Observables in the General Boundary Formulation

Oeckl, Robert

doi:10.1007/978-3-0348-0043-3_8

Cited by 22 publications

(79 citation statements)

References 15 publications

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“…In fact, this formalism can be viewed as a discrete incarnation of the 'general boundary formulation' of quantum theory [37,38,39,40]. Also there the Hilbert spaces and amplitude maps depend crucially on the space-time region and its boundary under consideration.…”

Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

“…In the 'general boundary formulation' [37,38,39,40] the so-called amplitude map essentially encodes the information about the dynamics. In the present formalism this amplitude map ρ 0→1 : H phys 0→1 → C can be defined via the unitary isomorphism U 0→1 and is given by (4.24)…”

Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

“…In order to obtain the amplitude map for the composition of the moves 0 → 1 and 1 → 2 one can translate axiom (T5) of [39,40] into the discrete. Using the state matching of procedure (i) in section 6.1 one can choose an orthonormal basis {ξ phys 1i } i∈I in H phys 1 (the matched pre-and post-physical Hilbert space at n = 1).…”

Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

“…The formalism developed here offers a systematic method for tracing and regularizing divergences arising in the construction of a path integral for variational discrete systems. In close analogy to the 'general boundary formulation' of quantum theory [37,38,39,40], we shall deal with transition amplitudes between different discrete time steps which, in a space-time context, can correspond to very general (not necessarily spatial) discretized boundaries of space-time regions.…”

Section: Introductionmentioning

confidence: 99%

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Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces

Hoehn

2014

Journal of Mathematical Physics

125

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A temporally varying discretization often features in discrete gravitational systems and appears in lattice field theory models subject to a coarse graining or refining dynamics. To better understand such discretization changing dynamics in the quantum theory, an according formalism for constrained variational discrete systems is constructed. While the present manuscript focuses on global evolution moves and, for simplicity, restricts to flat configuration spaces R N , a companion article [1] discusses local evolution moves. In order to link the covariant and canonical picture, the dynamics of the quantum states is generated by propagators which satisfy the canonical constraints and are constructed using the action and group averaging projectors. This projector formalism offers a systematic method for tracing and regularizing divergences in the resulting state sums. Non-trivial coarse graining evolution moves lead to non-unitary, and thus irreversible, projections of physical Hilbert spaces and Dirac observables such that these concepts become evolution move dependent on temporally varying discretizations. The formalism is illustrated in a toy model mimicking a 'creation from nothing'. Subtleties arising when applying such a formalism to quantum gravity models are discussed.

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Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

Section: A Discrete 'General Boundary Formulation'mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces

Hoehn

2014

Journal of Mathematical Physics

125

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“…where μ(x) has compact support in the interior of the right wedge, has been quantize using the Feynman prescription 13 within the GBF for the theories in Minkowski and in Rindler spaces. The notion of expectation value of an observable F in the GBF is defined in terms of a map, called observable map and denoted as ρ F , given in the Schrödinger-Feynman representation by…”

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confidence: 99%

General boundary treatment of the Unruh effect

Colosi¹

2017

The Fourteenth Marcel Grossmann Meeting

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The general boundary formulation (GBF) of quantum theory is a new axiomatic formulation that has emerged as a powerful tool to describe the dynamics of quantum fields. Within the GBF we consider the Unruh effect by studying the relation between the quantum field theories of a massive scalar field in 2d Minkowski and Rindler spaces. In particular we underline the existence of an obstruction in the identification of the usual Minkowski vacuum state with a superposition of multiparticle states defined in Rindler space. However, we show that these two states can be related by comparing expectation values of Weyl observables with compact support on the right wedge of Minkowski space. Finally, the quantum state in Rindler space responsible for this local version of the Unruh effect results to be unique.Keywords: Unruh effect; general boundary formulation.The general boundary formulation (GBF) of quantum theory represents an extension of the standard formulation; it is based on two main ingredients: The mathematical framework of Topological Quantum Field Theory (TQFT) and a generalization of the Born rule providing the probability interpretation 1,2 . As in TQFT, a set of axioms associates algebraic structures to geometric ones. In particular, state spaces (usually Hilbert spaces) are defined on hypersurfaces, i.e. oriented manifold of dimension d − 1, and amplitude maps are associated to space-time region, i.e. oriented manifold of dimension d, with boundaries. Such amplitude maps generalize the standard notion of transition amplitudes, usually defined for space-time regions bounded by Cauchy surfaces (in particular, by the disjoint union of an initial Cauchy surface and a final one). In the GBF, the boundary of the region where dynamics takes place is not required to be given by (the disjoint union of) Cauchy surfaces. Arbitrary hypersurfaces are admissible boundaries. For example, in Minkowski space a timelike connected hypersurface given by an hypercylinder (namely, a three ball in space extended over all of times) was considered in Ref. 3. The general interacting theory of a massive Klein-Gordon field defined in the hypercylinder region was shown to be equivalent with the one considered in the region bounded by two equal-Minkowski-time hyperplanes 4,5,a . In the hypercylinder region dynamics is not described in terms of an evolution from an initial state of the system defined at initial time to a final state at a final time. States on the hypercylinder (belonging to the Hilbert space associated to the hypercylinder hypersurface) are defined for all of times. Moreover, the connectedness of the hypersurface prevents an a priori a The same hypercylinder region considered in Anti-de Sitter space was used to propose a solution to the problem of defining the S-matrix of a field in Anti-de Sitter space where the lack of temporal asymptotic regions prevents the standard construction of the in− and out−states 6,7 . The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com by 34.218.44.141 on 05/13/18....

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