We have two aims. The main one is to expound the idea of renormalization in quantum field theory, with no technical prerequisites. Our motivation is that renormalization is undoubtedly one of the great ideas—and great successes--of twentieth-century physics. Also it has strongly influenced in diverse ways, how physicists conceive of physical theories. So it is of considerable philosophical interest. Second, we will briefly relate renormalization to Ernest Nagel's account of inter-theoretic relations, especially reduction. One theme will be (...) a contrast between two approaches to renormalization. The old approach, which prevailed from ca. 1945 to 1970, treated renormalizability as a necessary condition for being an acceptable quantum field theory. On this approach, it is a piece of great good fortune that high energy physicists can formulate renormalizable quantum field theories that are so empirically successful. But the new approach to renormalization explains why the phenomena we see, at the energies we can access in our particle accelerators, are described by a renormalizable quantum field theory. For whatever non-renormalizable interactions may occur at yet higher energies, they are insignificant at accessible energies. Thus the new approach explains why our best fundamental theories have a feature, viz. renormalizability, which the old approach treated as a selection principle for theories. That is worth saying since philosophers tend to think of scientific explanation as only explaining an individual event, or perhaps a single law, or at most deducing one theory as a special case of another. Here we see a framework in which there is a space of theories. And this framework is powerful enough to deduce that what seemed “manna from heaven” is to be expected: the good fortune is generic. We also maintain that universality, a concept stressed in renormalization theory, is essentially the familiar philosophical idea of multiple realizability; and that it causes no problems for reductions of a Nagelian kind. (shrink)

Renormalization group explanations account for the astonishing phenomenon that microscopically very different physical systems display the same macro-behavior when undergoing phase-transitions. Among philosophers, this explanandum phenomenon is often described as the occurrence of a particular kind of multiply realized macro-behavior. In several recent publications, Robert Batterman denies that RG explanations account for this explanandum phenomenon by following the commonality strategy, i.e. by identifying properties that microscopically very different physical systems have in common. Arguing against Batterman’s claim, I defend the (...) view that RG explanations are in accord with the commonality strategy. (shrink)

One realist response to the pessimistic meta-induction distinguishes idle theoretical wheels from aspects of successful theories we can expect to persist and espouses realism about the latter. Implementing the response requires a strategy for identifying the distinguished aspects. The strategy I will call renormalization group realism has the virtue of directly engaging the gears of our best current physics—perturbative quantum field theories. I argue that the strategy, rather than disarming the skeptical possibilities evinced by the pessimistic meta-induction, forces them (...) to retreat a level. I also suggest that those skeptical possibilities continue to carry force. (shrink)

Providing a precise statement of their position has long been a central challenge facing the scientific realist. This paper draws some morals about how realism ought to be formulated from the renormalization group framework in high energy physics.

This is an introduction to renormalization group methods in quantum field theory aimed at philosophers of science. review path integral methods, the relationship between early renormalization theory and renormalization group methods, and conceptual shifts in thinking about quantum field theory spurred by the development of renormalization group methods.

In this paper, I propose a general framework for understanding renormalization by drawing on the distinction between effective and continuum Quantum Field Theories (QFTs), and offer a comprehensive account of perturbative renormalization on this basis. My central claim is that the effective approach to renormalization provides a more physically perspicuous, conceptually coherent and widely applicable framework to construct perturbative QFTs than the continuum approach. I also show how a careful comparison between the two approaches: (i) helps to (...) dispel the mystery surrounding the success of the renormalization procedure; (ii) clarifies the various notions of renormalizability; and (iii) gives reasons to temper Butterfield and Bouatta's claim that some continuum QFTs are ripe for metaphysical inquiry (Butterfield & Bouatta, 2014). (shrink)

Quantum anomalies in the inverse square potential are well known and widely investigated. Most prominent is the unbounded increase in oscillations of the particle’s state as it approaches the origin when the attractive coupling parameter is greater than the critical value of 1/4. Due to this unphysical divergence in oscillations, we are proposing that the interaction gets screened at short distances making the coupling parameter acquire an effective (renormalized) value that falls within the weak range 0–1/4. This prevents the oscillations (...) form growing without limit giving a lower bound to the energy spectrum and forcing the Hamiltonian of the system to be self-adjoint. Technically, this translates into a regularization scheme whereby the inverse square potential is replaced near the origin by another that has the same singularity but with a weak coupling strength. Here, we take the Eckart as the regularizing potential and obtain the corresponding solutions (discrete bound states and continuum scattering states). (shrink)

In quantum field theory divergent expressions are "discarded", leaving finite expressions which provide the best predictions anywhere in science. In fact, this "renormalization procedure" involves no mystery or illegitimate operations. This paper explains, in terms accessible to non-experts, how the procedure really works and explores some different ways in which physicists have suggested that one understand it.

This paper discusses the alleged reduction of Thermodynamics to Statistical Mechanics. It includes an historical discussion of J. Willard Gibbs' famous caution concerning the connections between thermodynamic properties and statistical mechanical properties---his so-called ``Thermodynamic Analogies.'' The reasons for Gibbs' caution are reconsidered in light of relatively recent work in statistical physics on the existence of the thermodynamic limit and the explanation of critical behavior using the renormalization group apparatus. A probabilistic understanding of the renormalization group arguments allows for (...) a kind of unification of Gibbs' approach with contemporary understanding of the reduction problem. (shrink)

Since 1980 effective field theories (EFT's) have been the focus of much research by quantum field theorists but their philosophical implications have gone mostly unnoticed. Some authors claim EFT's are approximations to some fundamental theory. Others claim EFT's are ends in themselves, not approximations to some fundamental theory, and that we can use them to bypass the problem of renormalization. In the present work I argue that the EFT programme can bypass the problem if ontological commitments only come from (...) theoretical predictions. Since the history of QFT suggests some form of entity realism, the EFT programme does not allow us to bypass the problem of renormalization. (shrink)

The fact that the goals and methods of science, as well as its empirical conclusions, are subject to change, is shown to allow at once for: (a) the objectivity of warrant for knowledge claims; (b) the absence of a priori standards from epistemology; (c) the normative character of epistemology; and (d) the rationality of axiological innovation. In particular, Laudan's attempt to make axiological constraints undercut epistemic realism is confuted.

It is commonly claimed that the universality of critical phenomena is explained through particular applications of the renormalization group. This article has three aims: to clarify the structure of the explanation of universality, to discuss the physics of such RG explanations, and to examine the extent to which universality is thus explained. The derivation of critical exponents proceeds via a real-space or a field-theoretic approach to the RG. Building on work by Mainwood, this article argues that these approaches ought (...) to be distinguished: while the field-theoretic approach explains universality, the real-space approach fails to provide an adequate explanation. (shrink)

We revisit the construction of the gravitational functional renormalization group equation tailored to the Arnowitt–Deser–Misner formulation emphasizing its connection to the covariant formulation. The results obtained from projecting the renormalization group flow onto the Einstein–Hilbert action are reviewed in detail and we provide a novel example illustrating how the formalism may be connected to the causal dynamical triangulations approach to quantum gravity.

Analogies between classical statistical mechanics and quantum field theory played a pivotal role in the development of renormalization group methods for application in the two theories. This paper focuses on the analogies that informed the application of RG methods in QFT by Kenneth Wilson and collaborators in the early 1970's. The central task that is accomplished is the identification and analysis of the analogical mappings employed. The conclusion is that the analogies in this case study are formal analogies, and (...) not physical analogies. That is, the analogical mappings relate elements of the models that play formally analogous roles and that have substantially different physical interpretations. Unlike other cases of the use of analogies in physics, the analogical mappings do not preserve causal structure. The conclusion that the analogies in this case are purely formal carries important implications for the interpretation of QFT, and poses challenges for philosophical accounts of analogical reasoning and arguments in defence of scientific realism. Analysis of the interpretation of the cutoffs is presented as an illustrative example of how physical disanalogies block the exportation of physical interpretations from from statistical mechanics to QFT. A final implication is that the application of RG methods in QFT supports non-causal explanations, but in a different manner than in statistical mechanics. (shrink)

In previous work, I described several examples combining reduction and emergence: where reduction is understood a la Ernest Nagel, and emergence is understood as behaviour that is novel. Here, my aim is again to reconcile reduction and emergence, for a case which is apparently more problematic than those I treated before: renormalization. My main point is that renormalizability being a generic feature at accessible energies gives us a conceptually unified family of Nagelian reductions. That is worth saying since philosophers (...) tend to think of scientific explanation as only explaining an individual event, or perhaps a single law, or at most deducing one theory as a special case of another. Here we see a framework in which there is a space of theories endowed with enough structure that it provides a family of reductions. (shrink)

Despite the close connection between the central limit theorem and renormalization group (RG) methods, the latter should be considered fundamentally distinct from the kind of probabilistic framework associated with statistical mechanics, especially the notion of averaging. The mathematics of RG is grounded in dynamical systems theory rather than probability, which raises important issues with respect to the way RG generates explanations of physical phenomena. I explore these differences and show why RG methods should be considered not just calculational tools (...) but the basis for a physical understanding of complex systems in terms of structural properties and relations. (shrink)

A recently popular formulation of the Higgs naturalness principle prohibits delicate cancellations between running renormalized Higgs mass parameters and EFT matching corrections, by contrast with the principle’s original formulation, which prohibits delicate cancellations between the bare Higgs mass parameter and its quantum corrections. While the need for this latter cancellation is sometimes viewed as unproblematic since bare parameters are thought by some to be divergent and unphysical, renormalized parameters are finite and measurable, and the need for delicate cancellations between the (...) renormalized Higgs mass parameter and EFT matching corrections is therefore considered by some to constitute a more salient formulation of the Higgs naturalness problem. Here, we argue that to the contrary, the need for fine tuning of the renormalized Higgs mass parameter is an eliminable, unphysical artifact of renormalization scheme, and that this severely weakens the grounds for regarding it as a problematic instance of fine tuning. In doing so, we highlight what we take to be a number of important conceptual lessons about the physical interpretation of model parameters in QFT. (shrink)

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG explanations: linearization* property; (...) I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable. (shrink)

Renormalization group (RG) methods are an established strategy to explain how it is possible that microscopically different systems exhibit virtually the same macro behavior when undergoing phase-transitions. I argue – in agreement with Robert Batterman – that RG explanations are non-causal explanations. However, Batterman misidentifies the reason why RG explanations are non-causal: it is not the case that an explanation is non- causal if it ignores causal details. I propose an alternative argument, according to which RG explanations are non-causal (...) explanations because their explanatory power is due to the application mathematical operations, which do not serve the purpose of representing causal relations. (shrink)

The renormalization group is used to analyze the behavior of certain gravitationally significant renormalized coupling constants under a scaling of the spacetime curvature. After discussing a simple example, the results are summarized for a class of grand unified theories.

The renormalization group has been characterized as merely a coarse-graining procedure that does not illuminate the microscopic content of quantum field theory, but merely gets us from that content, as given by axiomatic QFT, to macroscopic predictions. I argue that in the constructive field theory tradition, RG techniques do illuminate the microscopic dynamics of a QFT, which are not automatically given by axiomatic QFT. RG techniques in constructive field theory are also rigorous, so one cannot object to their foundational (...) import on grounds of lack of rigor. (shrink)

The “sloppy models” program originated in systems biology, but has seen applications across a range of fields. Sloppy models are dependent on a large number of parameters, but highly insensitive to the vast majority of parameter combinations. Sloppy models proponents claim that the program may explain the success of science. I argue that the sloppy models program can at best provide a very partial explanation. Drawing a parallel with renormalization group realism, I argue that it would only give us (...) grounds for a minimal kind of scientific realism. Nonetheless, the program can offer certain epistemic virtues. (shrink)

Recently, many have argued that there are certain kinds of abstract mathematical explanations that are noncausal. In particular, the irrelevancy approach suggests that abstracting away irrelevant causal details can leave us with a noncausal explanation. In this paper, I argue that the common example of Renormalization Group explanations of universality used to motivate the irrelevancy approach deserves more critical attention. I argue that the reasons given by those who hold up RG as noncausal do not stand up to critical (...) scrutiny. As a result, the irrelevancy approach and the line between casual and noncausal explanation deserves more scrutiny. (shrink)

In this paper we pay attention to the inconsistency in the derivation of the symmetric electromagnetic energy–momentum tensor for a system of charged particles from its canonical form, when the homogeneous Maxwell’s equations are applied to the symmetrizing gauge transformation, while the non-homogeneous Maxwell’s equations are used to obtain the motional equation. Applying the appropriate non-homogeneous Maxwell’s equations to both operations, we obtained an additional symmetric term in the tensor, named as “compensating term”. Analyzing the structure of this “compensating term”, (...) we suggested a method of “gauge renormalization”, which allows transforming the divergent terms of classical electrodynamics (infinite self-force, self-energy and self-momentum) to converging integrals. The motional equation obtained for a non-radiating charged particle does not contain its self-force, and the mass parameter includes the sum of mechanical and electromagnetic masses. The motional equation for a radiating particle also contains the sum of mechanical and electromagnetic masses, and does not yield any “runaway solutions”. It has been shown that the energy flux in a free electromagnetic field is guided by the Poynting vector, whereas the energy flux in a bound EM field is described by the generalized Umov’s vector, defined in the paper. The problem of electromagnetic momentum is also examined. (shrink)

Kenneth Wilson won the Nobel Prize in Physics in 1982 for applying renormalization group, which he learnt from quantum field theory (QFT), to problems in statistical physics—the induced magnetization of materials (ferromagnetism) and the evaporation and condensation of fluids (phase transitions). See Wilson (1983). The renormalization group got its name from its early applications in QFT. There, it appeared to be a rather ad hoc method of subtracting away unwanted infinities. The further allegation was that the procedure is (...) so horrendously complicated that one cannot see the forest for the trees. The second allegation is justified in the applications that made it famous. But it is not true of the following example, which appears in Chowdhury and Stauffer (2000, 486-488). (shrink)

The renormalization group has been characterized as merely a coarse-graining procedure that does not illuminate the microscopic content of quantum field theory but merely gets us from that content, as given by axiomatic QFT, to macroscopic predictions. I argue that in the constructive field theory tradition, RG techniques do illuminate the microscopic dynamics of a QFT, which are not automatically given by axiomatic QFT. RG techniques in constructive field theory are also rigorous, so one cannot object to their foundational (...) import on grounds of lack of rigor. (shrink)

This paper investigates the consequences for our understanding of physical theories as a result of the development of the renormalization group. Kadanoff's assessment of these consequences is discussed. What he called the ``extended singularity theorem'' poses serious difficulties for philosophical interpretation of theories. Several responses are discussed. The resolution demands a philosophical rethinking of the role of mathematics in physical theorizing.

This paper examines the justifications for using infinite systems to 'recover' thermodynamic properties, such as phase transitions (PT), critical phenomena (CP), and irreversibility, from the micro-structure of matter in bulk. Section 2 is a summary of such rigorous methods as in taking the thermodynamic limit (TL) to recover PT and in using renormalization (semi-) group approach (RG) to explain the universality of critical exponents. Section 3 examines various possible justifications for taking TL on physically finite systems. Section 4 discusses (...) the legitimacy of applying TL to the problem of irreversibility and assesses the repercussions for its legitimacy on its home turf. (shrink)

This paper examines the justifications for using infinite systems to ‘recover’ thermodynamic properties, such as phase transitions, critical phenomena, and irreversibility, from the micro-structure of matter in bulk. Section 2 is a summary of such rigorous methods as in taking the thermodynamic limit to recover PT and in using renormalization group approach to explain the universality of critical exponents. Section 3 examines various possible justifications for taking TL on physically finite systems. Section 4 discusses the legitimacy of applying TL (...) to the problem of irreversibility and assesses the repercussions for its legitimacy on its home turf. (shrink)

In their Every Thing Must Go, Ladyman and Ross defend a novel version of Neo- Russellian metaphysics of causation, which falls into three claims: (1) there are no fundamental physical causal facts (orthodox Russellian claim), (2) there are higher-level causal facts of the special sciences, and (3) higher-level causal facts are explanatorily emergent. While accepting claims (1) and (2), I attack claim (3). Ladyman and Ross argue that higher-level causal facts are explanatorily emergent, because (a) certain aspects of these higher-level (...) facts (their universality) can be captured by renormalization group (RG) explanations, and (b) RG explanations are not reductive explanations. However, I argue that RG explanation should be understood as reductive explanations. This result undermines Ladyman and Ross’s RG-based argument for the explanatory emergence of higher-level causal facts. (shrink)

Discussions of the foundations of Classical Equilibrium Statistical Mechanics (SM) typically focus on the problem of justifying the use of a certain probability measure (the microcanonical measure) to compute average values of certain functions. One would like to be able to explain why the equilibrium behavior of a wide variety of distinct systems (different sorts of molecules interacting with different potentials) can be described by the same averaging procedure. A standard approach is to appeal to ergodic theory to justify this (...) choice of measure. A different approach, eschewing ergodicity, was initiated by A. I. Khinchin. Both explanatory programs have been subjected to severe criticisms. This paper argues that the Khinchin type program deserves further attention in light of relatively recent results in understanding the physics of universal behavior. (shrink)

Dans cet article nous analysons des écrits réflexifs de fin de formation de mentors, afin d’identifier les marques de transformation de l’identité professionnelle et de l’agir professionnel. Nous proposons une analyse thématique et langagière des textes en tant que porteurs d’un rapport au monde professionnel en cours d’élaboration. Nous mettons en évidence que ces textes contiennent des traces de renormalisations issues des apprentissages conceptuels et formels autant que des apprentissages expérientiels réalisés durant la formation. Cette analyse interroge au bout du (...) compte la formation elle-même avec ses multiples attendus. (shrink)

We revisit the implications of Haag’s theorem in the light of the renormalization group. There is still some lack of discussion in the literature about the possible impact of the theorem on the standard (as opposite of axiomatic) quantum field theory, and we try to shed light in this direction. Our discussion then deals with the interplay between Haag’s theorem and renormalization. While we clarify how perturbative renormalization (for the sub-class of interactions that are renormalizable) marginalizes its (...) impact when the coupling is formally small, we argue that a non-perturbative and non-ambiguous renormalization cannot be built if there is any reference to the interaction picture with free fields. In other words, Haag’s theorem should be regarded as a no-go theorem for the existence of a non-ambiguous analytic continuation from perturbative to non-perturbative QFT. (shrink)

We explicate recent results that shed light on the obscure and troubling problem of renormalization in Quantum Field Theory (QFT). We review how divergent predictions arise in perturbative QFT, and how they are renormalized into finite quantities. Commentators have worried that there is no foundation for renormalization, and hence that QFTs are not logically coherent. We dispute this by describing the physics behind liquid diffusion, in which exactly analogous divergences are found and renormalized. But now we are looking (...) at a problem that is physically and formally well-defined, proving that the problems of renormalization, by themselves, cannot refute QFT. (shrink)