field Theories Jonathan Bain The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter reviews effec tive field theory (EFT) tec hniques, foc using on the intertheoretic relations between low- energy effec tive theories and their high-energy c ounterparts. It disc usses how EFT c an be interpreted and desc ribes the steps in the c onstruc tion of an EFT in the top-down and bottom-up approac hes. The c hapter also analyz es the dependenc e of EFT on renormaliz ation sc hemes, and evaluates the extent to whic h the intertheoretic relation between an EFT and its high-energy theory can be described in terms of emergence. K ey words: effecti v e fi el d th eory , i n terth eoreti c rel ati on s, ren ormal i zati on sch eme, h i gh -en ergy th eory , emergen ce 1. Intro ductio n An effec tive field theory (EFT) of a physic al system is a theory of the dynamic s of the system at energies small c ompared to a given c utoff. For some systems, low-energy states with respec t to this c utoff are effec tively independent of (“decoupled from”) states at high energies. Hence one may study the low-energy sector of the theory without the need for a detailed description of the high-energy sector. Systems that admit EFTs appear in both relativistic quantum field theories (RQFTs) and c ondensed matter physic s. When an underlying high-energy theory is known, an effec tive theory may be obtained in a “top-down” approac h by a proc ess in whic h high- energy effec ts are systematic ally eliminated. When an underlying high-energy theory is not known, it may still be possible to obtain an EFT by a “bottom-up” approac h in whic h symmetry and “naturalness” c onstraints are imposed on c andidate Lagrangians. In both c ases, the intertheoretic relation between the EFT and its (possibly hypothetic al) high-energy theory is c omplic ated, and, arguably, c annot be desc ribed in terms of traditional ac c ounts of reduc tion. This has suggested to some authors that the EFT intertheoretic relation (and/or the phenomena associated with it) should be described in terms of a notion of emergence. Other authors have desc ribed the proc ess of c onstruc ting an EFT as one in whic h idealiz ations are made in order to produc e a c omputationally trac table, yet inherently approximate, theory that is empiric ally equivalent (to a given range of ac c urac y) to a typic ally c omputationally more c omplex, but c omplete, high-energy theory. One suc h c laim is that, in the c ontext of RQFTS, the set of possible worlds assoc iated with an EFT are ones in whic h spac e is disc rete and finite. This essay reviews effective field theory techniques, focusing on the intertheoretic relation that links an EFT with its (possibly hypothetic al) high-energy theory. The goal is to c ontribute to disc ussions on how EFTs c an be interpreted, and, in partic ular, to investigate the extent to whic h a notion of emergenc e is viable in suc h interpretations. Sec tion 2 sets the stage by reviewing the general steps in the c onstruc tion of an EFT in the top- down and bottom-up approaches. Section 3 then reviews the extent to which an EFT can be said to be empirically equivalent to its high-energy theory: typic al EFTs are nonrenormaliz able and thus break down at high energies; however, this has not stopped physic ists from using them to derive predic tions for low-energy phenomena. Sec tion 4 indic ates the extent to whic h the explic it form of an EFT depends on the type of renormaliz ation sc heme one Page 1 of 21

Effective Field Theories employs to handle divergent integrals that c an arise when one uses the EFT to c alc ulate the values of observable quantities. It is argued that the c hoic e of renormaliz ation sc heme is irrelevant for c alc ulating suc h values. However, to the extent that this c hoic e determines the explic it form of the EFT, arguably, it has nontrivial c onsequenc es when it c omes to the question of how the EFT c an be interpreted. These c onsequenc es are investigated in sec tion 5. Finally, sec tion 6 takes up the task of assessing the extent to whic h the intertheoretic relation between an EFT and its high-energy theory c an be desc ribed in terms of emergenc e. 2. The Nature o f EFTs The c onstruc tion of an EFT follows one of two general proc edures, top-down and bottom-up, depending on whether a high-energy theory is known. Both proc edures are based on an expansion of the effec tive ac tion (whic h formally represents the EFT) in terms of a (possibly infinite) sum of loc al operators, c onstrained by symmetry and “naturalness” c onsiderations. They differ on how the effec tive ac tion is obtained: the top-down approac h obtains it by eliminating degrees of freedom from the ac tion of the high-energy theory, whereas the bottom-up approac h c onstruc ts it from sc ratc h. 2.1 Top-Down The top-down approac h starts with a known theory and then systematic ally eliminates degrees of freedom assoc iated with energies above some c harac teristic energy sc ale E0 . The prac tic al goal is to obtain a low-energy theory that allows one to more easily c alc ulate the values of observable quantities assoc iated with energies below E0 than in the original theory. Intuitively, c alc ulations in suc h a low-energy “effec tive” theory have fewer parameters to deal with (namely, all those parameters assoc iated with high-energy degrees of freedom) and thus are simpler than c alc ulations in the original theory. However, the c onstruc tion of a low-energy effec tive theory that accomplishes this is not just a matter of simply ignoring the high-energy degrees of freedom, for they may be intimately tangled up with the low-energy degrees of freedom in nontrivial ways. (As we will see below, one way to distinguish a “renormaliz able” theory from a “nonrenormaliz able” theory is that in the former, the high-energy degrees of freedom are independent of the low-energy degrees of freedom, whereas in the latter, they are not.) One method of disentangling the high-energy and low-energy degrees of freedom was pioneered by Wilson and others in the 1970s. In the following, I will refer to it as the Wilsonian approac h to EFTs. It typic ally involves two steps: (I) The high-energy degrees of freedom are identified and integrated out of the ac tion. These high-energy degrees of freedom are referred to as the high momenta, or “heavy,” fields. The result of this integration is an effective ac tion that desc ribes nonloc al interac tions between the low-energy degrees of freedom (the low momenta, or “light,” fields). (II) To obtain a loc al effec tive ac tion (i.e., one that desc ribes loc al interac tions between low-energy degrees of freedom), the effec tive ac tion from Step I is expanded in terms of loc al operators. The following desc ribes these steps in slightly more tec hnic al detail.1 (I) Given a field theory desc ribed by an ac tion S and possessing a c harac teristic energy sc ale E, suppose we are interested in the physic s at a lower sc ale E ≪ E0 . First c hoose a c utoff Λ at or slightly below E0 and divide the fields ϕ into high and low momenta parts with respec t to Λ: ϕ = ϕ H + ϕ L, where ϕ H have momenta k 〉 Λ and ϕL have momenta k 〈 Λ. Now integrate out the high momenta fields. In the path integral formalism, one does the integral over the ϕ H . Sc hematic ally (1) where the Wilsonian effective action is given by eiSΛ [ϕL] = ∫ DϕHeiS[ϕL,ϕH] . The effective Lagrangian density Leff is thus given by SΛ [ϕL] = ∫ dD xLeff[ϕL], where D is the dimension of the spacetime. (II) Typic ally, the integration over the heavy (i.e., high momenta) fields will result in a nonloc al effec tive ac tion (i.e., one in whic h terms oc c ur that c onsist of operators and derivatives that are not all evaluated at the same spac etime point). In prac tic e, this is addressed by expanding the effec tive ac tion in a set of loc al operators: (2) Oi Page 2 of 21

Effective Field Theories where the sum runs over all local operators Oi allowed by the symmetries of the initial theory, and the gi are c oupling c onstants. Assuming weak c oupling, the expansion point S0 may be taken to be the free ac tion of the initial theory, so that g∗ = 0. To see how the effec tive ac tion relates to the initial high-energy ac tion, one c an perform a dimensional analysis on the operators that appear in (2). This allows one to obtain information about their behavior as the c utoff Λ is lowered. This analysis involves three steps: (i) Choose units in whic h the ac tion is dimensionless (ħ= 1, c = 1). In suc h units, energy has dimension +1 while length has dimension −1. The free ac tion c an now be used to determine units for the field operators.2 This then determines units for the c oupling c onstants, and subsequently, for terms in the expansion (2). For instance, if an operator Oi has been determined to have units Eδi (thus dimension δi), then its coupling constant gi has units ED−δi, and the magnitude of the ith term is ∫ dD xOi ∼ Eδi −D . (ii) To make the c utoff dependenc e of the terms explic it, one c an define dimensionless c oupling c onstants by λi = Λδi−Dgi . The order of the ith term in (2) is then (3) (iii) The terms in the expansion (2) c an now be c lassified into three types: • Irrelevant: δi 〉 D. This type of term falls at low energies as E → 0. Suc h terms are suppressed by powers of E /Λ. • Relevant: δi 〈 D. This type of term grows at low energies as E → 0. • Marginal: δi = D. This type of term is c onstant and equally important at low and high energies (insofar as quantum effec ts c an modify its sc aling behavior toward either relevanc y or irrelevanc y). This dimensional analysis indic ates that, in c ases of physic al relevanc e, there will only be a finite number of relevant and marginal terms in (2).3 In suc h c ases, the low-energy EFT will only depend on the underlying high- energy theory through a finite number of parameters. It is typic al in the literature on EFTs to elevate these c onsiderations to a princ iple. Polc hinski (1993, 6) artic ulates suc h a princ iple in the following way: The low energy physic s depends on the short distanc e theory only through the relevant and marginal c ouplings, and possibly through some leading irrelevant c ouplings if one measures small enough effec ts. Note that arbitrarily many irrelevant terms c an oc c ur in (2), but they are suppressed at low energies by powers of E/Λ. Moreover, the c utoff c an be used as a regulator for any divergenc es assoc iated with these terms in c alc ulations of the values of observable quantities. Thus, “even though the [effec tive] Lagrangian may c ontain arbitrarily many terms, and so potentially arbitrarily many c oupling c onstants, it is nonetheless predic tive so long as its predic tions are only made for low-energy proc esses, for whic h E/Λ ≪ 1” (Burgess 2004, 17). A little more will be said on this matter in sec tion 3 below. Finally, in addition to symmetry c onsiderations, a further c onstraint is typic ally applied to the expansion (2) of the effec tive ac tion: one assumes that the dimensionless c oeffic ients λ i are of order 1. This is assoc iated with a hypothesis of “naturalness,” insofar as it rules out the presenc e of very large or very small numbers, relative to the c utoff, in the expansion.4 Intuitively, a “natural” EFT should only involve quantities that are small, but not too small, relative to the c utoff. An immediate c onsequenc e of this is that mass terms, whic h have c oeffic ients proportional to powers of the c utoff, c annot appear in (2). Thus naturalness is typic ally formulated in terms of the following c ondition: 5 EFTs must be natural, meaning that all possible masses must be forbidden by symmetries. Note that this does not prec lude the existenc e of massive objec ts (fields, partic les, etc .) in an EFT desc ription of low-energy phenomena. Rather, it c onstrains suc h desc riptions to those in whic h mass terms are generated by broken high-energy symmetries. Thus, ac c ording to the Standard Model, massive vec tor bosons (the W and Z bosons) exist at low energies (with respec t to the appropriate c utoff) due to elec troweak symmetry breaking, even though gauge invarianc e prohibits massive terms in the elec troweak ac tion, and massive fermions exist similarly due to c hiral symmetry breaking.6 Page 3 of 21

Effective Field Theories At this point, the following qualific ations should be made c onc erning the above Wilsonian approac h to top-down EFTs: (a) First, in Step (I), the identific ation of the appropriate heavy and light field variables is not always self- evident. For example, the weakly c oupled EFT for quantum c hromodynamic s (QCD) known as c hiral perturbation theory is written in terms of pion fields as opposed to quark and gluon fields; and Landau's Fermi liquid theory of c onduc tors c an be written as an EFT of weakly interac ting quasipartic les, as opposed to strongly interac ting elec trons (Manohar 1997, 321).7 (b) Sec ond, in Step (I), the path integral over the heavy fields is typic ally performed in prac tic e using a saddle-point approximation (Dobado et al. 1997, 4). This involves expanding the ac tion of the high-energy theory about a given c onfiguration of the heavy fields c hosen to be a “saddle point” (or “stationary point”; i.e., a global extremum). To sec ond order in this expansion, the integral over the heavy fields takes the form of a Gaussian integral, whic h has a well-defined solution. (c) Third, in Step (II), the dimensional method of justifying the finite dependenc e of an EFT on its high-energy theory is based on using the free theory to determine units, and this assumes the high-energy theory is weakly c oupled. Strong interac tions may have the effec t of c hanging the sc aling behavior of terms. (d) Finally, the dimensional assignments in Step (II) work when using the EFT to make simple “tree-level” c alc ulations of the values of observed quantities. However, for higher-order “loop” c orrec tions to suc h c alc ulations, sc aling based on dimensional analysis may break down, and one may have to appeal to a partic ular renormaliz ation sc heme in order to justify the explic it relianc e of an EFT on a finite number of parameters (see sec tion 4 below).8 2.2 Bottom-Up The proc edure outlined above for c onstruc ting an EFT requires having in one's possession the ac tion (or Lagrangian density) of a high-energy theory. In some c ases of interest, the fundamental high-energy theory is not known, but an EFT is, nonetheless, still c onstruc tible. One simply begins with the operator expansion (2) and inc ludes all terms c onsistent with the naturalness c onstraint and with the symmetries and interac tions assumed to be relevant at the given energy sc ale. One c an then determine how these terms sc ale when a given c utoff is raised (as opposed to lowered). Examples of suc h “bottom-up” EFTs inc lude the Fermi theory of low-energy weak interac tions (as it was originally c onstruc ted); and, in the view of many physic ists, the Standard Model itself (see, e.g., Hartmann 2001 for disc ussion). Another example is effec tive field theoretic formulations of general relativity in whic h the Hilbert ac tion is identified as the first term of the expansion (2) (Burgess 2004). 2.3 Example: Low-Energy Superfluid Helium-4 Film An example of a top-down EFT c onstruc ted via the method of sec tion 2.1 is the low-energy theory of a superfluid helium-4 (4 He) film. The effec tive Lagrangian density that desc ribes this system is formally identic al to the Lagrangian density for (2+1)-dimensional quantum elec trodynamic s (QED3 ). This is somewhat surprising, given that the underlying “high-energy” theory is nonrelativistic . This example, in whic h a relativistic EFT might be said to emerge from a nonrelativistic high-energy theory, will be instruc tive in the disc ussion of the c onc ept of emergenc e in EFTs in section 6 below. At low temperatures, the liquid state of 4 He bec omes a superfluid c harac teriz ed by dissipationless flow and quantiz ed vortic es. The phase transition between the normal liquid and superfluid states is enc oded in an order parameter that takes the form of a macroscopic wave function φ0 = (ρ0 )1/2 eiθ describing the coherent ground state of a Bose c ondensate with density ρ0 and c oherent phase θ. An appropriate Lagrangian density desc ribes nonrelativistic neutral bosons (viz ., 4 He atoms) interac ting via a spontaneous symmetry breaking potential with coupling constant K (Zee 2003, 175, 257), (4) Here m is the mass of a 4 He atom, and the term involving the c hemic al potential μ enforc es partic le number c onservation. This is a thoroughly nonrelativistic Lagrangian density invariant under a global U(1) symmetry and Galilean transformations. We now c onsider (4) as representing an underlying “high-energy” theory and seek to c onstruc t a low-energy EFT via the top-down approac h outlined above. Page 4 of 21

Effective Field Theories To investigate the low-energy behavior of (4), one first needs to identify appropriate dynamic al variables by means of whic h a distinc tion c an be made between high- and low-energy degrees of freedom.9 Sinc e the ground state φ 0 is a func tion only of the phase, low-energy exc itations take the form of phase fluc tuations. This suggests rewriting the field variable φ in terms of density and phase variables φ = (ρ)1/2 eiθ and identifying the high-energy degrees of freedom with the density ρ. The next task is to integrate the density field out of (4).10 This c an be done by expanding the variables as ρ = ρ0 + δρ, θ = θ0 + δθ, where δρ,δθ represent small fluc tuations in the density and phase about their stationary ground state values ρ0,θ0. Substituting these into (4), one obtains the equivalent of (2) for the effective Lagrangian density: L4He = L0 [ρ0 , θ0 ] + L ′4He [δρ, δθ], where the first term reproduces (4) without the interac tion term, and the sec ond term desc ribes fluc tuation c ontributions. The high-energy fluc tuations δρ c an be eliminated by deriving the Euler-Lagrange equations of motion for the density variable and solving for δρ. After substitution back into L ′4He , and in two dimensions, the result is, (5) with δθ replac ed by θ for the sake of notation. Ignoring the higher-order terms, (5) formally desc ribes a sc alar field propagating at speed c2 = 2kρ0 /m. For units in whic h c = 1, it c an be rewritten as (6) where ημν is the (2+1)-dim Minkowski metric . Equation (6) is manifestly Lorentz invariant; in fac t, it is formally identic al to the Lagrangian density for a mass-less sc alar field propagating in (2+1)-dim Minkowski spac etime. To obtain QED3, c onsider the first term in (5). Insofar as this represents the kinetic energy density, one c an identify a superfluid veloc ity variable by vi ≡ (1/m) ∂iθ. The fac t that the mac rosc opic wave func tion is unique up to phase then entails that the superflow in a multiply-c onnec ted domain is quantiz ed, (7) around a c losed path enc irc ling a “hole,” where q is an integer. Suc h holes may be interpreted as vortices—points where the real and imaginary parts of φ0 vanish.11 Then (7) entails that the superflow about a vortex is quantiz ed. More important in this c ontext, (7) suggests an analogy with Gauss's Law in whic h a vortex plays the role of a c harge c arrier and the superfluid veloc ity plays the role of the elec tric field. To further c ash out this analogy, note that in two dimensions, the magnetic field is a sc alar, whereas the elec tric field is a 2-vec tor. This motivates the following identifications: (8a) (8b) in whic h the magnetic field is identified with the density, and the elec tric field with the superfluid veloc ity (here εij is the skew volume 2-form). Substituting into (6), one obtains the Lagrangian density for sourc eless QED3 (9) where Fμν = ∂μAν, − ∂νAμ, with the potential Aμ defined by Ei = ∂0Ai − ∂iA0, B = ∂1 A2 − ∂2A1. One may further note vathosa rttth e(ex7 )0c etuhnr rtcaeoinlmst pitsho atnhte etnh dte uo daf ela no vsf oitthyrte ef oxer l ce“ucertlrreoemmneta njgtμvanre=yti”c( 1 vc/ou2rrtπrice)en∈st, μ(inνqλs =o∂ fν±a∂r1 λa)θ si,s a wgdhivdeeirnneg bεaμy νs λ(o 1ius/r 2tchπee) t∂se ⃗kr×mew ∂A θ⃗ vμ.o jTluμvhm itsoe c (3a9-n)f o abrnmed .id12e nTthifiised extremiz ing with respec t to Aμ produc es the Maxwell equations with a sourc e. In summary, we started with the nonrelativistic Lagrangian density (4) for a superfluid 4 He film and found that, to lowest order, its EFT takes the form of the relativistic Lagrangian density for (2+1)-dim quantum elec trodynamic s. This was motivated by the formal similarity between vortex quantiz ation (7) and Gauss's Law. This similarity was exploited in terms of a duality transformation under whic h vortic es bec ome the sourc es of a gauge field formally identic al to the Maxwell field. Under a literal interpretation of this dual representation (8), low-energy exc itations of a superfluid 4 He film take the form of elec tric and magnetic fields, the former being given by the superfluid veloc ity, and the latter being given by the superfluid density. Moreover, topologic al defec ts (i.e., elementary vortic es) take the form of c harge-c arrying elec trons. 3. Reno rmalizability and Predictability Page 5 of 21

Effective Field Theories Historic ally, the Wilsonian approac h to EFTs outlined in sec tion 2.1 had its origin in the development of renormaliz ation group (RG) tec hniques by Wilson and others in the 1970s (see, e.g., Huggett and Weingard 1995; Cao and Sc hweber 1993). These tec hniques were originally developed to study the low-energy behavior of c ondensed matter systems and were subsequently applied to the problem of renormaliz ation in relativistic quantum field theories; i.e., the appearanc e of integrals that diverge at high energies when one uses a quantum field theory to c alc ulate the values of observable quantities. This is related to the issue of predic tability, insofar as a theory that “blows up” at high energies c annot be used to make high-energy predic tions. This sec tion c onsiders the issues of renormaliz ability and predic tability in the c ontext of EFTs. In partic ular, given that typic al EFTs are not renormaliz able, how does this affec t their ability to make predic tions? In the RG approac h to renormaliz ation, the intent is to analyz e the behavior of a theory at different energy sc ales s. One thus uses a sc ale-dependent momentum c utoff Λ (s) as the basis for an initial distinc tion between high- and low-energy modes, and the heavy modes with respec t to an initial energy Λ are then integrated out of the theory. The c utoff is now lowered to Λ(s) = sΛ and the parameters of the theory are then resc aled to formally restore the c utoff bac k to Λ. Suc c essive iterations of this proc edure generate a flow in the spac e of parameters of the theory. Sc ale-dependent parameters c an then be c lassified as relevant (shrinking in the high-energy limit as s→ ∞), irrelevant (growing as s→ ∞), or marginal (c onstant under sc ale transformation).13 A theory is now said to be renormaliz able if it c ontains no irrelevant parameters. Intuitively, suc h a theory is c utoff independent, insofar as its parameters bec ome independent of Λ(s) in the high-energy limit s → ∞. A nonrenormaliz able theory, on the other hand, is one in whic h there are (sc ale-dependent) irrelevant parameters. Suc h parameters c annot be ignored at high energies and thus c ontribute to ultraviolet divergent integrals. EFTs c an be either renormaliz able or nonrenormaliz able in the above sense, depending on whether they c ontain irrelevant terms, although typic ally the c onstruc tion outlined in sec tion 2.1 above produc es an infinite number of the latter. However, as eluded to in sec tion 2.1, the appearanc e of an infinite number of irrelevant terms in an EFT need not signal a breakdown in predic tability. After Manohar (1997, 322), the effec tive Lagrangian density assoc iated with (2) c an be represented sc hematic ally by the sum: (10) where L≤D contains terms with dimension ≤ D,  LD+1 contains terms with dimension D + 1, and soon, and, as in sec tion 2.1 above, D is the dimension of spac etime (rec all that an operator with dimension δ is deemed irrelevant, relevant, or marginal, depending on whether δ is greater than, less than, or equal to D, respec tively). In this sum, eac h summand c ontains a finite number of terms with c oeffic ients that are powers of the ratio (s/Λ). The first summand c onsists of a finite number of relevant and/or marginal terms to order z ero in (s/Λ) (thus suc h terms are sc ale-independent). Eac h summand thereafter c ontains a finite number of irrelevant terms to a higher order in (s/Λ) (thus suc h terms are sc ale-dependent). A renormaliz able Lagrangian density c onsists of only the first summand, thus when it is used to derive predic tions, they will be sc ale-independent. Nonrenormaliz able Lagrangians inc lude irrelevant terms, and predic tions derived from them will be sc ale-dependent. In general, to c ompute the value of an observable quantity to a given order r in (s/Λ), one should retain terms up to LD+r . To c onsider how this analysis of renormaliz ability relates to predic tability, note first that renormaliz ability, as defined above, is predic ated on the property of being sc ale-independent. A renormaliz able theory is independent of energy sc ale, whereas a nonrenormaliz able theory is not. So in order to artic ulate the relation between renormaliz ability and predic tability, one needs to artic ulate the relation between sc ale-independenc e and predic tability. An extreme view might require sc ale-independenc e (and henc e renormaliz ability) to be a nec essary c ondition for predic tability. The argument might run something like this: if a theory is sc ale-dependent, then using a c utoff to regulate divergent integrals will be of no help, insofar as (a) the c utoff must be taken to infinity at the end of the day; and (b) doing so will c ause sc ale-dependent terms (whic h are well-behaved at low-energies with respec t to the c utoff) to blow up. One intuition underlying this argument is that the c utoff must, in fac t, be taken to infinity at the end of the day; otherwise, we would not end up with the c ontinuum theory we began with. This appears to be the argument underlying Huggett and Weingard's response to their “Problem Number two of understanding renormaliz ation,” namely, “why do ac tual physic al theories depend on only a finite number of parameters? A slogan: why is the world renormalisable?” (Huggett and Weingard 1995, 179). Their answer to this problem is the following: purely relevant trajec tories terminate in the c ontinuum limit—c all this “asymptotic safety,”. Any irrelevant Page 6 of 21

Effective Field Theories dependent theories …do not terminate in this way and are not asymptotic ally safe. They generate indefinitely long trajec tories with ever varying [parameters], either periodic ally or ever growing. Either way, unphysic al singularities are likely. Thus, while asymptotic ally safe relevant theories are potentially physic al, irrelevant theories are not—just the result we hoped for to answer the sec ond question. (Huggett and Weingard 1995, 183) If one takes “potentially physic al” to mean “sc ale-independent,” then Huggett and Weingard's c laim that irrelevant (i.e., nonrenormaliz able) theories are not potentially physic al is c orrec t. However, if one takes “potentially physic al” to mean “c apable of produc ing finite predic tions,” then Huggett and Weingard's c laim does not go through: nonrenormaliz able theories are c apable of produc ing finite predic tions, with the qualific ation that suc h predic tions are sc ale-dependent. Manohar suggests that there is nothing wrong with suc h a notion of predic tability, insofar as there is no reason to expec t potentially physic al theories to be sc ale-independent: A non-renormaliz able theory is just as good as a renormaliz able theory for c omputations, provided one is satisfied with a finite ac c urac y. … While exac t c omputations are nic e, they are irrelevant. Nobody knows the exac t theory up to infinitely high energies. Thus any realistic c alc ulation is done using an effec tive field theory. (Manohar 1997, 322) Burgess likewise suggests that the distinc tion between a renormaliz able (viz ., sc ale-independent) theory and a nonrenormaliz able (viz ., sc ale-dependent) theory is a matter of degree rather than kind: Bec ause … only a finite number of terms in Leff c ontributes to any fixed order in [s/Λ], and these terms need appear in only a finite number of loops, it follows that only a finite amount of labor is required to obtain a fixed ac c urac y in observables. Renormaliz able theories represent the spec ial c ase for whic h it suffic es to work only to z eroth order in the ratio [s/Λ]. This c an be thought of as the reason why renormaliz able theories play suc h an important role throughout physic s.… Thus, although an effec tive Lagrangian is not renormaliz able in the traditional sense, it nevertheless is predic tive in the same way a renormaliz able theory is. (Burgess 2007, 349) The suggestion here is that, to the extent that sc ale-dependent predic tions and sc ale-independent predic tions are both c alc ulated in the same manner (i.e., by applying an appropriate renormaliz ation sc heme to divergent integrals), they are of the same kind. Thus, “nonrenormaliz able theories are not fundamentally different from renormaliz able ones. They simply differ in their sensitivity to more mic rosc opic sc ales whic h have been integrated out” (Burgess 1998, 13). This tolerant view of nonrenormaliz able theories, and in partic ular, the predic tability of EFTs, has arguably bec ome the norm among physic ists. What it implies about the ontology of EFTs, and in partic ular, the nature of the Wilsonian c utoff Λ, will have to wait until sec tion 5. Sec tion 4 provides a brief review of two explic it ways of deriving predic tions using EFTs. My ultimate c laim in sec tion 5 will be that the method one c hooses to derive predic tions from an EFT (i.e., the renormaliz ation sc heme one adopts) will influenc e the possible ways of interpreting it. 4. On Reno rmalizatio n Schemes and Types o f EFTs As Manohar (1997, 326) observes, knowing the Lagrangian density of a quantum field theory is not enough to c alc ulate the values of observable quantities. To ac c omplish the latter (using perturbative tec hniques) requires expanding the Green's func tion that represents a partic ular observable quantity in an infinite series in whic h, typic ally, divergent integrals appear.14 This is the problem of renormaliz ation in quantum field theory. Thus, in addition to knowing the Lagrangian density of a quantum field theory, one needs to spec ify a renormaliz ation sc heme. This is a method that spec ifies (1) a means of regulating divergent integrals and (2) a means of subtrac ting the assoc iated infinities in a systematic way. There are a number of different methods that ac c omplish this, two of whic h are important in the c ontext of interpreting EFTs. The first adopts momentum c utoff regulariz ation and a mass-dependent method of subtrac tion and is used (at least implic itly) in the Wilsonian approac h to c onstruc ting EFTs (outlined above in sec tion 2). The sec ond adopts dimensional regulariz ation and a mass- dependent method of subtrac tion and is assoc iated with what Georgi (1992, 1; 1993, 215) has c alled “c ontinuum EFTs.” Page 7 of 21

Effective Field Theories 4.1 Mass-Dependent Schemes and Wilsonian EFTs In a Wilsonian EFT, the explic it appearanc e of the c utoff L that defines the border between the low-energy physic s and the high-energy physic s suggests employing it as a means to regulate the partic ular type of divergent integrals that appear in c alc ulations of the values of observable quantities. Given suc h a divergent integral of the sc hematic form ∫ ∞ dD pK (p), where D is the dimension of spacetime and κ(p) is a particular function of momentum p, one 0 c an insert the c utoff Λ and rewrite the integral as the sum of a finite piec e and an infinite piec e: (11) For the types of divergent integrals under c onsideration, the infinite piec e c an be absorbed into a redefinition of the parameters of the theory through the introduc tion of renormaliz ation c onstants. It turns out that, in this method of regulariz ation, these c onstants are dependent on the heavy masses that appear in the high-energy theory; henc e, the manner in whic h they are defined is referred to as a mass-dependent subtrac tion sc heme.15 There are two main advantages of employing this type of renormaliz ation sc heme in the c ontext of EFTs. First, it is conceptually consistent with the image of an EFT as a low-energy approximation to a high-energy theory based on a restric tion of the latter to a partic ular energy sc ale. This sc ale is explic itly represented by the c utoff Λ, whic h thus plays a double role in designating the appropriate energy sc ale and in c utting off divergent integrals. The sec ond advantage of using this renormaliz ation sc heme is that it guarantees that the Dec oupling Theorem holds, given a few other assumptions. The Dec oupling Theorem is due to Appelquist and Caraz z one (1975). Hartmann (2001) desc ribes it thusly: For two c oupled systems with different energy sc ales m1 and m2 (m2 〉 m1) and desc ribed by a renormaliz able theory, there is always a renormaliz ation c ondition ac c ording to whic h the effec ts of the physic s at sc ale m2 c an be effec tively inc luded in the theory with the smaller sc ale m1 by c hanging the parameters of the c orresponding theory. (Hartmann 2001, 283) This theorem is a formal guarantee of the informal EFT “ideology” of Polc hinski (1993, 6), stated above in sec tion 2.1. Hartmann (2001, 284) is c areful to note that it requires that there is an underlying high-energy theory that is renormaliz able and that different mass sc ales exist in this theory. Moreover, as Georgi (1992, 3) indic ates, the renormaliz ation c ondition that the theorem refers to is, in fac t, a mass-dependent subtrac tion sc heme. The above advantages of c utoff regulated, mass-dependent renormaliz ation sc hemes are balanc ed by the following disadvantages: (1) A momentum c utoff regulariz ation method violates Poinc aré invarianc e of the underlying high-energy theory, as well as any gauge invarianc e it may possess. (2) Mass-dependent subtrac tion sc hemes typic ally prevent the justific ation, based on dimensional analysis, that allows one to ignore the potentially infinite number of irrelevant terms in the effec tive ac tion (2) from being extended from tree-level c alc ulations to higher-order loop c orrec tions. The reason is that in mass- dependent sc hemes, the simple tree-level dependenc e of irrelevant terms on orders of 1/Λ c an break down when doing higher-order loop c orrec tions. In partic ular, in these higher-order c orrec tions, the dependenc e of irrelevant terms on the c utoff may be of order 1 (in general, suc h terms have a power law dependenc e on the c utoff), and thus suc h terms c annot be ignored (Manohar 1997, 327– 328; Pic h 1998, 14). Note that this does not prevent loop c alc ulations from proc eeding in mass-dependent sc hemes; rather it makes them more difficult. (Manohar 1997, 329) 4.2 Mass-independent Schemes and Continuum EFTs To address problem (2), many authors suggest adopting a mass-independent renormaliz ation sc heme. In this type of sc heme, the dimensional parameter μ (analogous to the momentum c utoff Λ in the c utoff approac h) only appears in loop c orrec tions in logarithms, and not powers, thus the relevant integrals are small at sc ales muc h smaller than the heavy fields (Manohar 1997, 238; Pic h 1998, 15). This allows one to effec tively ignore the Page 8 of 21

Effective Field Theories contributions of irrelevant terms, not only at tree-level as naive dimensional analysis allows but also for higher- order loop c orrec tions as well. Mass-independent renormaliz ation sc hemes are typic ally assoc iated with the method of regulating divergent integrals known as dimensional regulariz ation. This method takes advantage of the mathematic al fac t that the partic ular types of divergent integrals that arise in quantum field theoretic c alc ulations, again represented schematically by ∫ ∞ dD pK (p), will converge for sufficiently small values of D. Formally, one lets D = 4 − ε in the 0 integral (where ε is a very small c onstant), and then analytic ally c ontinues D to 4. This proc ess pic ks up poles in D-dimensional momentum spac e, and these c an be absorbed into a redefinition of the parameters of the theory. In this c ase, suc h redefinitions are independent of the masses, henc e the term mass-independent subtraction scheme. In the c ontext of EFTs, there are two main advantages of employing a mass-independent renormaliz ation sc heme based on dimensional regulariz ation. First, dimensional regulariz ation respec ts Poinc aré and gauge invarianc e. Sec ond, as indic ated above, mass-independent renormaliz ation sc hemes allow one to trunc ate the effec tive ac tion (2) to a finite list of terms, not only for tree-level c alc ulations but also for higher-order loop c alc ulations. However, it turns out that this simplific ation c omes at the c ost of having terms that explic itly inc lude the heavy fields appear in this finite list (see, e.g., Burgess 2004, 19). This has the following c onsequenc es: (1′) Many authors c onsider mass-independent sc hemes to violate the “spirit” of an EFT, to the extent that the latter is based on the notion of a c utoff, below whic h the physic s is explic itly desc ribed by only the light fields (Burgess 2004, 19; Burgess 2007, 343; Polc hinski 1993, 5). (2′) Perhaps more importantly, the presenc e of heavy field terms in an effec tive ac tion employing a mass- independent renormaliz ation sc heme prevents the applic ation of the Dec oupling Theorem (Georgi 1993, 225; Manohar 1997, 329). As mentioned above in sec tion 3.1, the latter holds only for mass-dependent sc hemes. It turns out that problem (2′) c an be addressed by inserting dec oupling by hand into an EFT that employs a mass- independent sc heme, but this requires a slight rec onc eptualiz ation of the nature of an EFT. This results in what Georgi (1992, 1; 1993, 215) refers to as a “c ontinuum EFT.” How c an an EFT be c onstruc ted without initial appeal to a c utoff? Briefly, for top-down c onstruc tions, the initial momentum splitting of the fields in a Wilsonian EFT and the integration over the heavy modes is replac ed in a continuum EFT by the following steps (after Georgi 1993, 228; see also Burgess 2004, 19–20; Burgess 2007, 344): w(Ih′)e Sreta Lrt w(ϕith) dae dsicmreibnessio tnhael llyig rhet gfiuelladrsiz aendd t hLeHor(yχ ,wϕit)h dLeasgcrarinbgeisa ne vdeernysthitiyn gL eHls(eχ ,(wϕh) e+reL x( aϕr)e atth ea hlaeragvey s cfiealldes s, of mass M). Now evolve the theory to lower sc ales using the renormaliz ation group: This allows you to go from sc ale s to sc ale s − ds without c hanging the c ontent of the theory. (II′) When s gets below M, the effec tive theory is c hanged to a new one without the heavy fields: L (ϕ) + δL (ϕ), where δL (ϕ) encodes a “matching correction” that includes any new nonrenormalizable interac tions that may be required. The matc hing c orrec tion is made so that the physic s of the light fields is the same in the two theories at the boundary s = M. To explicitly calculate δL (ϕ), one expands it in a complete set of local operators in the same manner that the expansion (2) for Wilsonian EFTs is performed: (12) Dimensional analysis c an now be applied to determine the sc aling behavior of the terms in (12), in the same way it is applied in Wilsonian EFTs. Again, in the c ase of a c ontinuum EFT, this analysis is valid not just for tree-level c alc ulations but also for higher-order loop c alc ulations as well. To summariz e and c ompare, in the c onstruc tion of a Wilsonian EFT, the heavy fields are first integrated out of the underlying high-energy theory and the resulting Wilsonian effec tive ac tion is then expanded in a series of loc al operator terms. The c utoff Λ in a Wilsonian EFT plays a double role: first, through the definition of the heavy and light fields, in explic itly demarc ating the low-energy physic s from the high-energy physic s; and sec ond, in regulating divergent integrals in the c alc ulation of observable quantities. In the c onstruc tion of a c ontinuum EFT, the heavy fields are initially left alone in the underlying high-energy theory, whic h is first evolved down to the Page 9 of 21

Effective Field Theories appropriate energy sc ale. The c ontinuum EFT is then c onstruc ted by c ompletely removing the heavy fields from the high-energy theory, as opposed to integrating them out; and this removal is c ompensated for by an appropriate matc hing c alc ulation (this latter is ultimately responsible for the appearanc e of heavy modes into the operator expansion (12)). In a c ontinuum EFT, the first role of the Wilsonian c utoff Λ is played by the renormaliz ation sc ale s that demarc ates the low-energy physic s from the high-energy physic s. The sec ond role of the Wilsonian c utoff is dropped, the proc edure of dimensional regulariz ation taking its plac e. This last observation suggests the motivation for Georgi's phrase “c ontinuum EFT.” In a Wilsonian EFT, the regulariz ation of divergent integrals is performed by restric ting the range of momentum variables in integrals over momentum spac e. The Fourier transform equivalent of this proc edure is a restric tion of the range of c oordinate variables in integrals over c oordinate spac e. Henc e regulariz ation in a Wilsonian EFT is analogous to plac ing a high-energy c ontinuum theory on a disc rete lattic e, and this is the reason why momentum c utoff regulariz ation violates Poinc aré invarianc e. In c ontrast, in a “c ontinuum EFT,” the regulariz ation of divergent integrals is performed by c alc ulating them in a c ontinuous spac etime of dimension D (the Fourier transform equivalent of D- dimensional momentum spac e). This is the reason why dimensional regulariz ation does not violate Poinc aré invariance. 5. Onto lo gical Implicatio ns Different renormaliz ation sc hemes ultimately all agree on the values of physic al quantities. In partic ular, both mass- dependent and mass-independent sc hemes will, at the end of the day, agree on all empiric ally measured quantities.16 Thus a Wilsonian EFT for a given physic al system is empiric ally equivalent to that system's c ontinuum EFT. On the other hand, the fac t that these types of EFT plac e different emphasis on the nature of the c utoff suggests they c an be interpreted as telling us different things about the world. I now c onsider some of the implic ations this has for debates over the ontologic al status of EFTs. 5.1 Decoupling and Quasi-Autonomous Domains Cao and Sc hweber emphasiz e the role of the Dec oupling Theorem in understanding the ontologic al signific anc e of EFTs: Thus, with the dec oupling theorem and the c onc ept of EFT emerges a hierarc hic al pic ture of nature offered by QFT [quantum field theory], one that explains why the desc ription at any one level is so stable and is not disturbed by whatever happens at higher energies, and thus justifies the use of suc h desc riptions. (Cao and Schweber 1993, 64) In this pic ture, the [physic al world] c an be c onsidered as layered into quasi-autonomous domains, eac h layer having its own ontology and assoc iated ‘fundamental law’. (Cao and Sc hweber 1993, 72) They further suggest that EFTs entail “an antifoundationalism in epistemology and an antireduc tionism in methodology” (Cao and Sc hweber 1993, 69). Huggett and Weingard (1995, 187) interpret this as a view that “holds that nature is desc ribed by a genuinely never-ending tower of theories, and that the c ompeting possibilities of unific ation and new physic s should be abandoned.” Hartmann (2001, 298) c laims that “Cao and Sc hweber's talk of quasi-autonomous domains rests on the validity of the dec oupling theorem,” and then rightly points out that the latter is not nec essarily valid in all c ases in whic h EFTs exist. In partic ular, it requires the existenc e of an underlying high-energy renormaliz able theory with different mass sc ales. This vitiates Cao and Sc hweber's antifoundationalism and/or antireduc tionism, if they are taken to entail that there is no underlying theory. On the other hand, Castellani (2002) appears to endorse an aspec t of the “quasi-autonomous domain” interpretation: “The EFT approac h in its extreme version provides a level struc ture (‘tower’) of EFTs, eac h theory c onnec ted with the prec eding one (going‘up’ in the tower) by means of the [renormaliz ation group] equations and the matching conditions at the boundary” (Castellani 2002, 263). It appears that the partic ipants in this debate are talking past eac h other, having implic itly adopted different notions Page 10 of 21

Effective Field Theories of an EFT. Cao and Sc hweber, in partic ular, are sometimes ambiguous on what c onc ept of EFT they are employing. On the one hand, in their emphasis on the Dec oupling Theorem and momentum c utoff regulariz ation, they appear to adopt a Wilsonian notion of an EFT. For instanc e, in the c ontext of disc ussing the ontologic al signific anc e of momentum c utoff regulariz ation, they observe that “the following c an be stated for other regulariz ation sc hemes … but not for dimensional regulariz ation, whic h is more formalistic and irrelevant to the point disc ussed here” (Cao and Sc hweber 1993, 92 n. 17). On the other hand, immediately before introduc ing the EFT-inspired hierarc hic al pic ture of nature, they desc ribe an EFT in the following terms: “The EFT c an be obtained by deleting all heavy fields from the c omplete renormaliz able theory and suitably redefining the c oupling c onstants, masses, and the sc ale of the Green's func tions, using the renormaliz ation group equations” (Cao and Sc hweber 1993, 64). This appears to be a desc ription of the c onstruc tion of a c ontinuum EFT as outlined in sec tion 3.2. Hartmann (2001) makes it evident in his c ritique of Cao and Sc hweber that he implic itly has adopted the Wilsonian notion of an EFT. Finally, Castellani implic itly adopts a c ontinuum notion of an EFT, desc ribing its c onstruc tion as one based on matching conditions (see, e.g., Castellani 2002, 262). Note that the notion of EFT one adopts should be important in this debate. The Dec oupling Theorem was proven in the c ontext of Wilsonian EFTs of a partic ular kind; namely, those for whic h there exists an underlying renormaliz able high-energy theory with different mass sc ales. Thus, if by “EFT” Cao and Sc hweber mean “Wilsonian EFT,” then Hartmann's c ritique goes through: Wilsonian EFTs do not, by themselves, support an ontology of quasi-autonomous domains. On the other hand, the Dec oupling Theorem fails for c ontinuum EFTs, but, arguably this does not prevent them from supporting a well-defined notion of quasi-autonomous domains. This is bec ause in c ontinuum EFTs, dec oupling is inserted “by hand” in the form of matc hing c alc ulations. Thus, if by “EFT” Cao and Sc hweber mean “c ontinuum EFT,” then, arguably, Hartmann's c ritique does not go through: c ontinuum EFTs are, by themselves, c apable of supporting an ontology of quasi-autonomous domains. Henc e, provided by “EFT,” Castellani means “c ontinuum EFT,” her endorsement of suc h an ontology is justified.17 5.2 Realistic Interpretations of the Cutoff In typic al expositions of EFTs, emphasis is plac ed on a realistic interpretation of the c utoff. Many authors c laim that suc h a realistic interpretation is what separates the c ontemporary c onc ept of an EFT from older views of renormaliz ation (Hartmann 2001, 282; Castellani 2002, 261; Grinbaum 2008, 37). Under these older views, a c utoff, when it oc c urred in ac c ounts of QFTs, only played a role as a regulator of integrals and was taken to infinity at the end of the day (if this resulted in a finite theory, then the theory was deemed to be renormaliz able). The c ontemporary c onc ept of an EFT, so the story goes, is based on viewing the c utoff realistic ally in a different role; namely, as a means of demarc ating low-energy physic s from high-energy physic s. By now it should be obvious from the disc ussion in sec tion 4 that this standard ac c ount is only half the story; it, perhaps unfairly, privileges Wilsonian EFTs and the double role the c utoff plays in their c onstruc tion, over c ontinuum EFTs. This is not to say that a realistic interpretation of the c utoff c annot be made in the c ontext of c ontinuum EFTs. Rec all that in c ontinuum EFTs, the role that the Wilsonian c utoff Λ plays in demarc ating low-energy physic s from high-energy physic s is played by a sc aling variable s (i.e., the renormaliz ation sc ale that appears in the renormaliz ation group equations). Certainly this sc aling variable c an be realistic ally interpreted, perhaps as a basis for an ontology of quasi-autonomous domains; and it might even be referred to as a c utoff, in this c ontext. The important point is that, in a c ontinuum EFT, this sc aling variable does not play the sec ond role that the Wilsonian cutoff Λ plays; namely, as a regulator of divergent integrals. Thus to realistic ally interpret the c utoff in an EFT c ould mean one of two things: (a) The Wilsonian regulator Λ should be realistic ally interpreted. (b) The c onstant that demarc ates low-energy physic s from high-energy physic s (given by Λ in Wilsonian EFTs and by a partic ular value of s in c ontinuum EFTs) should be realistic ally interpreted. As the disc ussion at the end of sec tion 4.2 suggests, adopting (a) might motivate an ontology in whic h spac e is disc rete: momentum c utoff regulariz ation is analogous to plac ing a c ontinuum theory on a disc rete lattic e. But suc h an ontology is not forc ed upon us by a realistic interpretation of the c utoff. Simply put, (b) does not entail (a). One c an adopt a realistic interpretation of the c utoff in a c ontinuum EFT and, at the same time, an ontology in whic h Page 11 of 21

Effective Field Theories spacetime is continuous. This c onc lusion affec ts part of a rec ent debate over interpretations of quantum field theory (QFT). Wallac e (2006) adopts an interpretation of QFT in whic h a c utoff is inserted and realistic ally interpreted and justifies it by appealing to features of EFTs (see, e.g., 43). Fraser (2009) c ritic iz es this “c utoff variant” of QFT in the following way: “If the c utoffs are taken seriously, then they must be interpreted realistic ally; that is, spac e is really disc rete and of finite extent ac c ording to the c utoff variant of QFT” (Fraser 2009, 552). Fraser suggests this makes c utoff QFT unsatisfac tory: if the c utoff is not taken seriously, then c utoff QFT reduc es to “infinitely renormaliz ed QFT,” whic h is the standard textbook ac c ount in whic h c utoffs, when they appear, are taken to infinity in renormaliz ation sc hemes. An appeal to Haag's theorem then indic ates this textbook ac c ount is inc onsistent. On the other hand, if the c utoff is taken seriously, then c utoff QFTs avoid Haag's theorem (whic h, among other things, requires Poinc aré invarianc e); but they entail spac e is disc rete and finite, and “nobody defends the position that QFT provides evidenc e that spac e is disc rete and the universe is finite” (Fraser 2009, 552).18 Onc e again, being c lear on the type of EFT one adopts on whic h to base the c utoff variant of QFT will make a differenc e in this debate. If by “c utoff QFT” one means “Wilsonian EFT,” then Fraser's c ritique, arguably, goes through. However, if by “c utoff QFT” one means “c ontinuum EFT,” then the above argument does not go through: c ontinuum EFTs support an ontology in whic h spac etime is c ontinuous.19 Thus, provided one c an demonstrate that c ontinuum EFTs avoid Haag's theorem, a c utoff version of QFT based on c ontinuum EFTs is a viable alternative to the “formal variant” (i.e., axiomatic QFT) that Fraser (2009, 538) advoc ates. 5.3 EFTs and Approximation Finally, two further examples of how the distinc tion between types of EFTs is important for issues of interpretation involve the notions of idealiz ation and approximation. The fac t that Wilsonian EFTs support an ontology in whic h spac e is disc rete suggests to Fraser (2009, 564) that the c utoff variant of QFT is an indispensable idealiz ation: “[It] …is an idealiz ation in the sense that the possible worlds in whic h QFT is true are presumed to be worlds in whic h spac e is c ontinuous and infinite,” and “[t]his idealiz ation is indispensable insofar as it is not possible to remove the c utoffs entirely” (sinc e this would turn the c utoff variant into the infinitely renormaliz ed variant). But, again, c ontinuum EFTs do not idealiz e spac e as disc rete, henc e a version of c utoff QFT based on c ontinuum EFTs is not an idealiz ation of this type. Relatedly, Castellani (2002, 260, 263) suggests that EFTs are “intrinsic ally approximate and c ontext-dependent.” An EFT, under this view, is an approximation of an underlying high-energy theory and is valid only within a spec ified energy range. Now, arguably, Cao and Sc hweber's ontology of quasi-autonomous domains is intended in part to address c laims of this type. Under Cao and Sc hweber's view, an EFT desc ribes a quasi-autonomous domain by means of a c omplete desc ription of phenomena within a given energy range, independent for the most part of desc riptions at higher or lower energies (Cao and Sc hweber 1993, 64, are c areful to explain how high-energy effec ts do make themselves present in relevant and marginal terms of an effec tive Lagrangian density). Thus, the disc ussion in sec tion 5.1 entails that EFTs need not be interpreted as intrinsic ally approximate, provided one adopts c ontinuum EFTs as the objec t of one's interpretation. 6. EFTs and Emergence In the example in sec tion 2.3 above, the EFT of a superfluid 4 He film took the form (to lowest order) of quantum electrodynamics in (2 + 1) dimensions. The duality transformations (8) suggested that, at low energies, the density of a superfluid 4 He film behaves like a magnetic field, its veloc ity behaves like an elec tric field, and vortic es behave like c harge-c arrying elec trons. This example of a relativistic EFT of a c ondensed matter system, and others like it, have suggested to some physic ists that novel phenomena (fields, partic les, symmetries, spac etime, etc .) emerge in the low-energy limit of these systems.2 0 On the other hand, in the physic s literature, referenc es to emergenc e are typic ally not assoc iated with EFTs of relativistic QFTs.2 1 In the philosophy of physic s literature, the c onverse is true: philosophers of physic s have c onsidered notions of emergenc e related to EFTs of relativistic QFTs, but have paid little attention to emergenc e in EFTs of c ondensed matter systems.2 2 In this sec tion I take it as a given that the formal nature of an EFT is identic al in both c ontexts and c onsider the extent to whic h notions of emergenc e are applic able, regardless of c ontext. Page 12 of 21

Effective Field Theories Consider, first, the view from philosophy of physic s: Cao and Sc hweber, for instanc e, assoc iate the antireduc tionism of their quasi-autonomous domains interpretation of EFTs with a notion of emergenc e: “taking the dec oupling theorem and EFT seriously would entail c onsidering the reduc tionist …program an illusion, and would lead to its rejec tion and to a point of view that ac c epts emergenc e, henc e to a pluralist view of possible theoretic al ontologies” (Cao and Sc hweber 1993, 71). Likewise, Castellani (2002, 263) suggests that the EFT approac h “provides a level structure of theories where the way in which a theory emerges from another …is in principle desc ribable by using RG [Renormaliz ation Group] methods and matc hing c onditions at the boundary.” On the other hand, Castellani argues that the EFT approac h does not imply antireduc tionism, insofar as antireduc tionism is to be assoc iated with the denial of some type of intertheoretic relation: “The EFT sc hema, by allowing definite c onnec tions between theory levels, ac tually provides an argument against the basic antireduc tionist c laim” (Castellani 2002, 265). Note that while Cao and Schweber take emergence to be descriptive of ontologies (properties, objec ts, etc .), Castellani suggests emergenc e be viewed as a relation between theories. In both c ases, however, the emphasis is on autonomy. Cao and Sc hweber's emergent ontologies are restric ted to quasi- autonomous domains, each described by a distinct EFT. Castellani's emergent theories stand in “definite c onnec tions” with eac h other, but assumedly not so definite as to warrant the label of reduc tion. This sec tion will only be c onc erned with the extent to whic h the intertheoretic relation between a top-down EFT and its high-energy theory supports a notion of emergenc e; thus the approac h taken will be to view emergenc e as a relation between theories. Batterman (2002, 115) has suggested that, under a rec eived view in the philosophic al literature, suc h a relation holding between an emergent theory T′ and an underlying theory T c an mean any or all of the following: (a) The phenomena of T′ cannot be reduced to T. (b) The phenomena of T′ cannot be predicted by T. (c) The phenomena of T′ are causally independent of those of T. (d) The phenomena of T′ cannot be explained by T. The initial task of this sec tion will be to c onsider the extent to whic h the intertheoretic relation between an EFT and its high-energy theory supports these notions of autonomy. 6.1 The EFT Inter theoretic Relation Sec tions 2.1 and 4.2 above desc ribed the general steps in the c onstruc tion of a Wilsonian and a c ontinuum EFT, respec tively. Although differing in their details, these steps have the following general form: (I) One first identifies and then systematically eliminates high-energy degrees of freedom; and then (II) one expands the resulting effec tive Lagrangian density (or effec tive ac tion) in terms of loc al operators. The intertheoretic relation defined by this proc edure has one very important c harac teristic in the c ontext of a disc ussion of notions of emergenc e; namely, its relata are distinct theories. To see this, c onsider the following c onsequenc es of Steps (I) and (II): (1) First, the low-energy degrees of freedom of the EFT are typic ally formally distinc t from the high-energy degrees of freedom. This suggests they admit distinc t ontologic al interpretations (rec all, for instanc e, some of the examples from sec tion 2: pions versus quarks, quasipartic les versus elec trons, and elec tric and magnetic fields versus 4He atoms). (2) Sec ond, the EFT Lagrangian density typic ally is formally distinc t from the high-energy Lagrangian density. Manohar makes (2) clear in the context of the Fermi EFT of the weak force: It is important to keep in mind that the effective theory is a different theory from the full theory. The full theory of the weak interac tions is a renormaliz able field theory. The effec tive field theory is a non- renormaliz able field theory, and has a different divergenc e struc ture from the full theory. (Manohar 1997, 327) Page 13 of 21

Effective Field Theories Figure 6.1 The relation between the initial Lagrangian and the effective Lagrangian for superfluid helium. As another example of (2), c onsider the EFT of a superfluid 4 He film desc ribed in sec tion 2.3. Above a c ritic al temperature, the system c onsists of a nonrelativistic normal liquid. As the temperature is lowered below the c ritic al value, a phase transition oc c urs, ac c ompanied by a spontaneously broken symmetry, and the system enters the superfluid phase. If the temperature is lowered further, its c onstituents c an be desc ribed in terms of a relativistic EFT. Importantly, both the normal liquid and the superfluid, as well as the phase transition and the spontaneously broken symmetry, are all enc oded in a single Lagrangian density (4). All of these states and proc esses c an thus be said to be desc ribed by a single theory. On the other hand, the low-energy relativistic system is enc oded in the effec tive Lagrangian density (9), whic h is suffic iently formally distinc t from (4) to warrant viewing it as a different theory (see figure 6.1, after Bain 2008, 313). Note that the c laim that an EFT and its high-energy theory are distinc t theories is not intended to be based simply on the fac t that there is a formal distinc tion between their respec tive Lagrangian densities. It is not the c ase that, in general, there is a 1– 1 c orrespondenc e between Lagrangian densities and theories. For instanc e, simply c hanging the interac tion term in a Lagrangian density does not, arguably, c hange the theory it is intended to represent (c onsider the theory of Newtonian partic le dynamic s applied to different interac tions). However, in the c ase of an EFT and its high-energy theory, the differenc e between the two Lagrangian densities is substantial enough to warrant the assumption that one is dealing with two distinc t theories. In the 4 He c ase, the c ontrast is between a nonrelativistic Lagrangian density and a relativistic Lagrangian density: whereas in Manohar's example, the c ontrast is between a renormaliz able Lagrangian density and a nonrenormaliz able Lagrangian density. Moreover, as (1) indic ates, in both c ases, the dynamic al variables of the EFT are distinc t from those of the high-energy theory. With the above proviso in mind, in the Lagrangian formalism, a differenc e in the form of the Lagrangian density entails a differenc e in the Euler-Lagrange equations of motion for the relevant dynamic al variables. One might thus argue that an EFT T′ is derivationally independent from its assoc iated high-energy theory T, insofar as a spec ific ation of the equations of motion of T (together with pertinent initial and/or boundary c onditions) will fail to spec ify solutions to the equations of motion of T′. More generally, the steps involved in the c onstruc tion of an EFT typic ally involve approximations and heuristic reasoning. In the c ase of Wilsonian EFTs, rec all that the initial integral over the heavy fields typic ally involves a saddle-point approximation about the free theory, and even before suc h an approximation c an be c onstruc ted, in both the Wilsonian and c ontinuum EFT c ases, the task of identifying the relevant high-energy variables must be ac c omplished. This suggests that, in general, it will be diffic ult, if not impossible, to reformulate the steps involved in the c onstruc tion of an EFT (of either the Wilsonian or continuum types) in the form of a derivation. 6.2 Senses of Autonomy Given that the intertheoretic relation between an EFT and its assoc iated high-energy theory is c harac teriz ed by derivational independenc e, what does this suggest about the sense in whic h the former is autonomous from the latter? (a) Reductive Autonomy. One might first argue that the relation between an EFT and its high-energy theory c annot be c harac teriz ed by notions of reduc tion based on derivability. On the standard (Nagelian) ac c ount of reduc tion, for instanc e, a nec essary c ondition for a theory T′ to reduc e to another T is that T′ be a definitional extension of T (see, e.g., Butterfield and Isham 1999, 115). This requires first that T and T′ admit formulations as deduc tively c losed sets of sentenc es in a formal language (i.e., it assumes a syntactic c onc eption of theories), and sec ond that an extension T* of T c an be c onstruc ted suc h that the theorems of T′ are a subset of the theorems of T* (i.e., it requires that T′ is a sub-theory of T*). Formally, T* is c onstruc ted by adding to T a definition of eac h of the nonlogic al symbols of T′. One might now argue that this c annot be done in the c ase of a high-energy theory and its EFT. As noted above, in the Lagrangian formalism, differenc es in the Lagrangian densities representing two theories entail differenc es in the theories' Euler-Lagrange equations of motion. If one adopts the view that suc h equations represent the theory's dynamic al laws, then the dynamic al laws of an EFT and its high-energy theory are different, and a differenc e in dynamic al laws entails a differenc e in theorems derived from these laws. Thus, an EFT is not a sub-theory of its high-energy theory; henc e, one c annot say that an EFT reduc es to its high-energy theory on this view of reduc tion.2 3 Page 14 of 21

Effective Field Theories Note that the above argument does not depend essentially on a syntac tic c onc eption of theories. For instanc e, under a semantic c onc eption of theories, a typic al c laim is that a theory reduc es to another just when models of the first c an be embedded in models of the sec ond. This will not suffic e to reduc e an EFT to its high-energy theory so long as the embedding is required to preserve dynamic al laws (and if it is not, then it is unc lear whether the term “reduc tion” for suc h an embedding is appropriate2 4 ). (b) Predictive Autonomy. Predic tive autonomy between an EFT and its high- energy theory would seem to be another c onsequenc e of derivational independenc e. Given that the relation between an EFT and its high- energy theory T c annot be desc ribed in terms of a derivation in whic h T is implic ated, the phenomena that the EFT desc ribes c annot be derived, and henc e predic ted, on the basis of T. (c) Causal Autonomy. Whether derivational independenc e of an EFT from its high-energy theory entails c ausal independenc e will depend on one's c onc ept of c ausation. To demonstrate the c ausal independenc e of an EFT from its high-energy theory, one would have to provide an ac c ount of how the phenomena governed by the EFT are not implic ated in the c ausal mec hanisms assoc iated with the relevant high-energy phenomena. The example of superfluid 4 He films is instruc tive here. Under one interpretation, this EFT suggests that low-energy “ripples” of a superfluid 4 He film behave like relativistic elec tric and magnetic fields. Insofar as ripples in a substrate are implic ated in the c ausal mec hanisms that govern the substrate, this suggests causal links between the phenomena of the EFT and the high-energy theory. On the other hand, if one's view of c ausation is suc h that the existenc e of a c ausal relation requires the existenc e of a nomic c onnec tion (embodied in a dynamic al law, say), then one might argue that to the extent to whic h an EFT and its high-energy theory are nomic ally independent (in the sense, perhaps, of possessing distinc t dynamic al laws), they are c ausally independent, too. (d) Explanatory Autonomy. Whether the phenomena desc ribed by an EFT c an be explained in terms of the high-energy theory will obviously depend on the notion of explanation one adopts. Arguably, explanatory autonomy will obtain on any ac c ount of explanation that requires the explanandum to be the produc t of a derivation in whic h the explanans is implic ated; and to the extent to whic h an EFT is c ausally independent of its high-energy theory T, its phenomena c annot be c ausally explained by T. 6.3 Emergence and Limiting Relations Evidently there is room to maneuver in addressing the question of whether the intertheoretic relation between an EFT and its high-energy theory c an be desc ribed in terms of a notion of emergenc e, at least if suc h a notion is related to standard ac c ounts of reduc tion, predic tion, c ausation, and/or explanation. On the other hand, Batterman (2002) has offered a nonstandard ac c ount of emergenc e based on the failure of a limiting relation between two theories. This sec tion c onsiders its applic ability in the c ontext of an EFT and its high-energy theory. Batterman's notion of emergenc e is assoc iated with the failure of what he refers to as the “physic ists' sense” of reduc tion. Under this notion, a “c oarse” theory Tc reduc es to a “more refined” theory Tf provided a limit of the sc hematic form lim ε→0 Tf = Tc c an be shown to exist, where e is a relevant parameter.2 5 The novelty of emergent properties, according to Batterman, is “a result of the singular nature of the limiting relationship between the finer and coarser theories that are relevant to the phenomenon of interest” (2002, 121). This singular nature, when it exists, is indic ative of the existenc e of a real “physic al singularity,” ac c ording to Batterman. Thus: “The proposed account of emergent properties has it that genuinely emergent properties, as opposed to ‘merely’ resultant properties, depend on the existenc e of physic al singularities” (Batterman 2002, 125). For Batterman, then, there are two nec essary c onditions for the existenc e of an emergent property in the c ontext of a fundamental (more refined) theory T and a less fundamental (c oarse) theory T′: (a) The physic ists' notion of reduc tion must hold between T and T′; i.e., there must be a limiting relation between T and T′. (b) The limiting relation must fail in the c ontext with whic h the emergent property is identified; in partic ular, there must be a physic al singularity assoc iated with the emergent property. As an example of two theories that satisfy these c onditions, Batterman c onsiders thermodynamic s (TD) and statistic al mec hanic s (SM). With some qualific ation, it is possible to define an intertheoretic relation between the TD and SM desc riptions of a physic al system in terms of the thermodynamic limit N, ν → ∞ while N/ν = c onstant, where N and ν are the number of partic les and volume of the system (Batterman 2002, 123). This limit fails for a Page 15 of 21

Effective Field Theories thermodynamic system at a c ritic al point at whic h it undergoes a phase transition. At suc h a point, the c orrelation length assoc iated with the system (roughly, the measure of the c orrelation between spatially separated states) bec omes infinite. For Batterman, this is an example of a physic al singularity, and he is thus motivated to identify properties assoc iated with phase transitions as emergent properties. (In the 4 He example of sec tion 2.3, suc h properties would c orrespond to the highly c orrelated phenomena assoc iated with superfluidity.) Importantly, the intertheoretic relation between TD and SM in this c ontext c an be modeled by renormaliz ation group (RG) tec hniques. The thermodynamic limit generates an RG flow in the parameter spac e of a TD system. This is analogous to how a sc ale-dependent momentum c utoff L(s) generates an RG flow in the parameter spac e of an RQFT, as desc ribed above in sec tion 3. A TD system at a c ritic al point is then represented by a fixed point in its RG flow. This is a point at whic h the parameters of the theory remain unc hanged under further RG resc aling; i.e., they bec ome sc ale invariant. As Batterman explains, at a c ritic al point, there is a loss of a c harac teristic length sc ale. This leads to the hypothesis of sc ale invarianc e and the idea that the large sc ale features of a system are virtually independent of what goes on at a mic rosc opic level. In the thermodynamic c ase we see that the bulk properties of the thermodynamic systems are independent of the detailed mic rosc opic , molec ular c onstitution of the physic al system. (Batterman 2005, 243) In the disc ussion in sec tion 3 above, a fixed point in the RG flow assoc iated with an RQFT is the explic it indic ation that the theory is sc ale-independent and henc e renormaliz able. For suc h a theory, the low-energy properties of its c onstituents are independent of its detailed high-energy c onstitution. As in the TD/SM c ase, this also represents the loss of a c harac teristic sc ale, in this c ase an energy sc ale. Thus, the features of a renormaliz able theory are independent of what goes on at large energies. And just as in the TD/SM c ase, sc ale invarianc e in a renormaliz able RQFT is assoc iated with the existenc e of physic al singularities. In this c ase, a physic al singularity is assoc iated with an observable quantity (like a sc attering c ross sec tion) that is represented by a divergent Green's func tion. This analogy between the intertheoretic relation between TD and SM, on the one hand, and the relation between the low-energy and high-energy sec tors of a renormaliz able RQFT, on the other, suggests that Batterman's notion of emergenc e might be applic able in the latter c ase. To make this analogy more explic it, c onsider the following summaries of the relevant features of these examples: Example 1: T = statistic al mec hanic s (SM). T′ = thermodynamic s (TD). The limiting relation is the thermodynamic limit: N, ν → ∞ while N/ν = c onstant. (i) The thermodynamic limit fails at a fixed point in the assoc iated RG flow in the sense that, at a fixed point, there is no link between the bulk TD properties and the mic rosc opic SM properties. This is a manifestation of sc ale independenc e. (ii) A physic al singularity assoc iated with the failure of the thermodynamic limit is a diverging c orrelation length. Emergent properties are properties assoc iated with the system at the fixed point. Example 2: T = renormaliz able c ontinuum RQFT. T′ = c utoff-regulated RQFT. The limiting relation is the c ontinuum limit: Λ(s) → ∞. More prec isely, to further the analogy with Example 1, the c ontinuum limit c an be given sc hematic ally by Λ(s) → ∞, [bare parameters] → ∞, while [renormaliz ed parameters] = [bare parameters]/Λ(s) = c onstant.2 6 (i) The c ontinuum limit fails at a fixed point in the assoc iated RG flow in the sense that, at a fixed point, there is no link between the low-energy c utoff theory and the high-energy c ontinuum theory. This is a manifestation of sc ale independenc e. (ii) A physic al singularity assoc iated with the failure of the c ontinuum limit is represented by a diverging Green's func tion. Emergent properties are properties assoc iated with the system at a fixed point. In princ iple, these are properties c onstruc ted out of relevant operators. Fraser has pointed out the following disanalogy between Examples 1 and 2: “whereas the desc ription of a system as c ontaining an infinite number of partic les furnished by [statistic al mec hanic s] is taken to be false, the desc ription of space as continuous and infinite that is furnished by QFT with an infinite number of degrees of freedom is taken to be true” (Fraser 2009, 565). Thus, in Example 1, the limiting relation is taken to be an idealiz ation, whereas in Page 16 of 21

Effective Field Theories Example 2 it is not. It would appear, however, that this disanalogy is not relevant to Batterman's notion of emergenc e, to the extent that the latter is assoc iated with the nec essary c onditions (a) and (b) above. Condition (a) requires simply that a limiting relation exist, but it says nothing about the status of this relation; in partic ular, whether it is taken to be an idealiz ation or not. This suggests that the properties assoc iated with the values of observable quantities c onstruc ted from the Green's func tions of a renormaliz able RQFT are emergent in Batterman's sense. The question now is: To what extent does Example 2 offer insight into the nature of emergenc e in the c ontext of EFTs? Two observations appear to be relevant in this c ontext. First, not all EFTs are assoc iated with renormaliz able high- energy theories. For those that are not, Batterman's notion of emergenc e c annot be supported without further ado. Sec ond, even in the c ase of an EFT with an assoc iated renormaliz able high-energy theory, the EFT will typic ally be formally distinc t from the latter. This is a result of the sec ond step in the c onstruc tion of an EFT (for both Wilsonian and c ontinuum versions) in whic h the effec tive Lagrangian density is c onstruc ted via a loc al operator expansion. In an RG analysis of a renormaliz able c ontinuum RQFT, this step is replac ed with a parameter-resc aling proc edure by means of whic h the low-energy c utoff Lagrangian density is transformed bac k into the initial form of the original Lagrangian density. The upshot is that T and T′ in Example 2 are formally identic al, whereas an EFT and its high- energy theory are not. Simply put, the c utoff-regulated RQFT of Example 2 is not the same mathematic al objec t as an EFT assoc iated with a renormaliz able high-energy theory. This suggests that further work needs to be done if Batterman's notion of emergenc e is to be applied in the c ontext of an EFT and its high-energy theory. 7. Co nclusio n Two general c onc lusions seem appropriate from this review of effec tive field theory. First, the disc ussion in sec tions 4 and 5 suggests that, in order to understand how EFTs c an be interpreted, one needs to understand the methods that physic ists use in applying them. By foc using attention on different renormaliz ation sc hemes that prac tic ing physic ists ac tually employ, one c an disc ern two types of empiric ally equivalent EFTs—Wilsonian EFTs and c ontinuum EFTs. These are nontrivial examples of empiric ally equivalent theories insofar as, in the c ontext of a given high-energy theory, they make the same low-energy predic tions, but they suggest different ontologies. Continuum EFTs support an ontology of quasi-autonomous domains, whereas Wilsonian EFTs do not. Continuum EFTs support an ontology that inc ludes a c ontinuous spac etime, whereas Wilsonian EFTs require spac e to be disc rete and finite. These features of Wilsonian EFTs have c ontributed to the view that EFTs in general engage in idealiz ation and are inherently approximate. The fac t that c ontinuum EFTs do not engage in suc h idealiz ations suggests that EFTs do admit interpretations in whic h they are not c onsidered inherently approximate. The sec ond c onc lusion one may draw from this review is that, if one desires to assoc iate (some aspec t of) the intertheoretic relation between an EFT and its (possibly hypothetic al) high-energy theory with a notion of emergenc e, then more work has to be done. In the c ontext of standard ac c ounts of emergenc e, the relevant feature of the EFT intertheoretic relation is that it supports a notion of derivational autonomy (i.e., an EFT c annot be said to be a derivation of its assoc iated high-energy theory). But just how derivational autonomy c an be linked with a notion of emergenc e will depend on suc h things as how one artic ulates additional c onc epts suc h as reduc tion, explanation, and/or c ausation. Sec tion 6 also demonstrated that the EFT intertheoretic relation does not support Batterman's (2002) more formal notion of emergenc e based on the failure of a limiting relation between two theories. Suc h a failure of a limiting relation does oc c ur between a renormaliz able high-energy RQFT and a c utoff- regulated theory obtained from it by renormaliz ation group tec hniques, but this is a different c ontext than the one in which an EFT is obtained from a high-energy theory. Again, the relevant property of the EFT intertheoretic relation here is that it is a relation between formally distinc t, derivationally autonomous, theories. References Appelquist, T., and J. Caraz z one (1975). Infrared singularities and massive fields. Physical Review D11: 2856– 2861. Page 17 of 21

Effective Field Theories Bain, J. (2008). Condensed matter physic s and the nature of spac etime. In The Ontology of Spacetime, Vol. 2, ed. D. Dieks, 301–329. Amsterdam: Elsevier Press. Batterman, R. (2002). The devil in the details: Asymptotic reasoning in explanation, reduction, and emergence. Oxford: Oxford University Press. ———. (2005). Critic al phenomenon and breaking drops: Infinite idealiz ation in physic s. Studies in History and Philosophy of Modern Physics 36: 225–244. Burgess, C. P. (1998). “An Ode to Effec tive Lagrangians”. Available online at arXiv: http://arxiv.o rg/abs/hep- ph/9812470. Burgess, C. P. (2004). Quantum gravity in everyday life: General relativity as an effec tive field theory. Living Reviews in Relativity. http://www.livingreviews.o rg/lrr- 2004- 5. ———. (2007). An introduc tion to effec tive field theory. Annual Review of Nuclear and Particle Science 57: 329– 367. Butterfield, J., and C. Isham (1999). On the emergenc e of time in quantum gravity. In The Arguments of Time, ed. J. Butterfield, 111–168. Oxford: Oxford University Press. Campbell-Smith, A. and N. Mavromatos (1998). “Effec tive Gauge Theories, the Renormaliz ation Group, and High-Tc Superconductivity,” Acta Physica Polonica B 29: 3819–3870. Cao, T., and S. Sc hweber (1993). The c onc eptual foundations and the philosophic al aspec ts of renormaliz ation theory. Synthese 97: 33–108. Castellani, E. (2002). Reduc tionism, emergenc e, and effec tive field theories. Studies in History and Philosophy of Modern Physics 33: 251–267. Dobado, A., A. Gomez -Nic ola, A. L. Maroto, and J. P. Pelaez (1997). Effective Lagrangians for the standard model. Berlin: Springer. Fraser, D. (2009). Quantum field theory: Underdetermination, inc onsistenc y, and idealiz ation. Philosophy of Science 76: 536–567. Georgi, H. (1992). Thoughts on effec tive field theory. Nuclear Physics B (Proc eedings Supplements) 29B, C: 1– 10. ———. (1993). Effec tive field theory. Annual Review of Nuclear and Particle Science 43: 209– 252. Grinbaum, A. (2008). On the eve of the LHC: Conc eptual questions in high-energy physic s. philsci- archive.pitt.edu, deposited on June 27, 2008. http://philsci-archive.pitt.edu/archive/00004088/. Hartmann, S. (2001). Effec tive field theories, reduc tionism and sc ientific explanation. Studies in History and Philosophy of Modern Physics 32: 267–304. Huggett, N., and R. Weingard (1995). The renormaliz ation group and effec tive field theories. Synthese 102: 171– 194. Manohar, A. (1997). Effec tive field theories. In Perturbative and nonperturbative aspects of quantum field theory, Lec ture Notes in Physic s, Vol. 479/1997, 311– 362. Berlin: Springer. Available online at arXiv: http://arxiv.o rg/abs/hep- ph/9606222. Neubert, M. (2006). Effec tive field theory and heavy quark physic s. In Physics in D ≥ 4. TASI 2004. Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, ed. J. Terning, C. Wagner, and D. Zeppenfeld, 149–194. Singapore: World Sc ientific . Available online at arXiv: http://arxiv.o rg/abs/hep- ph/0512222. Pic h, A. (1998). Effec tive field theory. arXiv: http://arxiv.o rg/abs/hep- ph/9806303. Polc hinski, J. (1993). Effec tive field theory and the Fermi surfac e. In Proceedings of 1992 Theoretical Advanced Page 18 of 21

Effective Field Theories Studies Institute in Elementary Particle Physics, ed. J. Harvey and J. Polc hinski. Singapore: World Sc ientific . arXiv: http://arxiv.o rg/abs/hep- th/9210046. Rothsein, I. (2004). TASI Lec tures on effec tive field theories. arXiv: http://arxiv.o rg/abs/hep- ph/0308266. Schakel, A. (2008). Boulevard of broken symmetries: Effective field theories of condensed matter. Singapore: World Scientific. Stone, M. (2000). The physics of quantum fields. Berlin: Springer. Wallac e, D. (2006). In defenc e of naivete: The c onc eptual status of Lagrangian quantum field theory. Synthese 151: 33–80. Wen, X.-G. (2004). Quantum field theory of many-body systems. Oxford: Oxford University Press. Weinberg, S. (1996). The Quantum Theory of Fields, Volume 2: Modern Applications. Cambridge: Cambridge University Press. Zee, A. (2003). Quantum field theory in a nutshell. Princ eton: Princ eton University Press. Zhang, S.-C. (2004) To see a world in a grain of sand. In Science and ultimate reality: Quantum theory, cosmology and complexity, ed. J. D. Barrow, P. C. W. Davies, and C. L. Harper, 667–690. Cambridge: Cambridge University Press. Notes: (1) This exposition is based on Polc hinski (1993), and Campbell-Smith and Mavromatos (1998). See also Burgess (2004, 2007), Dobado et al. (1997), Manohar (1997), Pic h (1998), Rothsein (2004), and Sc hakel (2008). (2) Consider, for instanc e, a sc alar field theory with free ac tion S = (1/2) ʃ dD x(∂μφ )2 . This ac tion c ontains D powers of the spac etime c oordinate from dD x (with total energy units E−D), and −2 powers from the two oc c urrenc es of the spac etime derivative ∂μ ≡ ∂/∂xμ (with total units E2 ). Thus, in order for the ac tion to be dimensionless (with units E0), the field φ must have units Ey satisfying E−D E2 Ey Ey = E0, and thus dimension y = D/2 − 1. (3) Consider, again, sc alar field theory From note 2, the dimension of a sc alar fields is given by D/2 − 1; henc e, in general, an operator Oi constructed from Mϕ's and N derivatives will have dimension δi = M(D/2 − 1) + N. For D ≥ 3, there are only a finite number of ways in whic h Si 〈 D and δi = D. (4) See, e.g., Neubert (2006, 155). (5) Polc hinski (1993, 9). Another way to motivate this restric tion is by noting that mass terms c orrespond to gaps in the energy spec trum insofar as suc h terms desc ribe exc itations with finite rest energies that c annot be made arbitrarily small. These gaps c reate problems when taking a smooth low-energy limit (in the sense of a smooth renormaliz ation group evolution of parameters). Thus, for Weinberg (1996, 145), renormaliz ation group theory c an only be applied to EFTs that are massless or nearly massless. (6) While these aspec ts of the Standard Model suggest it c an be viewed as a natural EFT, other aspec ts famously prec lude this view. In partic ular, terms representing massive sc alar partic les like the Higgs boson are not protec ted by any symmetry and thus should not appear in an EFT. That they do, and that the order of the Higgs term is proportional to the elec troweak c utoff, generates the “hierarc hy problem” for the Standard Model. (7) Another example is Non-Relativistic QCD (NRQCD), whic h is an EFT of quark/gluon bound systems for whic h the relative velocity is small. The low-energy fields are obtained by splitting the gluon field into four modes and identify three of these as light variables. Rothsein (2003, 61) desc ribes this proc ess of identific ation as an “art form” as opposed to a systematic proc edure. (8) Tree-level c alc ulations are c ontributions to the perturbative expansion of a physic al quantity (like a sc attering c ross-sec tion) that do not involve integrating over the internal momenta of virtual proc esses. Loop c alc ulations, on Page 19 of 21

Effective Field Theories the other hand, involve possibly divergent integrals over internal momenta and are typic ally assoc iated with higher-order c orrec tions to tree-level c alc ulations. (The terminology is based on the graphic al representation of perturbative c alc ulations by Feynman diagrams.) (9) The following exposition draws on Wen (2004, 82–83; 259–264) and Zee (2003, 257–258; 314–316). (10) Formally this involves calculating the functional integral eiSeft[θ] = ∫ DρeiS4He [θ,ρ] , where Seff[θ] is the effective low-energy action, and S4He[θ, ρ] = ∫ d4 xL4He is the action of the high-energy theory As mentioned at the end of sec tion 2.1, suc h integrals c an be c alc ulated using a saddle-point approximation, whic h, in this c ontext, is equivalent to the semic lassic al expansion method outlined above (Sc hakel 2008, 75). (11) More prec isely, vortic es are soliton solutions to the equations of motion of (4) c harac teriz ed by π = f(r)eiθ, with boundary c onditions f(0) = 0 and f(r) → ψ0 , as r→ ∞. Intuitively, these c onditions desc ribe a loc aliz ed wave with finite energy that does not dissipate over time. (12) Note that the form of jvμ contracts over skew and symmetric indices; however, it is not identically zero, since for vortic es, θ is not a globally defined func tion. (13) In this c ontext, relevant terms are c alled “super renormaliz able,” irrelevant terms are c alled “nonrenormaliz able,” and marginal terms are c alled “renormaliz able.” (14) A Green's func tion is a vac uum expec tation value of field operators. (15) More prec isely, a mass-dependent subtrac tion sc heme is one in whic h anomalous dimensions and renormaliz ation group β func tions explic itly depend on μ/M, where μ is the renormaliz ation sc ale and M is the heavy mass (Georgi 1993, 221). (16) Burgess (2004, 20) explains this in the following way: “the differenc e between the c utoff- and dimensionally regulariz ed low-energy theory c an itself be parameteriz ed by appropriate loc al effec tive c ouplings within the low- energy theory Consequently, any regulariz ation-dependent properties will nec essarily drop out of final physic al results, onc e the (renormaliz ed) effec tive c ouplings are traded for physic al observables.” (17) Grinbaum (2008, 40), notably, makes a distinc tion between the “stric t” form of dec oupling assoc iated with the Appelquist-Caraz z one theorem, and a “milder” empiric al dec oupling thesis, whic h evidently is to be assoc iated with the matc hing c alc ulations involved in the c onstruc tion of c ontinuum EFTs. (18) See also Huggett and Weingard (1995, 178) for similar intuitions. (19) This suggests that the c ategories of empiric ally equivalent variants of QFT that Fraser (2009, 538) identifies should be expanded. Her “c utoff QFT” c ategory might be split into “c utoff regulariz ed QFT” and “dimensionally regulariz ed QFT.” (20) For instanc e, Zhang (2004, 669) reviews “examples of emergenc e in c ondensed matter systems” that take the form of relativistic EFTs, inc luding the QED3 c ase. These and other examples in the c ondensed matter literature are discussed in Bain (2008). (21) See Castellani (2002) for a review of the 1960s?1970s debates between solid-state physic ists and partic le physic ists over the c onc epts and status of reduc tion and emergenc e. (22) Both philosophers and physic ists have c onsidered notions of emergenc e in c ondensed matter systems exhibit ing spontaneously broken symmetries. But, as sec tion 6.1 below suggests, this c ontext is distinc t from the c ontext in whic h emergenc e might be assoc iated with EFTs. (23) Butterfield and Isham (1999, 122) observe that the standard definition of supervenienc e c an be c harac teriz ed in terms of an infinitistic definitional extension; thus neither c an it be said that an EFT supervenes (in this sense) on its associated high-energy theory. Page 20 of 21

Effective Field Theories (24) Admittedly this assumes that, whatever else reduc tion amounts to, it is essentially nomic in nature. (25) Batterman's (2002, 78) example is the reduc tion of spec ial relativity to c lassic al mec hanic s in the limit ν/c → 0, where ν is the veloc ity of a given physic al system and c is the speed of light. (26) See Stone (2000, 204) for the c ondensed matter c ontext. The bare parameters are the parameters of the theory before resc aling is performed to restore the c utoff bac k to its initial value after one iteration of the RG transformations. The renormaliz ed parameters are the resc aled parameters. Jonathan Bain J onathan Bain is Associate Professor of philosophy of science at the Polytechnic Institute of New York University. His research interests include philosophy of spacetim e, scientific realism , and philosophy of quantum field theory.

The Tyranny of Scales Robert Batterman The Oxford Handbook of Philosophy of Physics Edited by Robert Batterman Abstract and Keywords This c hapter addresses the problem in applied mathematic s and physic s c onc erning the behavior of materials that display radic ally different, dominant behaviors at different length sc ales. It offers strategies for upsc aling from theories or models at small sc ales to those at higher sc ales, and disc usses the philosophic al c onsequenc es of having to c onsider struc tures that appear at sc ales intermediate between the mic ro and the mac ro. The c hapter also c onsiders why the Navier-Cauc hy equations for isotropic elastic solids work so well in desc ribing the bending behavior of steel beams at the macroscale. K ey words: appl i ed math emati cs, ph y si cs, beh av i or of materi al s, ph i l osoph i cal con sequ en ces, Nav i er-Cau ch y equ ati on s, el asti c sol i ds, steel beams, macroscale 1. Intro ductio n In this essay I will foc us on a problem in physic s and applied mathematic s. This is the problem of modeling ac ross sc ales. Many systems, say a steel girder, manifest radic ally different, dominant behaviors at different length sc ales. At the sc ale of meters, we are interested in its bending properties, its buc kling strength, etc . At the sc ale of nanometers or smaller, it is c omposed of many atoms, and features of interest inc lude lattic e properties, ionic bonding strengths, etc . To design advanc ed materials (suc h as c ertain kinds of steel), materials sc ientists must attempt to deal with physic al phenomena ac ross 10+ orders of magnitude in spatial sc ales. Ac c ording to a rec ent (2006) NSF researc h report, this “tyranny of sc ales” renders c onventional modeling and simulation methods useless as they are typic ally tied to partic ular sc ales (Oden 2006, p. 29). “Confounding matters further, the princ ipal physic s governing events often c hanges with sc ale, so the models themselves must c hange in struc ture as the ramific ations of events pass from one sc ale to another” (Oden, pp. 29– 30). Thus, even though we often have good models for material behaviors at small and large sc ales, it is often hard to relate these sc ale-based models to eac h other. Mac rosc ale models represent the integrated effec ts of very subtle fac tors that are prac tic ally invisible at the smallest, atomic , sc ales. For this reason it has been notoriously diffic ult to model realistic materials with a simple bottom-up-from-the-atoms strategy. The widespread failure of that strategy forc ed physic ists interested in overall mac ro-behavior of materials toward c ompletely top-down modeling strategies familiar from traditional c ontinuum mec hanic s.1 A response to the problem of the “tyranny of sc ales” would attempt to exploit our rather ric h knowledge of intermediate mic ro- (or meso-) sc ale behaviors in a manner that would allow us to bridge between these two dominant methodologies. Mac rosc opic sc ale behaviors often fall into large c ommon c lasses of behaviors suc h as the c lass of isotropic elastic solids, c harac teriz ed by two phenomenologic al parameters—so-c alled elastic moduli. Can we employ knowledge of lower sc ale behaviors to understand this universality—to determine the moduli and to group the systems into c lasses exhibiting similar behavior? This is related to engineering c onc erns as well: Can we employ our smaller sc ale knowledge to better design systems for optimal mac rosc opic performanc e Page 1 of 23

The Tyranny of Scales characteristics? The great hope that has motivated a lot of rec ent researc h into so-c alled “homogeniz ation theory” arises from a c onvic tion that a “between-sc ales” point of view, suc h as that developed by Kadanoff, Fisher, and Wilson in the renormaliz ation group approac h to c ritic al phenomena in fluids and magnets, may very well be the proper methodologic al strategy with whic h to begin to overc ome the tyranny of sc ales. A number of philosophers have rec ently c ommented on the renormaliz ation group theory, but I believe their foc us has overlooked what is truly novel about the methodologic al perspec tive that the theory employs. Philosophic al disc ussions of the applic ability of mathematic s to physic s have not, in my opinion, paid suffic ient attention to c ontemporary work on this problem of modeling ac ross sc ales. In many instanc es, philosophers hold on to some sort of ultimate reduc tionist pic ture: whatever the fundamental theory is at the smallest, basic sc ale, it will be suffic ient in princ iple to tell us about the behavior of the systems at all sc ales. Continuum modeling on this view represents an idealization—as Feynman has said,“a smoothed-out imitation of a really muc h more c omplic ated mic rosc opic world” (Feynman, Leighton, and Sands 1964, p. 12). Furthermore, the suggestion is that suc h models are in princ iple eliminable. There is a puz z le however. Continuum model equations suc h as the Navier-Stokes equations of hydrodynamic s or the equations for elastic solids work despite the fac t that they c ompletely (ac tually, almost c ompletely—this is c ruc ial to the disc ussion below) ignore small sc ale or atomistic details of various fluids. The rec ipe (I c all it “Euler's c ontinuum rec ipe”) by whic h we c onstruc t c ontinuum models is safe: if we follow it, we will most always be led to empiric ally adequate suc c essful equations c harac teriz ing the behavior of systems at the mac rosc opic level. Why? What explains the safety of this rec ipe? Surely this requires an answer. Surely, the answer must have something to do with the physic s of the modeled systems at smaller sc ales. If suc h an answer c annot be provided, we will be left with a kind of skeptic ism: without suc h an answer, we c annot expec t anything like a unified c onc eption of applied mathematic s' use of c ontinuum idealiz ations.2 If an answer is forthc oming, then we have to fac e the reduc tionist pic ture mentioned above. Will suc h an answer—an answer that explains the robustness and safety of employing c ontinuum modeling—support the view that c ontinuum models are mere c onvenienc es, only pragmatic ally justified, given the powerful simplific ations gained by replac ing large but finite systems with infinite systems? As noted, many believe that a reduc tionist/eliminitivist pic ture is the c orrec t one. I maintain that even if we c an explain the safety and robustness of c ontinuum modeling (how this c an be done is the foc us of this essay), the reduc tionist pic ture is mistaken. It is a mistaken pic ture of how sc ienc e works. My foc us here is on a philosophic al investigation that is true to the ac tual modeling prac tic es of sc ientists. (I am not going to be addressing issues of what might be done in princ iple, if not in prac tic e.) The fac t of the matter is that sc ientists do not model the mac rosc ale behaviors of materials using pure bottom-up tec hniques.3 I suggest that muc h philosophic al c onfusion about reduc tion, emergenc e, atomism, and antirealism follows from the absolute c hoic e between bottom-up and top-down modeling that the tyranny of sc ales apparently forc es upon us. As noted, rec ent work in homogeniz ation theory is beginning to provide muc h more subtle desc riptive and modeling strategies. This new work c alls into question the stark dic hotomy drawn by the “do it in a completely bottom-up fashion” folks and those who insist that top-down methods are to be preferred. The next sec tion disc usses the proposal that the use of c ontinuum idealiz ations present no partic ular justific atory worries at all. Rec ent philosophic al literature has foc used on the role of c ontinuum limits in understanding various properties of phase transitions in physic al systems suc h as fluids and magnets. Some authors, partic ularly Jeremy Butterfield (2011) and John Norton (2011), have expressed the view that there are no partic ularly pressing issues here: the use of infinite limits is perfec tly straightforwardly justified by appeal to pragmatic c onsiderations. I argue that this view misses an important differenc e in methodology between some uses of infinite limits and those used by renormaliz ation group arguments and homogeniz ation theory. In sec tion 3, I present an interesting historic al example involving nineteenth c entury attempts to derive the proper equations governing the behavior of elastic solids and fluids. A c ontroversy raged throughout that c entury c onc erning the merits of starting from bottom-up atomic desc ription of various bodies in trying to arrive at empiric ally adequate c ontinuum equations. It turns out that the bottom-up advoc ates lost the debate. Correc t equations apparently c ould only be ac hieved by esc hewing all talk of atomic or molec ular struc ture, advoc ating instead a top-down approac h supplemented, importantly, with experimentally determined data. In sec tion 4, I Page 2 of 23

The Tyranny of Scales formulate the tyranny of sc ales as the problem, just mentioned, of trying to understand the c onnec tion between rec ipes for modeling at atomic sc ales (Euler's disc rete rec ipe) and Euler's c ontinuum rec ipe appropriate for c ontinuum models. Finally, I present a general disc ussion of work on homogeniz ation that provides at least the beginning of an answer to the safety question and to the problem of bridging sc ales between the atomic and the c ontinuum. This researc h c an be seen as allaying skeptic al worries about a unified applied mathematic al methodology regarding the use of c ontinuum idealiz ations of a c ertain kind. 2. Steel Beams, Scales, Scientific Metho d Let us c onsider the steel girder in a bit more detail. In many engineering applic ations steel displays linear elastic ity. This is to say that it obeys Hooke's Law—its strain is linearly proportional to its stress. One phenomenologic al parameter related to its stress/strain (i.e., stiffness) properties is Young's modulus appearing in the equations of motion for solids, as well as in equilibrium and variational equations. At sc ales of 1 meter to 10 meters, say, the steel girder appears to be almost c ompletely homogeneous: z ooming in with a small mic rosc ope will reveal nothing that looks muc h different. In fac t, there appears to be a kind of loc al sc ale invarianc e here.4 So for behaviors that take plac e within this range of sc ales, the steel girder is well-modeled or represented by the Navier-Cauc hy equations: (1) The parameters λ and μ are the “Lamé” parameters and are related to Young's modulus. Now jump from this large-sc ale pic ture of the steel to its smallest atomic sc ale. Here the steel, for typic al engineering purposes, is an alloy that c ontains iron and c arbon. At this sc ale, the steel exhibits highly ordered c rystalline lattic e struc tures. It looks nothing like the homogeneous girder at the mac rosc ales that exhibits no c rystalline struc ture. Somehow between the lowest sc ale of c rystals on a lattic e and the sc ale of meters or millimeters, the low-level ordered struc tures must disappear. But that suggests that properties of the steel at its most basic , atomic level c annot, by themselves, determine what is responsible for the properties of the steel at mac rosc ales. I will disc uss this in more detail below. In fac t, the story is remarkably c omplex. It involves appeal to various geometric al properties that appear at microscales intermediate between the atomic and the mac ro,6 as well as a number of other fac tors suc h as martensitic transformations.7 The symmetry breaking is effec ted by a c ombination of point defec ts, line defec ts, slip disloc ations, and higher dimensional wall defec ts that c harac teriz e interfac ial surfac es. All of these c ontribute to the homogenization of the steel we see and manipulate at the mac rosc ale. And, of c ourse, in engineering c ontexts the macro features (bending properties, for example) are the most important—we do not want our buildings or bridges to collapse. 2.1 Reduction, Limits, Continuum Models A simpler c ase than steel involves trying to c onnec t the finite statistic al mec hanic al theory of a fluid at the atomic sc ale to the thermodynamic c ontinuum theory at mac ro sc ales.8 The relationship between statistic al mec hanic s and thermodynamic s has rec eived a lot of attention in the rec ent philosophic al literature. Debates about intertheoretic reduc tion, its possibility, and its nature have all appealed to examples from thermodynamic s and statistic al mec hanic s. Many of these disc ussions, in the rec ent literature, have foc used on the nature and potential emergenc e of phase transitions in the so-c alled thermodynamic limit9 (Butterfield 2011a; Menon and Callender 2012; Belot 2005; Bangu 2009). What role does the thermodynamic limit play in c onnec ting theories? What role does it play in understanding c ertain partic ular features of thermodynamic systems? It will be instruc tive to c onsider the role of this limit in a more general c ontext than that typic al of the literature. This is the c ontext in whic h we c onsider the generic problem of upsc aling from atomic to laboratory sc ales, as in the c ase of the steel girder disc ussed above. In doing this, I hope it will bec ome c lear that many of the rec ent philosophic al disc ussions miss c ruc ial features of the methodology of applying limits like the thermodynamic limit. Before turning to the debates about the use of the thermodynamic limit and the justific ation of using infinite limits to understand the goings on in finite systems, I think it is worthwhile to step bac k to c onsider, briefly, some general issues about theory reduc tion. As mentioned above, many philosophers and physic ists tac itly (and sometimes explic itly) maintain some sort of in princ iple reduc tionist point of view. I do not deny that maybe in some as yet to Page 3 of 23

The Tyranny of Scales be artic ulated sense there may be an in princ iple bottom-up story to be told. However, appeals to this possibility ignore ac tual prac tic es and furthermore are never even remotely filled out in any detail. Typic ally the c laim is simply: “The fundamental theory (whatever it is, quantum mec hanic s, quantum field theory, etc .), bec ause it is fundamental (whatever that ultimately means), must be able to explain/reduce everything.” Nagel's seminal work (1961) c onsidered the reduc tion of thermodynamic s to statistic al mec hanic s to be a straightforward and paradigm c ase of intertheoretic reduc tion. On his view, as is well known, one derives the thermodynamic laws from the laws of statistic al mec hanic s employing so-c alled bridge laws c onnec ting terms/predic ates appearing in the reduc ed theory with those appearing in the reduc ing theory.10 In several plac es I have argued that this Nagelian strategy and its variants fail for many c ases of so-c alled reduc tion (Batterman 1995, 2002). I have argued that a limiting sense of reduc tion in whic h, say, statistic al mec hanic s “reduc es to” thermodynamic s in an appropriate limit (if it does) provides a more fruitful c onc eption of intertheoretic reduc tion than the Nagelian strategies where the relation seems to go the other way around: on the Nagelian strategies one has it that thermodynamic s reduc es to statistic al mec hanic s, in the sense of deduc tive derivation. There are a number of reasons for thinking the nonNagelian, “limiting,” sense of reduc tion is a superior sense of reduc tion. For one, there is the diffic ulty of finding the required definitional c onnec tions that the bridge laws are meant to embody.11 But in addition, the kinds of c onnec tions established between theories by taking limits do not appear to be expressible as definitional extensions of one theory to another. In many c ases, the limits involved are singular, and even when they are not, the use of mathematic al limits invokes mathematic s well beyond that expressible in the language of first order logic —a c harac teristic feature of Nagel's view of reduc tion and of its neoNagelian refinements. Despite these arguments a number of authors have rec ently tried to argue that reduc tion should be understood in Nagelian terms; that is, as the definitional extension of one theory to another. Jeremy Butterfield and Naz im Bouatta, for example, …take reduc tion as a relation between theories (of the systems c onc erned). It is essentially deduc tion; though the deduc tion is usually aided by adding appropriate definitions linking two theories' voc abularies. This will be c lose to endorsing the traditional philosophic al ac c ount [Nagel's], despite various objec tions levelled against it. The broad pic ture is that the c laims of some worse or less detailed (often earlier) theory c an be deduc ed within a better or more detailed (often later) theory, onc e we adjoin to the latter some appropriate definitions of the terms in the former. …So the pic ture is, with D standing for the definitions: Tb&D ⇒ Tt. In logic ians' jargon Tt is a definitional extension of Tb. (Butterfield and Bouatta 2011) In the c urrent c ontext the more basic , better theory (statistic al mec hanic s) is Tb and the reduc ed, tainted theory (thermodynamic s) is Tt.12 Butterfield and Bouatta obviously are not moved by the objec tions to the Nagelian sc heme that I briefly mentioned above. I suggest though, as we delve a bit more deeply into the examples of phase transitions and of the steel girder, that we keep in mind the question of whether the c ontinuum ac c ount of the bending behavior of the steel c an be reduc ed to the theory of its atomic c onstituents in the sense that we c an derive that c ontinuum behavior from the “better,” “more detailed,” and “later” atomic theory. Even if we extend the logic ians' sense of deduc tion (as definitional extension) beyond that of first order logic so as to inc lude inferenc es involving mathematic al limits, will suc h a deduc tion/reduc tion be possible? So the real question, as both of these examples employ c ontinuum limits, c onc erns why the use of suc h limits is justified. The debate about the justific ation of the use of infinite limits and, ultimately, about reduc tion c onc erns whether the appeal to limits c an in the end be eliminated. It is a pressing debate, bec ause no party thinks that at the most fundamental level, the steel girder is a c ontinuum. And no party thinks that a tea kettle boiling on the stove c ontains an infinite number of molec ules. What justifies our employing suc h infinite idealiz ations in desc ribing and explaining the behaviors of those systems? For Butterfield there is a “Straightforward Justific ation” for the use of infinite limits in physic al modeling. This Justific ation c onsists of two obvious, very general, broadly instrumentalistic , reasons for using a model that adopts the limit N = ∞: mathematic al c onvenienc e, and empiric al adequac y (up to a required ac c urac y). So it also applies to many other models that are almost never c ited in philosophic al disc ussions Page 4 of 23

The Tyranny of Scales of emergenc e and reduc tion. In partic ular, it applies to the many c lassic al c ontinuum models of fluids and solids, that are obtained by taking a limit of a c lassic al atomistic model as the number of atoms N tends to infinity (in an appropriate way, e.g. keeping the mass density c onstant). (2011, p. 1080) He c ontinues by emphasiz ing two “themes” c ommon to the use of many different infinite models: The first theme is abstrac tion from finitary effec ts. That is: the mathematic al c onvenienc e and empiric al adequac y of many suc h models arises, at least in part, by abstrac ting from suc h effec ts. Consider (a) how transient effec ts die out as time tends to infinity; and (b) how edge/boundary effec ts are absent in an infinitely large system. The sec ond theme is that the mathematic s of infinity is often muc h more c onvenient than the mathematic s of the large finite. The paradigm example is of c ourse the c onvenienc e of the c alc ulus: it is usually muc h easier to manipulate a differentiable real func tion than some func tion on a large disc rete subset of ℝ that approximates it. (2011, p. 1081) The advantages of these themes are, ac c ording to Butterfield, twofold. First, it may be easier to know or determine the limit's value than the ac tual value primarily bec ause of the removal of boundary and edge effec ts. Sec ond, in many examples of c ontinuum modeling we have a func tion defined over the finite c ollec tion of atoms or lattic e sites that osc illates or fluc tuates and so c an take on many values. In order to employ the c alc ulus we often need to “have eac h value of the func tion defined as a limit (namely, of values of another func tion)” (pp. 1081– 82). Butterfield seems to have in mind the standard use of averaging over a “representative elementary volume” (REV)13 and then taking limits N → ∞, volume going to z ero, so as to identify a c ontinuum value for a property on the mac rosc ale. In fac t, he c ites c ontinuum models of solids and fluids as paradigm examples: For example, c onsider the mass density varying along a rod, or within a fluid. For an atomistic model of the rod or fluid, that postulates N atoms per unit volume, the average mass-density might be written as a func tion of both position x within the rod or fluid, and the side-length L of the volume L3 c entred on x, over whic h the mass density is c omputed: f(N,x,L). Now the point is that for fixed N, this func tion is liable to be intrac tably sensitive to x and L. But by taking a c ontinuum limit N → ∞, with L → 0 (and atomic masses going to z ero appropriately so that quantities like density do not “blow up”), we c an define a c ontinuous, maybe even differentiable, mass-density func tion ρ(x) as a func tion of position—and then enjoy all the c onvenienc e of the c alc ulus. So muc h by way of showing in general terms how the use of an infinite limit N = ∞ c an be justified—but not mysterious! At this point, the general philosophic al argument of this paper is c omplete! (p. 1082) So for Butterfield most of the disc ussions c onc erning the role, and partic ularly the justific ation, of the use of the thermodynamic limit in the debates about phase transitions have generated a lot of hot air. The justific ation, on his view, for employing suc h limits in our modeling strategies is largely pragmatic —for the sake of c onvenienc e. In addition, there is, as he notes, the further c onc ern that the use of suc h limits be empiric ally adequate—getting the phenomena right to within appropriate error bounds. Muc h of his disc ussion c onc erns showing that the use of suc h limits c an most always be shown to be empiric ally adequate in this sense (Butterfield 2011). Unfortunately, I think that sometimes things are more subtle than the straightforward justific ation admits. In fac t, there are good reasons to think that the use of the thermodynamic limit in the c ontext of the renormaliz ation group (RG) explanation of c ritic al phenomena—one of the c ases he highlights—fails to be justified by his own c riteria. It is a different methodology, one that does not allow for the sort of justific atory story just told. The straightforward story as desc ribed above c annot be told for the RG methodology for the simple reason that that story fails to be empiric ally adequate in those contexts. One c an begin to understand this by making a distinc tion between what might be c alled “ab initio” and “post fac to” c omputational strategies. Butterfield's remarks about the mass density in a rod (say a steel girder) in one sense appear to endorse the ab initio strategy. Consider a model of the rod at the sc ale of atoms where the atoms loc k together on a c rystal lattic e. The limit averaging strategy has us inc rease the siz e of the lattic e until we have, in effec t, a perfec t c rystal of infinite extent. This lets us ignore boundary effec ts as he notes. The limiting average density that we arrive at using this ab initio (atomic only) strategy will ac tually be grossly inc orrec t at higher sc ales. This is bec ause, at higher mic ro (meso) sc ales real iron c ontains many struc tures suc h as disloc ations, grain Page 5 of 23

The Tyranny of Scales boundaries, and other metastabilities that form within its mass and that energetic ally allow loc al portions of the material at these higher scales to settle into stable modes with quite different average densities. See figure 7.1. These average densities will be quite different than the ab initio c alc ulations from the perfec t c rystal. What spec ial features hidden within the quantum c hemistry of iron bond allow those struc tures to form? We really don't know. But until we gain some knowledge of how those struc tures emerge, we will not be able ac c urately to determine c omputationally the bulk features of steel girders in the way desc ribed. Values for Young's modulus and frac ture strength that we may try determine on the basis of this ab initio reasoning will be radic ally at varianc e with the ac tual measured values for real steel. On the other hand, if we possessed a realistic model of steel at all length sc ales, then we c ould c onc eivably define a simple average over a representative volume (at a muc h higher sc ale than the atomic ). But this post fac to c alc ulation would rely upon c omplete data about the system at all sc ales. No limits would be involved whatsoever. Perhaps some super genius may someday in princ iple propose an inc redibly detailed model of iron bonding that would allow the c alc ulation of the mac ro parameters like Young's modulus in a kind of ab initio mode imagined by Butterfield, but suc h a hypothetic al projec t is c ertainly not the aim of the RG tec hniques that are under c onsideration here. Figure 7.1 Microstructures of steel Suc h ab initio c alc ulations provide wrong answers bec ause they c annot “see” the energetic ally allowed loc al struc tural c onfigurations that steel manifests at larger sc ales. On the other hand, if we are investigating materials that (for whatever reason) display nic e sc aling relationships ac ross some range of sc ales (as steel does for sc ales 8– 10 orders of magnitude above the atomic ), then we will be able to employ RG type tec hniques to determine the various universality c lasses (c harac teriz ed by the phenomenologic al parameters—Young's modulus, e.g.) into whic h they must fall. Thus the RG methodology, unlike the ab initio REV averaging strategy, provides a rationale for evading extreme bottom-up c omputations so as to gain an understanding of why steel, for example, only requires a few effec tive parameters to desc ribe its behavior at mac rosc ales. While there surely are c ases in whic h averaging is appropriate, and the straight-forward justific ation may be plausible, there are other c ases, as I have been arguing, in whic h it is not. In order to further eluc idate this point, I will say a bit about what the RG argument aims to do. I will then give a very simple example of why one should, in Page 6 of 23

The Tyranny of Scales many instanc es, expec t the story involving averaging over a representative volume element (REV) to fail. In fac t, the failure of this story is effec tively the motivation behind Wilson's development of the distinc t RG methodology. More generally, if our concern is to understand why continuum models such as the Navier-Cauchy equation are safe and robust, the straightforward justific ation will miss what is most c ruc ial. I have disc ussed the RG in several public ations (Batterman 2002; 2005; 2011). Butterfield (2011) and Butterfield and Bouatta (2011) present c onc ise desc riptions as well. For the purposes here, as noted earlier, I am going to present some of the details with a different emphasis than these other disc ussions have provided. In partic ular, I want to stress the role of the RG as part of a methodology for upsc aling from a statistic al theory to a hydrodynamic /c ontinuum theory. In so doing, I follow a suggestion of David Nelson (2002, pp. 3– 4) who builds on a paper of Ken Wilson (1974). The suggestion is that entire phases of matter (not just c ritic al phenomena) are to be understood as determined by a “fixed point” reflec ting the fac t that “universal physic al laws [are] insensitive to mic rosc opic details” (2002, p. 3). Spec ific ally, the idea is to understand how details of the atomic sc ale physic s get enc oded (typic ally) into a few phenomenologic al parameters that appear in the c ontinuum equations governing the mac rosc opic behavior of the materials. In a sense, these phenomenologic al parameters (like visc osity for a fluid, and Young's modulus for a solid) c harac teriz e the appropriate “fixed point” that defines the c lass of material exhibiting universal behavior despite potentially great differenc es in mic rosc ale physic s. Let us c onsider a ferromagnet modeled as a set of c lassic al spins σi on a lattic e—the Ising model. In this model, neighboring spins tend to align in the same direc tion (either up or down: σi = ±1). Further, we might inc lude the effec t of an external magnetic field B. Then the Hamiltonian for the Ising model is given by with the first sum over nearest neighbor pairs of spins, μ is a magnetic moment. A positive value for the c oupling c onstant J reflec ts the fac t that neighboring spins will tend to be aligned, both up or both down. Figure 7.2 Spontaneous magnetization at Tc For ferromagnets we c an define an order parameter—a func tion of the net magnetiz ation for the system—whose derivative exhibits a disc ontinuity or jump at the so-c alled c ritic al or Curie temperature, Tc. Above Tc, in z ero magnetic field, the spins are not c orrelated due to thermal fluc tuations and so the net magnetiz ation is z ero. As the system c ools down to the Curie temperature, there is singularity in the magnetiz ation (defined as a func tion of the free energy). (See figure 7.2.) The magnetiz ation exhibits power law behavior near that singularity c harac teriz ed by the relation where t is the reduced temperature t = T− Tc . It is a remarkable fac t that physic ally quite distinc t systems—magnets Tc modeled by different Hamiltonians, and even fluids (whose order parameter is the difference between vapor and liquid densities in a c ontainer)—all exhibit the same power law sc aling near their respec tive c ritic al points: The number β is universal and c harac teriz es the phenomenologic al behavior of a wide c lass of systems at and near c ritic ality.14 The RG provides an explanation for this universal behavior; and in partic ular, it allows one theoretic ally to Page 7 of 23

The Tyranny of Scales determine the value for the exponent β. For the 3-dimensional Ising model, that theoretic al value is approximately.33. Experimentally determined values for a wide c lass of fluids and magnets are found in the range.31– .36. So-c alled “mean field” c alc ulations predic t a value of .5 for β (Wilson 1974, p. 120). A major suc c ess of the RG was its ability to c orrec t mean field theory and yield results in c lose agreement with experiment. In a mean field theory, the order parameter M is defined to be the magnetic moment felt at a lattic e site due to the average over all the spins on the lattic e. This averaging ignores any large-sc ale fluc tuations that might (and, in fac t, are) present in systems near their c ritic al points. The RG c orrec ts this by showing how to inc orporate fluc tuations at all length sc ales, from the atomic to the mac ro, that play a role in determining the mac rosc opic behavior (spec ific ally the power law dependenc e—M α \\t\\β) of the systems near c ritic ality. In fac t, near c ritic ality the lattic e system will c ontain “bubbles” (regions of c orrelated spins—all up or all down) of all siz es from the atomic to the system siz e. As Kadanoff notes, “[f]rom this pic ture we c onc lude that c ritic al phenomena are c onnec ted with fluc tuations over all length sc ales between ξ [essentially the system siz e] and the mic rosc opic distanc e between partic les” (Kadanoff 1976, p. 12). So away from c ritic ality, below the c ritic al temperature, say, the lattic e systems will look pretty muc h homogeneous.15 For a system with T ≪ Tc in figure 7.2 we would have relatively large c orrelated regions of spins pointing in the same direc tion. There might be only a few insignific antly small regions where spins are c orrelated in the opposite direc tion. This is what is responsible for there being a positive, nonz ero, value for M at that temperature. Now suppose we were interested in desc ribing a large system like this away from c ritic ality using the c ontinuum limit as understood by Butterfield above. We would c hoose a representative elementary volume of radius L around a point x. The volume is small with respec t to the system siz e ξ, but still large enough to c ontain many spins. Next we would average the quantity M(N,x,L) over that volume and take the limits N → ∞, L → 0 so as to obtain the proper c ontinuum value and so that we would be able to model the ac tually finite c ollec tion of spins using c onvenient c ontinuum mathematic s. But near the c ritic al temperature (near Tc ) the system will look heterogeneous—exhibiting a c omplic ated mixture of two distinc t phases as in figure 7.3. Now we fac e a problem. In fac t, it is the problem that effec tively undermined the mean field approach to critical phenomena. The averaging method employing a representative elementary volume element misses what is most important. For one thing, we will need to know how to weight the different phases as to their import for the mac rosc opic behavior of the system. In other words, were we to perform the REV averaging, all of the physic s of the fluc tuations responsible for the c oexisting bubbles of up spins and bubbles of down spins would be ignored. Here is a simple example to see why this methodology will often fail for heterogeneous systems (Torquato 2002, p. 11). Consider a c omposite material c onsisting of equal volumes of two materials, one of whic h is a good elec tric al c onduc tor and one of whic h is not. A c ouple of possible c onfigurations are shown in figure 7.4. Suppose that the dark, c onnec ted phase is the good c onduc tor. If we were to proc eed using the REV rec ipe, then, bec ause the volume frac tions are the same, we would grossly underestimate the bulk c onduc tivity of the material in the left c onfiguration and grossly underestimate its bulk insulating c apac ities in the right c onfiguration. REV averaging treats only the volume frac tion and c ompletely misses mic rostruc tural details that are relevant to the bulk (mac rosc ale) behavior of the material. In this simple example, the mic rostruc tural feature that is relevant is the topologic al c onnec tedness of the one phase vs. the other—that is, the details about the boundaries between the two phases. Note that the fac t that boundaries play an important role serves to undermine the first “theme” of the Straightforward Justific ation for the use of limits; namely, that taking the limits enable us to remove edge and boundary effec ts. To the c ontrary, these c an and do play very important roles in determining the bulk behavior of the materials. Page 8 of 23

The Tyranny of Scales Figure 7.3 Bubbles within bubbles within bubbles …(after Kadanoff 1976, pp. 11–12) Figure 7.4 50–50 volume mixture One might object that all one needs to do to save the REV methodology would be to properly weight the con- tribution of the different phases to the overall average. But this is not something that one can do a priori or through ab initio calculations appealing to details and properties of the individual atoms at the atomic scale. Even worse, note the partial blobs at the corners marked by the arrows in figure 7.4. How large are the complete blobs of which they are a part? We do not know because the limited scale of the window (size L of the REV) does not allow us to “see” what is happening at large scales. It is entirely possible (and in the case of critical phenomena actually the c ase) that these partial blobs will be part of larger c onnec ted regions only visible at greater sc ale lengths. They may be dreaded invaders from a higher scale.16 If suc h invaders are present, then we have another reason to be wary of limiting REV averaging methods—we will grossly fail to estimate the effec tive c onduc tivity of the material at mac rosc ales. On the other hand, if we have some nic e sc aling data about the behavior of material of the sort exploited by the RG, we may well gain enough of a handle on the material's overall behavior to plac e its c onduc tivity in a firm universality c lass with other materials that sc ale in similar ways. As noted above, in more c omplic ated situations, suc h as the steel girder with whic h we began, mic rostruc tural features inc lude mesosc ale disloc ations, defec ts of various kinds, and martensitic transformations. If we engaged in a purely bottom-up lattic e view about steel, paying attention only to the struc tures for the pure c rystal lattic e, then we would get c ompletely wrong estimates for its total energy, for its average density, and for its elastic properties. The relevant Hamiltonians require terms that simply do not appear at the smallest sc ales.17 The upshot, then, is that the straightforward justific ation for the use of infinite limits will miss exac tly what is important for understanding what is going on for systems at and near c ritic ality. There, they no longer appear homogeneous across a large range of scales. If we are to try to connect (and thereby extract) correct phe- nomenologic al mac rosc opic values for appropriate parameters (e.g., β) we need to c onsider struc tures that exist at sc ales greater than the fundamental/basic /atomic . Again, what does this say about the prospec ts for an overall reduc tionist understanding of the physic s of systems viewed at mac rosc ales? The RG considers such intermediate scales by including in the calculations the effects of fluctuations or equivalent- ly, the fact that bubbles within bubbles of different phases appear near criticality. We need methods that tell us how to homogenize heterogeneous materials. In other words, to extract a continuum phenomenology, we need a methodology that enables us to upscale models of materials that are heterogeneous at small scales to those that are homogeneous at macroscales, as is evidenced by the fact that only a very small number of phenomenological parameters are required to characterize their continuum level behaviors. It appears, then, that the straightforward justific ation of the use of c ontinuum limits needs to be rec onsidered or replac ed in those Page 9 of 23

The Tyranny of Scales c ontexts where the materials of interest exhibit heterogeneous mic rostruc tures. In sec tion 5 I will say a bit more about the nature and generality of this different methodology. In the next sec tion, I present a historic al disc ussion, one aim of whic h is to illustrate that this debate about modeling ac ross sc ales is not, in the least bit, new. Furthermore, the disc ussion should give pause to those who think c ontinuum models are ultimately unnec essary. This is the story of deriving appropriate c ontinuum equations for the behavior of elastic solids and gave rise to a c ontroversy that lasted for most of the nineteenth c entury. 3. Bridging acro ss Scales: A Histo rical Co ntro versy Why are the Navier-Stokes equations named after Navier and Stokes? The answer is not as simple as “they both, independently, arrived at the same equation.” In fac t, there are differenc es between the equation Navier first c ame up with and that derived by Stokes. The differenc es relate to the assumptions that eac h employed in his derivation, but more importantly, these different assumptions ac tually led to different equations. Furthermore, the differenc e between the equations was symptomatic of a c ontroversy that lasted for most of the nineteenth c entury (de Boer 2000, p. 86). While the Navier-Stokes equation desc ribes the behavior of a visc ous fluid, the c ontroversy has its roots in the derivation of equations for the behavior of an elastic solid. I intend to foc us on the latter equations and only at the end make some remarks about the fluid equations. The c ontroversy c onc erned the number of material c onstants that were required to desc ribe the behavior of elastic solids. Ac c ording to Navier's equation, a single c onstant marked a material as isotropic elastic . Ac c ording to Stokes and Green, two c onstants were required. For anisotropic elastic materials (where symmetries c annot be employed) the debate c onc erned whether the number of nec essary c onstants was 15 or 21. This dispute between, respec tively, “rari-c onstanc y” theorists and “multi-c onstanc y” theorists depended upon whether one's approac h to the elastic solid equations started from a hypothesis to the effec t that solids are c omposed of interac ting molec ules or from the hypothesis that solids are c ontinuous. Navier's derivation began from the hypothesis that the deformed state of an elastic body was to be understood in terms of forces acting between individual particles or molecules that make up the body. Under this assumption, he derived equations containing only one material constant ε. Navier's equations for an elastic solid are as follows (de Boer 2000, p. 80): (2) (3) (4) Here ε, Navier's material c onstant, reflec ts the molec ular forc es that are supposed to turn on when external forc es are applied to the body. x, y, z are the c oordinates representing the loc ation of a material point in the body.18 u, v, w are the displac ement c omponents in the direc tions x, y, z; X, Y, Z represent the external ac c elerations (forces) in the directions x, y, z; Δ =∂∂x22   +   ∂2   +   ∂2    is the Laplac e operator; Θ = ∂u   +  ∂v   +  ∂w    is the ∂y2 ∂z2 ∂x ∂y ∂z volume strain; and ρ is the material density. Cauc hy also derived an equation for isotropic elastic materials by starting from a molec ular hypothesis similar to Navier's. However, his equation c ontains the c orrec t number of material c onstants (two). It is instruc tive to write down Cauc hy's equations and to disc uss how, essentially, a mistaken, inc onsistent derivational move on his part yielded a more ac c urate set of equations than Navier. Page 10 of 23

The Tyranny of Scales Cauc hy's equations for an elastic solid are as follows (de Boer 2000, p. 81) (c ompare with equation (1)): (5) (6) (7) R, A are the two material c onstants. Cauc hy noted, explic itly, that when A = 0 his equations agree with Navier's when R = ε19 (de Boer 2000, p. 81). How did Cauc hy arrive at a different equation than Navier, despite starting, essentially, from the same molec ular assumptions about forc es? He did so by assuming that, despite the fac t that he is operating under the molec ular hypothesis, he c an, in his derivation replac e c ertain summations by integrations. In effec t, he ac tually employs a continuum condition c ontradic tory to his fundamental starting assumption.20 George Green, in 1839, published a study that arrived at the c orrec t equations—essentially (5)– (7)—by c ompletely esc hewing the molec ular hypothesis. He treated the entire body as c omposed of “two indefinitely extended media, the surfac e of junc tion when in equilibrium being a plane of infinite extent.”2 1 He also assumed that the material was not c rystalline and, henc e, isotropic . Then using a princ iple of the c onservation of energy/work he derived, using variational principles of Lagrangian mechanics, his multi-constant equation. Finally, following the disc ussion of Todhunter and Pearson (1960), we note that Stokes's work supported the multi- c onstanc y theory in that he was able to generaliz e his equations for the behavior of viscous fluids to the c ase of elastic solids by making no distinc tion between a visc ous fluid and a solid undergoing permanent—plastic — deformation. “He in fac t draws no line between a plastic solid and a visc ous fluid. The formulae for the equilibrium of an isotropic plastic solid would thus be bi-c onstant” (Todhunter and Pearson 1960, p. 500). This unific ation of c ontinuum equations lends further support to the multi-c onstanc y theory. The historic al debate represents just the tip of the ic eberg of the c omplexity surrounding both theoretic al and experimental work on the behavior of the supposedly simpler, isotropic , c ases of elastic solids. Nevertheless, the multi-c onstanc y theory wins the day for appropriate c lasses of struc tures. And, derivations that start from atomic assumptions fail to arrive at the c orrec t theory. It seems that here may very well be a c ase where a c ontinuum point of view is ac tually superior: bottom-up derivation from atomistic hypotheses about the nature of elastic solid bodies fails to yield c orrec t equations governing the mac rosc opic behavior of those bodies. There are good reasons, already well understood by Green and Stokes, for esc hewing suc h reduc tionist strategies. This c ontroversy is important for the c urrent projec t for the following reason. Green and Stokes were moved by the apparent sc aling or homogeneity observed in elastic solids and fluids. That is, as one z ooms in with reasonable powerful mic rosc opes one sees the steel to be the same at different magnific ations; likewise for the fluid. Green and Stokes then extrapolated this sc ale invarianc e to hold at even larger magnific ations—at even smaller sc ales. We now know (and likely they suspec ted) that this extrapolation is not valid beyond c ertain sc ale lengths—the atomistic nature of the materials will begin to show itself. Nevertheless, the c ontinuum modeling was dramatic ally suc c essful in that it predic ted the c orrec t number and the c orrec t c harac ter of the phenomenologic al c onstants. De Boer reflec ts on the reasons for why this c ontroversy lasted so long and was so heated: Why was so muc h time spent on molec ular theory c onsiderations, in partic ular, by the most outstanding mec hanic s spec ialists and mathematic ians of the epoc h? One of the reasons must have been the temptation of gaining the c onstitutive relation for isotropic and anisotropic elastic c ontinua direc tly from pure mathematic al studies and simple mec hanic al princ iples;2 2 It was only later realiz ed that Hooke's generalized law is an assumption, and that the foundation of the linear relation had to be supported by experiments. (2000, pp.86–87) Page 11 of 23

The Tyranny of Scales The upshot of this disc ussion is reflec ted in de Boer's emphasis that the c onstitutive equations or spec ial forc e laws (Hooke's law) are dependent, for their very form, on experimental results. So a simple dismissal of c ontinuum theories as “in princ iple” eliminable, as reduc ible, and merely pragmatic ally justified, is mistaken. Of c ourse, the phenomenologic al parameters, like Young's modulus (related to Navier's ε), must enc ode details about the ac tual atomistic struc ture of elastic solids. But it is naive, indeed, to think that one c an, in any straightforward way derive or deduc e from atomic fac ts what are the phenomenologic al parameters required for c ontinuum model of a given material. It is probably even more naive to think that one will be able to derive or deduc e from those atomic fac ts what are the actual values for those parameters for a given material. This historic al disc ussion and the intense nineteenth-c entury debate between the rari- and multi-c onstanc y theorists apparently supports the view that there is some kind of fundamental inc ompatibility between small sc ale and c ontinuum modeling prac tic es. That is, it lends support to the stark c hoic e one must apparently make between bottom-up and top-down modeling suggested by the tyranny of sc ales. A modern, more nuanc ed, and better informed view c hallenges this c onsequenc e of the tyranny of sc ales and will be disc ussed in sec tion 5. However, suc h a view will not, in my opinion, bring muc h c omfort to those who believe the use of c ontinuum models or idealiz ations is only pragmatic ally justified. A modern statement supporting this point of view can be found in (Phillips 2001): [M]any material properties depend upon more than just the identity of the partic ular atomic c onstituents that make up the material.…[M]ic rostruc tural features suc h as point defec ts, disloc ations, and grain boundaries c an eac h alter the measured mac rosc opic “properties” of a material. (pp. 5– 8) It is important to reiterate that, c ontrary to typic al philosophic al usage, “mic rostruc tural features” here is not synonymous with “atomic features”! Defec ts, disloc ations, etc . exist at higher sc ales. In the next sec tion I will further develop the stark dic hotomy between bottom-up modeling and top-down modeling as a general philosophic al problem arising between different rec ipes for applying mathematic s to systems exhibiting different properties ac ross a wide range of sc ales. 4. Euler's Recipes: Discrete and Co ntinuum 4.1 Discrete Applied mathematic al modeling begins with an attempt to write down an equation governing the system exhibiting the phenomenon of interest. In many situations, this aim is ac c omplished by starting with a general dynamic al princ iple suc h as Newton's sec ond law: F = ma. Unfortunately, this general princ iple tells us absolutely nothing about the material or body being investigated and, by itself, provides no model of the behavior of the system. Further data are required and these are supplied by so-c alled “spec ial forc e laws” or “c onstitutive equations.” A rec ipe, due to Leonhard Euler, for finding an appropriate model for a system of partic les proc eeds as follows (Wilson 1974): 1. Given the c lass of material (point partic les, say), determine the kinds of spec ial forc es that ac t between mimj them. Massive particles obey the constitutive gravitational force: FG   =  G r2ij . Charged partic les additionally will obey the Coulomb force law: FE  =  ke qi qj . r2ij 2. Choose Cartesian c oordinates along whic h one dec omposes the spec ial forc es. 3. Sum the forc es ac ting on eac h partic le along the appropriate axis. 4. Set the sum for each particle i equal to mi d2x to yield the total forc e on the partic le. dt2 This yields a differential equation that we then employ (= try to solve) to further understand the behavior of our point partic le system. Only rarely (for very few partic les or for spec ial symmetries) will this equation suc c umb to analytic al evaluation. In many instanc es, further simplific ation employing mathematic al strategies of variable reduc tion, averaging, etc . enable us to gain information about the behavior of interest. Page 12 of 23

The Tyranny of Scales 4.2 Continuum As we saw in sec tion 3, Cauc hy had a role in the derivation of equations for elastic solids. We note again that he was luc ky to have arrived at the c orrec t equations, given that he started with a bottom-up derivation in mind. Nevertheless, Cauc hy was an important figure in the development of c ontinuum mec hanic s: it turns out that at mac rosc ales, forc es within a c ontinuum c an be represented by a single sec ond-rank tensor, despite all of the details that appear at the atomic level. This is known as the Cauc hy stress tensor (Philips 2001, p. 39). The analog of Newton's sec ond law, for c ontinua is the princ iple of balanc e of linear momentum. It is a statement that “the time rate of c hange of the linear momentum is equal to the net forc e ac ting on [a] region Ω”: 2 3 (8) Here ∂Ω is the boundary of the region Ω, t is the trac tion vec tor representing surfac e forc es (squeez ings, for instanc e), and f represents the body forc es suc h as gravity. The left-hand side of equation (8) is the time rate of c hange of linear momentum. The material time derivative, D/Dt, is required bec ause in addition to explic it time dependenc e of the field, we need to c onsider the fac t that the material itself c an move into a region where the field is different. As with Euler's disc rete rec ipe, equation (8) requires input from c onstitutive equations to apply to any real system. Whether our interest is in the desc ription of injec ting polymers into molds, the evolution of Jupiter's red spot, the development of texture in a c rystal, or the formation of vortic es in wakes, we must supplement the governing equations of c ontinuum mec hanic s with some c onstitutive desc ription. (Phillips 2001, p. 51) For the c ase of a steel girder, c onsidered in the regime for use in c onstruc ting bridges or buildings we need the input that it obeys something like Hooke's law—that its stress is linearly related to its strain. In modern terminology, we need to provide data about the Cauc hy stress tensor. For isotropic linear elastic solids, symmetry c onsiderations c ome into play and we end up with equation (1)—the Navier-Cauc hy equation that c harac teriz es the equilibrium states of suc h solids: The “Lamé” parameters (related to Young's modulus) express the empiric al details about the material response to stress and strain. 4.3 Controversy A question of pressing c onc ern is why the c ontinuum rec ipe should work at all. We have seen in the historic al example that it does, and in fac t, we have seen that were we simply to employ the disc rete (point partic le) rec ipe, we would not arrive at the c orrec t results. In asking why the c ontinuum rec ipe works on the mac rosc ale, we are asking about the relationship between the dynamic al models that trac k the behavior of individual atoms and molec ules and equations like those of Navier, Stokes, Cauc hy, and Green that are applic able at the sc ale of millimeters. Put slightly differently, we would like an ac c ount of why it is safe to use the Cauc hy momentum equation in the sense that it yields c orrec t equations with the appropriate (few) parameters for broadly different c lasses of systems—from elastic solids to visc ous fluids. From the point of view of Euler's c ontinuum rec ipe, one derives the equations for elastic solids, or the Navier- Stokes equations, independently of any views about the molec ular or atomic makeup of the medium. (In the nineteenth c entury the question of whether matter was atomistic had yet to be settled.) To ask for an ac c ount of why it is safe to use the c ontinuum rec ipe for c onstruc ting mac rosc ale models is to ask for an ac c ount of the robustness of that methodology. The key physic al fac t is that the bulk behaviors of solids and fluids are almost c ompletely insensitive to the ac tual nature of the physic s at the smallest sc ale. The “almost” here is c ruc ial. The atomic details that we do not know (and, henc e, do not explic itly refer to) when we employ c ontinuum rec ipe are enc oded in the small number of phenomenologic al parameters that appear in the resulting equations—Young's modulus, the visc osity, etc . So the answer to the safety question will involve showing how to determine the “fixed points” c harac teriz ing broad c lasses of mac rosc opic materials—fixed points that are Page 13 of 23

The Tyranny of Scales c harac teriz ed by those phenomenologic al parameters. Rec all the statement by Nelson c ited above in sec tion 2.1. In the c ontext of c ritic al phenomena and the determination of the c ritic al exponent β, this upsc aling or c onnec tion between the Euler's disc rete and c ontinuum rec ipes is ac c omplished by the renormaliz ation group. In that c ontext, the idea of a c ritic al point and related singularities plays an important role. But Nelson's suggestion is that upsc aling of this sort should be possible even for c lasses of systems without c ritic al points. For example, we would like to understand why Young's modulus is the appropriate phenomenological parameter for classifying solids as linear elastic, despite rather severe differences in the atomic structure of members of that class. Finding answers to questions of this latter type is the purview of so-c alled “homogeniz ation” theory, of whic h one c an profitably think the RG to be a spec ial c ase. In the next sec tion, I will spend a bit more time on the RG explanation of the universality of c ritic al behavior, filling in some gaps in the disc ussion in sec tion 2.1. And, I will try to say something about general methodology of upsc aling through the use of homogeniz ation limits. 5. A Mo dern Reso lutio n To begin, c onsider a problem for a c orporation that owns a lot of c asinos. The CEO of the c orporation needs to report to the board of trustees (or whomever) on the expec ted profits for the c orporation. How is she to do it? Assuming (c ontrary to fac t) that c asino gaming is fair, she would present to the board a Gaussian or normal probability distribution showing the probabilities of various profits and losses, with standard deviations that would allow for statistic al predic tions as to expec ted profits and losses. She may also seek information as to how to manipulate the mean and varianc e so as to guarantee the likelihood of greater profits for less risk, etc . The Gaussian distribution is a func tion c harac teriz ed by two parameters—the mean μ and the varianc e σ2 . Where will the CEO get the values for the mean and varianc e? Most likely by empirically investigating the ac tual means and varianc es displayed over the past year by the various c asinos in the c orporation. Consider figure 7.5. Should the CEO look to the individual gambles or even to c ollec tions of individual gambles of different types in partic ular c asinos? A bottom-up reduc tionist would say that all of the details about the c orporation as a whole are to be found by c onsidering these details. But, in fac t, (i) she should not foc us too muc h on spatiotemporal loc al features of a single c asino: suppose someone hits the jac kpot on a slot mac hine. Likely, many people will run to that part of the c asino, diminishing profits from the roulette wheels and blac kjac k tables, and skewing the predic tion of the ac tual mean and varianc e she is after. Nor (ii) would it be wise to foc us too muc h on groups of c asinos say in a partic ular geographic area (suc h as Las Vegas) over c asinos owned in another area (suc h as Atlantic City). After all, different tax struc tures in these different states and munic ipalities play an important role as well. Suc h intermediate struc tures and environmental c onsiderations are c ruc ial—c onsider again the bubbles within bubbles struc tures that c harac teriz e the heterogeneities at lower sc ales in the c ase of the universality of c ritic al phenomena. The CEO needs to look at large groups of c ollec tions of c asinos where there is evident sc aling and self-similarity. Apparent sc aling behavior and self-similarity at large sc ales is an indic ation of homogeneity. Thus, as with our steel girder, empiric al data (at large sc ales) is required to determine the values of the relevant parameters. Page 14 of 23