CHAPTER 7. Relativistic Quantum Field Theory

Jean Zinn‐Justin

Abstract This chapter introduces the relativistic quantum field theory (QFT) of the neutral scalar boson field. It is a local, relativistic invariant, theory for a real field extension of the non-relativistic field theory of the Bose gas. Locality is a property that plays a central role in most of this work. The QFT is discussed both from the viewpoint of real-time evolution and statistical physics. The holomorphic formalism leads to representations of the S-matrix in terms of field integrals. The S-matrix elements are related to the continuation to real time of various kinds of Euclidean correlation functions. It is argued that the massive φ4 QFT has the quantum Bose gas with a pair potential, in the grand canonical formulation, as a non-relativistic limit.

P. A. M. Dirac, K. K. Gupta, George Sudershan

J. R. Claycomb

Jonathan Bain

Chapter 5 addresses the question, “What explains CPT invariance and the spin–statistics connection (SSC)?” It first considers why we should care about explaining CPT invariance and the SSC. It then argues that the CPT and spin–statistics theorems cannot be said to provide the sought after explanations. It offers instead an explanation of the SSC in interacting non-relativistic theories. This explanation explains by virtue of a derivation from non-fundamental principles in non-interacting relativistic quantum field theories (RQFTs), and an explication of intertheoretic relations between interacting RQFTs, non-interacting RQFTs, and interacting non-relativistic theories. The chapter ends by arguing that the presence of both the SSC and CPT invariance in realistic interacting RQFTs cannot be explained by even this type of explanation, and that currently, the existence problem precludes a full understanding of these properties in realistic interacting RQFTs.

Jonathan Bain

Chapter 4 evaluates attempts to derive CPT invariance and the spin–statistics connection (SSC) outside the framework of relativistic quantum theories. It first considers attempts at non-relativistic quantum mechanical derivations of the SSC. These attempts split into those that adopt the framework of finite-dimensional (i.e., non-field-theoretic) non-relativistic quantum mechanics, and fall under the general heading of “configuration space approaches,” and an approach that adopts the framework of non-relativistic quantum field theory. The chapter indicates how recent work in the philosophy of physics literature on the relation between permutation invariance and indistinguishability sheds light on the limitations of the former, while the discussion in Chapter 3 demonstrates where the latter fails. The chapter then considers attempts to derive CPT invariance in relativistic classical field theories from both a pragmatist (Lagrangian) and a purist (algebraic) perspective.