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arXiv:gr-qc/9812018

Symmetries and conservation laws in histories-based generalized quantum mechanics

Tulsi Dass, Yogesh N. Joglekar

Published Dec 5, 1998• 0 citations
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Abstract

Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved.

DOI
10.48550/arxiv.gr-qc/9812018

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Symmetries are defined in histories-based generalized quantum mechanics paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of temporal sequences' of events' using partial semigroups) and logic structure for single time histories'. Symmetries are classified into orthochronous (those preserving the temporal order' of `events') and non-orthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved.
Symmetries are defined in histories-based theories paying special attention to the class of history theories admitting quasitemporal structure (a generalization of the concept of `temporal sequences' of `events' using partial semigroups) and logic structure for `single-time histories'. Symmetries are classified into orthochronous (those preserving the `temporal order' of `events') and nonorthochronous. A straightforward criterion for physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence of histories considered by Gell-Mann and Hartle as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, a definition of conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved. The formalism evolved is applied to histories (of particles, fields or more general objects) in general curved spacetimes. Sharpening the definition of symmetry so as to include a continuity requirement, it is shown that a symmetry in our formalism implies a conformal isometry of the spacetime metric.
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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.
An axiomatic development of dynamics of systems in the framework of histories is given which contains the history versions of classical and traditional quantum mechanics as special cases. We consider theories which admit a quasitemporal structure (a generalization of the concept of time using partial semigroups) and whose “single time” propositions have the mathematical structure of a logic; isomorphism of logics at different “instants of time” is not assumed. The concept of directed partial semigroup is introduced to incorporate the concept of direction of flow of time in quasitemporal theories. Starting with a few simple axioms, the space of history propositions is explicitly constructed and shown to be an orthoalgebra (as envisaged in the scheme of Isham and Linden [J. Math. Phys. 35, 5452–5476 (1994)]); its subspace consisting of history filters (“homogeneous histories”) is a meet semilattice. The partial semigroups employed allow semi-infinite irreducible decompositions.
This chapter introduces the general concept of symmetry in both classical and quantum physics. The part devoted to classical physics stresses the importance of conservation laws and the role of Noether’s theorem for their understanding. A fairly general discussion of Noether’s theorem is offered, including ambiguities in the definition of the Noether current, possible existence of generalized, tensor conservation laws, and the inverse of Noether’s theorem. The Hamiltonian formalism is then used to motivate the transition from classical to quantum physics, and the definition of symmetry in the latter. In anticipation of the discussion of spontaneous symmetry breaking in the following chapter, the text here emphasizes the difference between implementing symmetry in terms of transformations on the algebra of physical observables, and in terms of unitary mappings on the Hilbert space of states.
The basic ingredients of the ‘‘consistent histories’’ approach to quantum theory are a space 𝒰𝒫 of ‘‘history propositions’’ and a space 𝒟 of ‘‘decoherence functionals.’’ In this article we consider such history quantum theories in the case where 𝒰𝒫 is given by the set of projectors 𝒫(𝒱) on some Hilbert space 𝒱. We define the notion of a ‘‘physical symmetry of a history quantum theory’’ (PSHQT) and specify such objects exhaustively with the aid of an analog of Wigner’s theorem. In order to prove this theorem we investigate the structure of 𝒟, define the notion of an ‘‘elementary decoherence functional,’’ and show that each decoherence functional can be expanded as a certain combination of these functionals. We call two history quantum theories that are related by a PSHQT ‘‘physically equivalent’’ and show explicitly, in the case of history quantum mechanics, how this notion is compatible with one that has appeared previously.