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arXiv:gr-qc/0311092 · Journal of Mathematical Physics

Histories electromagnetism

Aidan Burch

Published May 7, 2004• 5 citations
arXivHTMLPDF

Abstract

Working within the HPO (History Projection Operator) Consistent Histories formalism, we follow the work of Savvidou on (scalar) field theory [J. Math. Phys. 43, 3053 (2002)] and that of Savvidou and Anastopoulos on (first-class) constrained systems [Class. Quantum Gravt. 17, 2463 (2000)] to write a histories theory (both classical and quantum) of Electromagnetism. We focus particularly on the foliation-dependence of the histories phase space/Hilbert space and the action thereon of the two Poincaré groups that arise in histories field theory. We quantize in the spirit of the Dirac scheme for constrained systems.

DOI
10.1063/1.1723702

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Working within the HPO (History Projection Operator) Consistent Histories formalism, we follow the work of Savvidou on (scalar) field theory [J. Math. Phys. 43, 3053 (2002)] and that of Savvidou and Anastopoulos on (first-class) constrained systems [Class. Quantum Gravt. 17, 2463 (2000)] to write a histories theory (both classical and quantum) of Electromagnetism. We focus particularly on the foliation-dependence of the histories phase space/Hilbert space and the action thereon of the two Poincaré groups that arise in histories field theory. We quantize in the spirit of the Dirac scheme for constrained systems.
These lecture notes cover the important developments in histories approach to quantum mechanics with overall content and emphasis somewhat different from other reviews and books on the subject.The idea of Houtappel, Van Dam and Wigner of employing objects based on primitive concepts of physical theories is discussed in some detail and the fact that histories are such objects is emphasized. Application of histories formalism to the problem of understanding the quasiclassical domain is treated in some detail. Other topics discussed include generalized histories-based quantum mechanics and its application to the quantum mechanics of space-time,generalization of the notion of time sequences employing partial semigroups,quasitemporal structures, history projection operator (HPO) formalism, the algebraic scheme of Isham and Linden, an axiomatic scheme for quasitemporal histories-based theories and symmetries and conservation laws in histories-based theories.
For a quantum field living on a nonstatic space–time no instantaneous Hamiltonian is definable, for this generically necessitates a choice of inequivalent representation of the canonical commutation relations at each instant of time. This fact suggests a description in terms of time-dependent Hilbert spaces, a concept that fits naturally in a (consistent) histories framework. Our primary tool for the construction of the quantum theory in a continuous-time histories format is the recently developed formalism based on the notion of the history group. This we employ to study a model system involving a 1+1 scalar field in a cavity with moving boundaries. The instantaneous (smeared) Hamiltonian and a decoherence functional are then rigorously defined so that finite values for the time-averaged particle creation rate are obtainable through the study of energy histories. We also construct the Schwinger–Keldysh closed-time-path generating functional as a “Fourier transform” of the decoherence functional and evaluate the corresponding n-point functions.
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.
We discuss the use of histories labelled by a continuous time in the approach to consistent-histories quantum theory in which propositions about the history of the system are represented by projection operators on a Hilbert space. This extends earlier work by two of us [C. J. Isham and N. Linden, J. Math. Phys. 36, 5392–5408 (1995)] where we showed how a continuous time parameter leads to a history algebra that is isomorphic to the canonical algebra of a quantum field theory. We describe how the appropriate representation of the history algebra may be chosen by requiring the existence of projection operators that represent propositions about the time average of the energy. We also show that the history description of quantum mechanics contains an operator corresponding to velocity that is quite distinct from the momentum operator. Finally, the discussion is extended to give a preliminary account of quantum field theory in this approach to the consistent histories formalism.
We develop a new algorithm for the quantization of systems with first-class constraints. Our approach lies within the (History Projection Operator) continuous-time histories quantization programme. In particular, the Hamiltonian treatment (either classical or quantum) of parameterized systems is characterized by the loss of the notion of time in the space of true degrees of freedom (i.e. the `problem of time'). The novel temporal structure of the HPO theory (two laws of time transformation that distinguish between the temporal logical structure and the dynamics) persists after the imposition of the constraints, hence the problem of time does not arise. We expound the algorithm for both the classical and quantum cases and apply it to simple models.