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.
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.
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.
This book seeks to answer the question "What explains CPT invariance and the spin-statistics connection?" These properties play foundational roles in relativistic quantum field theories (RQFTs), are supported by high-precision experiments, and figure into explanations of a wide range of phenomena, from antimatter, to the periodic table of the elements, to superconductors and superfluids. They can be derived in RQFTs by means of the famous CPT and Spin-Statistics theorems; but, the author argues, these theorems cannot be said to explain these properties, at least under standard philosophical accounts of scientific explanation. This is because there are multiple, in some cases incompatible, ways of deriving these theorems, and, secondly, because the theorems fail for the types of theories that underwrite the empirical evidence: non-relativistic quantum theories, and realistic interacting RQFTs. The goal of this book is to work towards an understanding of CPT invariance and the spin-statistics connection by first providing an analysis of the necessary and sufficient conditions for these properties, and second by advocating a particular account of explanation appropriate for this context.
A recent paper by Peshkin [1] has drawn attention again to the problem of understanding the spin statistics connection in non-relativistic quantum mechanics. Allen and Mondragon [2] has pointed out correctly some of the flaws in Peshkin's arguments which are based on the single valuedness under rotation of the wave functions of systems of identical particles. We examine carefully the claim that is made in the title of [2] that there can be ``no spin-statistics connection in non-relativistic quantum mechanics''. We show that we can derive the spin statistics connection for non-relativistic quantum field theories which have equations of motion of the Hamiltonian type based on SU(2) invariance of the Lagrangian. The formalism and machinary of non-relativistic quantum field theory as opposed to usual quantum mechanics is necessary for constructing our proof.
George Sudarshan was quite a bit ahead of the times when he sketched out a non-relativistic proof of the spin-statistics connection forty years back. A recent revival of interest in the role of relativistic arguments in proving the spin-statistics connection prompted us to revisit Sudarshan's original arguments and present them in simpler fashion. The proof is designed to work in three space dimensions. Sudarshan's observation was that if we impose SU (2) invariance, along with a few other restrictions, on the kinematic part of the Lagrangian of a field theory which is derivable from a Weiss-Schwinger type of the Principle of Least Action, then the spin-statistics connection automatically follows. The assumptions and simplifications that go into constructing the proof will be carefully laid out so as to identify the limitations of this approach.
Given the classical dynamics of a non-relativistic particle in terms of a Hamiltonian or an action, it is relatively straightforward to obtain the non-relativistic quantum mechanics (NRQM) of the system. These standard procedures, based on either the Hamiltonian or the path integral, however, do not work in the case of a relativistic particle. As a result we do not have a single-particle description of relativistic quantum mechanics (RQM). Instead, the correct approach requires a transmutation of dynamical variables from the position and momentum of a single particle to a field and its canonical momentum. Particles, along with antiparticles , reappear in a very nontrivial manner as the excitations of the field. The fact that one needs to adopt completely different languages to describe a relativistic and non-relativistic free particle implies that obtaining the NRQM limit of QFT is conceptually nontrivial. I examine this limit in several approaches (like, for e.g., Hamiltonian dynamics, Lagrangian and Hamiltonian path integrals, field theoretic description etc.) and identify the precise issues which arise when one attempts to obtain the NRQM from QFT in each of these approaches. The dichotomy of NRQM and QFT does not originate just from the square root in the Hamiltonian or from the demand of Lorentz invariance, as is sometimes claimed. The real difficulty has its origin in the necessary existence of antiparticles to ensure a particular notion of relativistic causality. Because of these conceptual issues, it turns out that one cannot , in fact, obtain some of the popular descriptions of NRQM by any sensible limiting procedure applied to QFT. To obtain NRQM from QFT in a seamless manner, it is necessary to work with NRQM expressed in a language closer to that of QFT. This fact has several implications, especially for the operational notion of space coordinates in quantum theory. A close examination of these issues, which arise when quantum theory is combined with special relativity, could offer insights in the context of attempts to combine quantum theory with general relativity.
The traditional standard theory of quantum mechanics is unable to solve the spin-statistics problem, i.e. to justify the utterly important “Pauli Exclusion Principle” but by the adoption of the complex standard relativistic quantum field theory. In a recent paper [E. Santamato and F. D. De Martini, Found. Phys. 45 (2015) 858] we presented a complete proof of the spin-statistics problem in the nonrelativistic approximation on the basis of the “Conformal Quantum Geometrodynamics” (CQG). In this paper, by the same theory, the proof of the spin-statistics theorem (SST) is extended to the relativistic domain in the scenario of curved spacetime. No relativistic quantum field operators are used in the present proof and the particle exchange properties are drawn from rotational invariance rather than from Lorentz invariance. Our relativistic approach allows to formulate a manifestly step-by-step Weyl gauge invariant theory and to emphasize some fundamental aspects of group theory in the demonstration. As in the nonrelativistic case, we find once more that the “intrinsic helicity” of the elementary particles enters naturally into play. It is therefore this property, not considered in the standard quantum mechanics (SQM), which determines the correct spin-statistics connection observed in Nature.