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This chapter begins with a philosophically-oriented outline of quantum field theory that looks at the different ways in which a quantum field can be constructed. The difficulties in articulating a particle ontology are discussed, and it is argued that any putative interpretation of the theory must accommodate the observed particle ‘grin’. A range of metaphysical options are explored, including trope theory and Davudson’s theory of events. The foundations of Fock space are presented in detail from the model-theoretic perspective, and it is suggested that quasi-set theory provides the appropriate formal foundation. Thus, the quanta of quantum field theory can also be accommodated in terms of the overall framework of this work.

This chapter begins the project of interpreting quantum gauge field theories. It rejects an interesting recent interpretation and explains why it is so difficult to arrive at a better one. The difficulty stems from the problems of interpreting any quantum field theory. Even the non-relativistic quantum mechanics of particles is a theory whose interpretation remains at best controversial, and at worst simply lacking. Interpretations of quantum field theory face the additional hurdle that it is not clear what the theory is about: neither a field nor a particle ontology is readily squared with the mathematics of the theory. The chapter explores the status of loop representations in several approaches toward the interpretation of quantum field theory.

This chapter reviews various interpretations of structural realism and then adopts a definition that allows both relations between things that are already individuated (‘relations between things’) and relations that individuate previously un-individuated entities (‘things between relations’). Since both spacetime points in general relativity and elementary particles in quantum theory fall into the latter category, the chapter proposes a principle of maximal permutability as a criterion for the fundamental entities of any future theory of ‘quantum gravity’; i.e., a theory yielding both general relativity and quantum field theory in appropriate limits. It reviews a number of current candidates for such a theory. The chapter ends by suggesting a new approach to the question of which spacetime structures should be quantized.

Technically, it is very difficult to construct a tenable quantum theory of gravity. The major concern of a philosopher, however, is with having a consistent strategy to guide technical moves. If we look at quantum gravity this way, we immediately face a question of theoretical constraints imposed by general relativity and quantum field theory, which are the two most successful theories in fundamental physics: one deals with gravity in a classical field-theoretical framework, the other deals with quantum fields. Since quantum gravity means a quantum theory of the gravitational field, what should we do so that we can secure a chance of success if we cannot meet these constraints in their original forms, and thus have to go beyond the two theories? It was previously argued that the trouble is that it is impossible to meet the constraints imposed by these two theories in a single theory without radically revising each of them. Briefly, quantum field theory requires a Minkowskian spacetime as a fixed background, which is rejected by general relativity; and the latter requires a continuous manifold that cannot stand violent quantum fluctuations. What is the way out? Before any attempt is made to address this crucial issue, we have to take a closer look at the notions of physical reality offered by general relativity and quantum field theory respectively. This question is interesting in its own right, in addition to its relevance for our construction of a tenable quantum theory of gravity, because confusions in this regard have to be cleared before we can have a correct understanding of general relativity and quantum field theory. This chapter argues that structural realism is a framework in which the aforementioned confusions can be cleared and constraints met satisfactorily, and thus a strategy in guiding technical moves for constructing a consistent quantum theory of gravity can be suggested.

Julian Schwinger had now scaled the peak of quantum electrodynamics (QED), not once, but three times, the last time by inventing a new approach to any quantum-mechanical system, the quantum dynamical principle. Now the task of the field theorist, as was already apparent in the 1930s, was to build upon this success of QED and apply the powerful machinery invented to understand the strong and weak nuclear interactions. This chapter describes the story of Schwinger's work in the central period between the quantum field theory revolutions of the late 1940s and the early 1970s, roughly during the period 1957 through 1965. Schwinger's work on the phenomenological field theory, dispersion relations, spin and the TCP theorem, Euclidean field theory, gauge invariance and mass, quantum gravity, and magnetic charge are examined.

Barely six months after the Shelter Island Conference, which reawakened his interest in quantum electrodynamics (QED), and just three months after returning to Harvard University from his extended honeymoon to the West Coast, Julian Schwinger published a one-page note in the Physical Review entitled ‘On quantum electrodynamics and the magnetic moment of the electron’. A preliminary account of this work was presented by Schwinger at the 10th Washington Conference on Theoretical Physics in November 1947, which attracted the interest of J. Robert Oppenheimer and Richard Feynman. This chapter looks at Schwinger's method of canonical transformations, his covariant approach to QED, Sin-itiro Tomonaga's covariant formulation of quantum field theory, Feynman's theory of positrons and his space-time approach to quantum electrodynamics, Freeman Dyson's research on the radiation theories of Schwinger, Tomonaga, and Feynman, and the synergism between the works of Feynman and Schwinger with respect to QED.

This introductory chapter presents the general motivations for constructing a quantum theory of gravity. The main argument is the conceptual incompleteness of present theoretical physics. The chapter presents the relevant length and energy scales, that is, the Planck scales and its relation to astrophysical scales, and addresses the experimental status of the relation between quantum theory and gravity. It then discusses the semiclassical Einstein equations and their shortcomings. The chapter concludes with an overview of the main approaches to quantum gravity discussed in this book. This chapter is written in a rather general and non-technical style and should be accessible to a wide audience of people interested in theoretical physics.

This chapter explains the idea of a loop representation of a quantum field theory. Loop representations are important because they offer the prospect of a ‘gauge-free’ formulation of a gauge theory. In such a formulation, the theory's representations do not admit gauge transformations for gauge symmetry cases even to be a formal property of the theory, and gauge redundancy has been removed. This raises the prospect that the conclusion of the first part of the book may continue to hold even for quantum Yang-Mills gauge theories — that these describe non-localized properties of loops in space.

In a remarkable lecture Julian Schwinger delivered at the University of Nottingham on July 14, 1993, on the occasion of his receiving an honorary degree, entitled ‘The Greening of quantum field theory: George and I’, he summarised the central role Green's function played throughout his career. Schwinger then went on to recount his experience at the Massachusetts Institute of Technology's Radiation Laboratory during World War II and traced the influences of George Green on his own works. This chapter chronicles Schwinger's research in relation to Green's function, his first trip to Europe, and his work on the gauge invariance and vacuum polarization, the quantum action principle, electrodynamic displacements of energy levels, quantum field theory, and condensed matter physics.

This chapter reviews the basic building blocks of the regularization of Quantum Field Theories (QFT) on a space-time lattice. It assumes some familiarity with QFT in the continuum. In an introductory section, the path integral formulation is reviewed, focusing on important aspects such as the transfer matrix, the relation of correlation functions and physical observables, the perturbative expansion, and the key issue of renormalization and the Wilsonian renormalization group. It then considers in detail the lattice formulation of scalar, fermion and gauge field theories, paying careful attention to their physical interpretation, and the continuum limit. The difficulty of discretizing chiral fermions is discussed in detail, and various fermion discretizations are described. The strong coupling expansion is introduced in the context of lattice Yang-Mills theory and the criteria for confinement and for the presence of a mass gap are presented. It concludes with a description of Wilson's formulation of lattice QCD and a brief overview of its applications.

To calculate scattering S-matrix elements, quantities relevant to Particle Physics, it is necessary to consider instead the quantum evolution operator in real time. This chapter begins by deriving the path integral representation of the evolution operator and the S-matrix in simple quantum mechanics. To illustrate the power of the formalism, it shows how to recover the perturbative expansion of the scattering amplitude, some semi-classical approximations, and the eikonal approximation. When the asymptotic states at large time are eigenstates of the harmonic oscillator, instead of free particles, the holomorphic formalism becomes useful. A simple generalization of the path integral of Section 5.1 leads to the corresponding path integral representation of the S-matrix. In the case of the Bose gas the evolution operator is then given by a holomorphic functional integral. Using the parallel formalism of Section 5.6, the chapter derives an analogous representation for the evolution operator of a system of non-relativistic fermions. It then begins the study of relativistic quantum field theory with the example of the self-coupled neutral scalar boson. It shows that the holomorphic formalism, in a form that extends the construction of Section 5.5 to relativistic real time evolution, leads to various representations of the S-matrix in terms of functional integrals. The chapter relates S-matrix elements to the continuation to real time of various kinds of euclidean correlation functions.

In a series of papers by Freed, Hopkins, and Teleman (2003, 2005, 2007a) the relationship between positive energy representations of the loop group of a compact Lie group G and the twisted equivariant K-theory K ^τ+dimG _G (G) was developed. Here G acts on itself by conjugation. The loop group representations depend on a choice of ‘level’, and the twisting τ is derived from the level. For all levels the main theorem is an isomorphism of abelian groups, and for special transgressed levels it is an isomorphism of rings: the fusion ring of the loop group andK ^τ+dimG _G (G) as a ring. For G connected with π₁G torsionfree, it has been proven that the ring K ^τ+dimG _G (G) is a quotient of the representation ring of G and can be calculated explicitly. In these cases it agrees with the fusion ring of the corresponding centrally extended loop group. This chapter explicates the multiplication on the twisted equivariant K-theory for an arbitrary compact Lie group G. It constructs a Frobenius ring structure on K ^τ+dimG _G (G). This is best expressed in the language of topological quantum field theory: a two-dimensional topological quantum field theory (TQFT) is constructed over the integers in which the abelian group attached to the circle is K ^τ+dimG _G (G).

This chapter reviews some equilibrium properties in Statistical Quantum Field Theory, that is, relativistic Quantum Field Theory (QFT) at finite temperature, a relativistic extension of the statistical quantum theories discussed in Sections 5.5, 5.6. It discusess, in particular, the limit of high temperature or the situation of finite temperature phase transitions. It emphasizes that additional physical intuition about QFT at finite temperature in (1; d - 1) dimensions can be gained by realizing that it can also be considered as a classical statistical field theory in d dimensions with finite size in one dimension. This identification allows, in particular, an analysis of finite temperature QFT in terms of the renormalization group and the theory of finite size effects of the classical theory. These ideas are illustrated with several standard examples, the φ⁴ field theory, the non-linear σ model, the Gross–Neveu model, some gauge theories. The corresponding effective reduced theories are constructed at one-loop order. In models where the field is a N-component vector, the large N expansion provides a specially convenient tool to study the complete crossover between low and high temperature, and, therefore, dimensional reduction.

This chapter focuses on locality in quantum field theory. Many of the ideas involved are well exemplified in the study of bundles on Riemann surfaces which Nigel Hitchin is famous for. The chapter begins, there, especially as the question of locality relates to an aspect of his work that has not been talked about so far, namely, its role in so-called ‘geometric Langlands theory’.

Chapter 12 argued that quantum statistical mechanics puts unitarily inequivalent representations to use in ways no rigid interpretation can make sense of. Two features of working QFTs which promise a quantum field theoretic realization of Chapter 12's argument are Goldstone bosons and the Higgs mechanism. This chapter explains why they're promising by presenting them as instance of broken symmetry. Then it tempers the promise by admitting that the working QFTs in which these features occur are less mathematically explicit than they need to be to persuasively realize the argument of Chapter 12. The chapter closes by extracting from this very circumstance a non-conclusive reason to lend the argument of Chapter 12 interpretive weight. The reason is that our best theories of physics are still under construction, and their successors could share with the models presented in Chapter 12 the features on which the argument of Chapter 12 hinged.