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CRC Press eBooks

The Particle-Operator and Particle States in Quantum Field Theory

Minoru Omote, S. Kamefuchi

Published Nov 30, 2003• 0 citations
Publisher

Abstract

By using the method of particle-operators, proposed previously, we examine the problem of particle states (spatially localized) with emphasis upon the relativistic case. A physical interpretation is also given of the results which seem to contradict locality, covariance and causality.
DOI
10.1201/9781482269079-113

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In order to discuss localization experiments and also to extend the consistent history interpretation of quantum mechanics to relativistic properties, the techniques introduced in a previous paper [J. Math. Phys. 38, 697 (1997)] are applied to the localization of a photon in a given region of space. An essential requirement is to exclude arbitrarily large wavelengths. The method is valid for a particle with any mass and spin. Though there is no proper position operator for a photon, one never needs one in practice. Causality is valid up to exponentially small corrections.
Relativistic quantum theory poses a number of conceptual problems over and above the non-relativistic mechanics. From a purely mathematical point of view it is also much more sophisticated. Even the kinematic theory poses considerable difficulties: locality, antimatter, negative energy, and charge are already systematically linked in ways that cannot simply be paraphrased. A further complication is that in the midst of this structure we have what is usually called the Newton-Wigner representation (or, in the case of the Dirac theory, the Foldy-Wouthuysen representation), in which fields and states are no longer covariantly described and where the “local” self-adjoint quantities (NW-local observables) do not obey microcausality, but only satisfy equal-time commutators with respect to a particular inertial frame. If we pass to this represenation, we obtain at a stroke the basic structure of non-relativistic quantum field theory; it is not too hard to descend from that to a many-particle mechanics, and to recover the usual definitions of localization.
In this paper I will present conceptions of state reduction and particle and/or system localization which render these subjects fully compatible with the general requirements of a relativistic, i.e. Lorentz invariant, quantum theory. In the case of state reduction, the concept presented has, in it's main features, been advanced by several investigators in recent years, (Giovannini 1983), (Aharonov and Albert 1984), (Malin 1984), (Dieks 1985). My own contribution to this topic has been to elaborate the concept in question and examine, in some detail, the often counterintuitive but never causally anomalous or internally inconsistent consequences that emerge from taking the concept seriously (Fleming 1985,1986). In the case of particle and system localization, the concepts are more nearly my own and have been pursued by me for some time (Fleming 1964, 1965, 1966) as a way of freeing the subject of localization in relativistic quantum systems from the conceptual problems pointed out by Newton and Wigner (1949) myself (Fleming 1965) and extensively studied by Hegerfeldt et. al. (1974, 1980) and Ruijsenaars (1981).
In relativistic quantum field theory the notion of a local operation is regarded as basic: each open space-time region is associated with an algebra of observables representing possible measurements performed within this region. It is much more difficult to accommodate the notions of events taking place in such regions or of localized objects. But how can the notion of a local operation be basic in the theory if this same theory would not be able to represent localized measuring devices and localized events? After briefly reviewing these difficulties we discuss a strategy for eliminating the tension, namely by interpreting quantum theory in a realist way. To implement this strategy we use the ideas of the modal interpretation of quantum mechanics. We then consider the question of whether the resulting scheme can be made Lorentz invariant.
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.
We have recently introduced a realistic, covariant, interpretation for the reduction process in relativistic quantum mechanics. The basic problem for a covariant description is the dependence of the states on the frame within which collapse takes place. A suitable use of the causal structure of the devices involved in the measurement process allowed us to introduce a covariant notion for the collapse of quantum states. However, a fully consistent description in the relativistic domain requires the extension of the interpretation to quantum fields. The extension is far from straightforward. Besides the obvious difficulty of dealing with the infinite degrees of freedom of the field theory, one has to analyse the restrictions imposed by causality concerning the allowed operations in a measurement process. In this paper we address these issues. We shall show that, in the case of partial causally connected measurements, our description allows us to include a wider class of causal operations than the one resulting from the standard way of computing conditional probabilities. This alternative description could be experimentally tested. A verification of this proposal would give stronger support to the realistic interpretations of the states in quantum mechanics.