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Il Nuovo Cimento B

Steps towards the axiomatic foundations of the relativistic quantum theory of fields: Properties of the fields

Gabriel D. Puccini, H. Vucetich

Published Jan 1, 2005• 1 citation
Publisher

Abstract

A realistic physical axiomatic formulation of the relativistic quantum theory of fields is presented. Based on the action principle of Schwinger, the approach is developed in an aaabstract ( it does not depend on any particular representation of field operators) and covariant way for fields satisfying global gauge invariance. The field equations and the explicit expressions for the generators are deduced without using the "correspondence principle". Unlike other proofs, crossing symmetry arises in an independent way of the causality requirement. By contrast to the mathematical systems of axioms, particular emphasis on the problem of attaching a physical meaning to the formalism is handled. Arguments for a realistic interpretation in terms of "quantum fields" are advanced. Our realistic view is illustrated by regarding quantum fields as "real things" with a number of properties (e.g., interaction, symmetry properties, etc.). As a result of this conception, the fundamental variables in terms of which our axiomatics is constructed are field operators, not particles states. The advantages of our axiomatic approach are finally discussed.

DOI
10.1393/ncb/i2003-10098-y

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A realistic physical axiomatic formulation of the relativistic quantum theory of fields is presented. Based on the action principle of Schwinger, the approach is developed in an aaabstract ( it does not depend on any particular representation of field operators) and covariant way for fields satisfying global gauge invariance. The field equations and the explicit expressions for the generators are deduced without using the "correspondence principle". Unlike other proofs, crossing symmetry arises in an independent way of the causality requirement. By contrast to the mathematical systems of axioms, particular emphasis on the problem of attaching a physical meaning to the formalism is handled. Arguments for a realistic interpretation in terms of "quantum fields" are advanced. Our realistic view is illustrated by regarding quantum fields as "real things" with a number of properties (e.g., interaction, symmetry properties, etc.). As a result of this conception, the fundamental variables in terms of which our axiomatics is constructed are field operators, not particles states. The advantages of our axiomatic approach are finally discussed.
The axiomatic formulation of an Algebraic Relativistic Quantum Field Theory leads naturally -via the Haag-Kastler axioms- to the fact that operators representing observables generate a von Neumann algebra localized in a space-time region in which said observations are made. I present and explain the axioms, as well as the mathematical preliminaries needed to understand them and apply them to the so-called causal diamond and Fermi's Two-Atom System. The latter shows the importance of formulating axioms concerning causality that try to make compatible the requirement from Special Relativity according to which no causal influence can travel faster than the speed of light and the quantum mechanical theory of measurement and preparability of states. For a relativistic field-theoretic setting, these concepts must be re-formulated, and this reasoning suggests that the operator algebra must be a type III von Neumann factor. I also introduce some philosophical problems concerning our usual notion of causality from Hume to contemporary theories of counterfactual causation and show how the algebraic structure of the theory is deeply related to the causal one and how this theory can be extended if we change the way we think about causality, measurements, and local preparability of states.
The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schr\"odinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.
The concepts of instantaneous observables and dynamical variables are analyzed and generalized to arbitrary spacelike hyperplanes. A formalism is developed which gives the basic equations of relativistic quantum mechanics for dynamical variables on arbitrary hyperplanes a manifestly covariant form. A covariant linear transformation on the Poincaré generators introduces the hyperplane generators which yield commutation relations displaying a clear separation of the kinematical and dynamical properties of dynamical variables. An axiomatic study of the center-of-mass position operator yields the uniqueness of the operator and completes the physical interpretation of the hyperplane generators. The Poincaré invariance and hyperplane independence of the scattering operator is related to asymptotic conservation laws in the hyperplane formalism, and finally, a nonlocal, hyperplane-dependent, field theory of free spinless particles is considered.
An axiomatic approach to quantum mechanics that has, as a theorem, the Schrödinger equation may be of enormous value to cope with interpretation issues, since all the interpretation constructs must be present in the axioms, or directly derived by them from their mathematical unfolding. Thus, it is critical to show that this axiomatic derivation is reliable beyond any possible doubt. To show this, it is possible to make generalizations and extensions of the axioms to derive the Schrödinger equation in the underlying generalized or extended formats. In previous papers, we have shown that the axiomatic approach we propose can be used to derive the Schrödinger equation as a direct axiom. Since then, we have also shown that it was possible to generalize that derivation to coordinate systems other than the Cartesian, as well as its relativistic extensions that lead to the relativistic wave equations. An extension to dissipative systems was also performed, allowing us to mathematically derive the Caldirola–Kanai equation from first principles. All these derivations were performed using pure states and in the absence of the electromagnetic field. This means that we can further generalize the approach to embrace these two possibilities. Being an axiomatic approach, we show that we need only to slightly modify the axioms to derive the Schrödinger equation for these two contexts. Despite being quite direct, the algebraic complexity of these derivations should give the reader the desired confidence in the proposed axioms.
An important way in which quantum field theory (QFT) differs from non-relativistic quantum mechanics is that in QFT systems generically possess an infinite number of degrees of freedom. This chapter examines the proposals for non-fundamental roles that the particle concept plays in particle physics. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism. One difficulty for the quanta interpretation of QFT is that the analysis in terms of quanta offered in the preceding paragraphs is restricted to free systems. The no-go results for interacting systems inspire a different approach to regarding particles as non-fundamental entities. The fields which feature in the QFT formalism are mathematical expressions; work needs to be done to determine the ontological properties of the entities that are represented by the mathematical field formalism.