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  • 1
    Electronic Resource
    Electronic Resource
    Quantum temporal logic and decoherence functionals in the histories approach to generalized quantum theory (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 5452-5476 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The recent suggestion that a temporal form of quantum logic provides the natural mathematical framework within which to discuss the proposal by Gell-Mann and Hartle for a generalized form of quantum theory based on the ideas of histories and decoherence functionals is analyzed and developed herein. Particular stress is placed on properties of the space of decoherence functionals, including one way in which certain global and topological properties of a classical system are reflected in a quantum history theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530759
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  • 2
    Electronic Resource
    Electronic Resource
    Continuous time and consistent histories (1998)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 39 (1998), S. 1818-1834 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: 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. © 1998 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532265
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  • 3
    Electronic Resource
    Electronic Resource
    Perennials and the group-theoretical quantization of a parametrized scalar field on a curved background (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3522-3538 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The perennial formalism is applied to the real, massive Klein–Gordon field on a globally-hyperbolic background space–time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras Scan and Sloc of elementary perennials are constructed. The elements of Scan correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of Sloc are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space–time, and by the time shifts that are generated by space–time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly. The construction of the quantum theory using the perennial formalism is shown to be equivalent to the Segal quantization of a Weyl system if the time shift automorphisms of the algebra Scan are used. In this way, the absence of any timelike Killing vector field in the background space–time leads naturally to the "problem of time'' for quantum field theory on a background space–time. Within the perennial formalism, this problem is formally identical to the problem of time for any parametrized system, including general relativity itself. Two existing strategies—the "scattering'' approach, and the "algebraic'' approach—for dealing with this problem in quantum field theory on a background space–time are translated into the language of the perennial formalism in the hope that this may give some insight into how the general problem can be solved. The non-unitary time evolution typical of the Hawking effect is shown to be due to global properties of the corresponding phase space: specifically, the time shifts map a global transversal surface to a non-global one. Thus, the existence of this effect is closely related to the global time problem. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531579
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  • 4
    Electronic Resource
    Electronic Resource
    The classification of decoherence functionals: An analog of Gleason's theorem (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6360-6370 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Gell-Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are "histories'' and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert space V; in effect, one seeks the history analog of Gleason's famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=trV⊗V(α⊗βX) for some operator X on the tensor-product space V⊗V. © 1994 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530679
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  • 5
    Electronic Resource
    Electronic Resource
    The symplectic geometry of a parametrized scalar field on a curved background (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 3505-3521 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We study the real, massive Klein–Gordon field on a C∞ globally-hyperbolic background space–time with compact Cauchy hypersurfaces. In particular, the parametrization of this system as initiated by Dirac and Kuchar is put on a rigorous basis. The discussion is focussed on the structure of the set of spacelike embeddings of the Cauchy manifold into the space–time, and on the associated e-tensor density bundles and their tangent and cotangent bundles. The dynamics of the field is expressed as a set of automorphisms of the space of initial data in which each pair of embeddings defines one such automorphism. Using these results, the extended phase space of the system is shown to be a weak-symplectic manifold, and the Kuchar constraint is shown to define a smooth constraint submanifold which is foliated smoothly by the constraint orbits. The pull-back of the symplectic form to the constraint surface is a presymplectic form which is singular on the tangent spaces to the constraint orbits. Thus, the geometric structure of this infinite-dimensional system is analogous to that of a finite-dimensional, first-class parametrized system, and hence many of the results for the latter can be transferred to the infinite-dimensional case without difficulty. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531578
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  • 6
    Electronic Resource
    Electronic Resource
    Continuous histories and the history group in generalized quantum theory (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 5392-5408 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We treat continuous histories within the histories approach to generalized quantum mechanics. The essential tool is the "history group:'' the analog, within the generalized history scheme, of the canonical group of single-time quantum mechanics. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531267
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  • 7
    Electronic Resource
    Electronic Resource
    Quantum logic and the histories approach to quantum theory (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 2157-2185 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An extensive analysis is made of the Gell-Mann and Hartle axioms for a generalized ‘histories' approach to quantum theory. Emphasis is placed on finding analogs of the lattice structure employed in standard quantum logic. Particular attention is given to ‘quasitemporal' theories in which the notion of time-evolution in conventional Hamiltonian physics is replaced by something that is much broader; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space–time. The quasitemporal structure is coded in a partial semigroup of ‘temporal supports' that underpins the lattice of history propositions. Nontrivial examples include quantum field theory on a non-globally-hyperbolic space–time, and a possible cobordism approach to a theory of quantum topology. A key result is the demonstration that the set of history propositions in standard quantum theory can be realized in such a way that each such proposition is represented by a genuine projection operator. This gives valuable insight into the possible lattice structure in general history theories.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530544
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  • 8
    Electronic Resource
    Electronic Resource
    Coherent states for n-dimensional Euclidean groups E(n) and their application (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 607-620 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The n-dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path-integral representation involving novel first-order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529402
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  • 9
    Electronic Resource
    Electronic Resource
    QUANTUM FIELD THEORY IN CURVED SPACETIME: AN OVERVIEW (1977)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 302 (1977), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1977.tb37043.x
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  • 10
    Electronic Resource
    Electronic Resource
    SUMMARY OF WORKSHOPS ON QUANTUM GRAVITY (1980)
    Oxford, UK : Blackwell Publishing Ltd
    Annals of the New York Academy of Sciences 336 (1980), S. 0 
    Details
    ISSN: 1749-6632
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Natural Sciences in General
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1749-6632.1980.tb15939.x
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