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arXiv:0905.1263 · Europhysics Letters (EPL)

Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field

PETER T. MORGAN

Published Aug 1, 2009• 7 citations
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Abstract

The difference between a Klein-Gordon random field and the complex Klein-Gordon quantum field is characterized, explicitly comparing the roles played by negative-frequency modes of test functions in creation and annihilation operator presentations of the two theories. The random field and the complex quantum field can both be constructed from the same creation and annihilation operator algebra, making them equivalent in that sense.

DOI
10.1209/0295-5075/87/31002

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The difference between a Klein-Gordon random field and the complex Klein-Gordon quantum field is characterized, explicitly comparing the roles played by negative-frequency modes of test functions in creation and annihilation operator presentations of the two theories. The random field and the complex quantum field can both be constructed from the same creation and annihilation operator algebra, making them equivalent in that sense.
A random field that is empirically equivalent to the quantized electromagnetic field is constructed. A mapping between the creation and annihilation operator algebras of a random field and of the quantized electromagnetic field provides a functor between the algebras and the Hilbert spaces generated by the vacuum states over those algebras. The functor inevitably does not extend to a functorial relationship between the local algebras generated by the random field and by the quantized electromagnetic field, but the empirical content provided by the vacuum state restores an empirical equivalence through the Hilbert spaces. The isomorphism from one creation and annihilation algebra to the other is not translation invariant because it depends on mapping positive frequency modes of one helicity to equivalent negative frequency modes, but the two theories taken independently are presented in equally well-defined and manifestly Lorentz and translation covariant ways.
Abstract The purpose of this article is to construct an explicit relation between the field operators in Quantum Field Theory and the relevant operators in Quantum Mechanics for a system of N identical particles, which are the symmetrised functions of the canonical operators of position and momentum, thus providing a clear relation between Quantum Field Theory and Quantum Mechanics. This is achieved in the context of the non-interacting Klein–Gordon field. Though this procedure may not be extendible to interacting field theories, since it relies crucially on particle number conservation, we find it nevertheless important that such an explicit relation can be found at least for free fields. It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations. The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton–Wigner position operator. We make some clarifications on the interpretation of Newton–Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations. From a complex Klein–Gordon field, position and momentum operators can be constructed for both particles and antiparticles.
Classical nonlocal field models consisting of probability densities over functions defined everywhere on Minkowski space are constructed, using functional methods. These models are equivalent to states of the quantized real Klein-Gordon field in the sense that the marginal probability density over real functions defined everywhere on a 3-dimensional hyperplane S, at all times and for all Lorentz boosts, is equal to the corresponding probability density over real functions on S that is given by a state of the quantized real Klein-Gordon field. This paper establishes a relationship between quantum field theory and classical statistical field theory different from the well-known relationship of analytic continuation. 1
Generalized parity [Formula: see text], time-reversal [Formula: see text], and charge-conjugation [Formula: see text] operators were initially defined in the study of the pseudo-Hermitian Hamiltonians. We construct a concrete realization of these operators for Klein–Gordon fields and show that in this realization [Formula: see text] and [Formula: see text] operators respectively correspond to the ordinary time-reversal and charge-grading operations. Furthermore, we present a complete description of the quantum mechanics of Klein–Gordon fields that is based on the construction of a Hilbert space with a relativistically invariant, positive-definite, and conserved inner product. In particular we offer a natural construction of a position operator and the corresponding localized and coherent states. The restriction of this position operator to the positive-frequency fields coincides with the Newton–Wigner operator. Our approach does not rely on the conventional restriction to positive-frequency fields. Yet it provides a consistent quantum mechanical description of Klein–Gordon fields with a genuine probabilistic interpretation.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.