A “general boundary” formulation for quantum mechanics and quantum gravity

Oeckl, Robert

doi:10.1016/j.physletb.2003.08.043

Cited by 153 publications

(273 citation statements)

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“…It was shown in [8,9] that this does not thwart the correct semiclassical limit for the case of a single simplex, but doubt remained as to whether this success would survive when more general triangulations are considered. In this section we address this question within the general framework of this paper, replacing at each simplex in the Regge path integral the exponential of the action by its spinfoam-inspired equivalent, and showing that the contribution from the non-Regge terms is suppressed in the semiclassical limit 13 .…”

Section: Semiclassical Regime Of Spin Foam Modelsmentioning

confidence: 99%

“…The starting point for this progress was the suggestion by Rovelli [8,9] of a procedure for computing the graviton propagator from Loop Quantum Gravity (LQG) [1,10,11] with the dynamics implemented covariantly in terms of a spin foam model. On the calculational side, the key ingredients are a boundary semiclassical spin network state peaked on large spins and an analytic expression for the large spin asymptotics of the spin foam vertex amplitude [12]; on the conceptual side, the framework is the boundary state formalism discussed in [1] and in [13,14,15,16,17], which prescribes how to compute observables in the boundary of a spacetime region with a path integral over the interior region only. IfÔ 1 ,Ô 2 are local boundary geometry observables (such as areas, dihedral angles, 3-volumes or lengths [18,19,20,21,22,23]) acting on a space of spin networks s , then the expectation value for their correlation in a boundary geometry q is given in the boundary state formalism by…”

Section: Introductionmentioning

confidence: 99%

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Semiclassical regime of Regge calculus and spin foams

Bianchi

Satz

2009

Nuclear Physics B

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Recent attempts to recover the graviton propagator from spin foam models involve the use of a boundary quantum state peaked on a classical geometry. The question arises whether beyond the case of a single simplex this suffices for peaking the interior geometry in a semiclassical configuration. In this paper we explore this issue in the context of quantum Regge calculus with a general triangulation. Via a stationary phase approximation, we show that the boundary state succeeds in peaking the interior in the appropriate configuration, and that boundary correlations can be computed order by order in an asymptotic expansion. Further, we show that if we replace at each simplex the exponential of the Regge action by its cosine -as expected from the semiclassical limit of spin foam models -then the contribution from the sign-reversed terms is suppressed in the semiclassical regime and the results match those of conventional Regge calculus.

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Section: Semiclassical Regime Of Spin Foam Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Semiclassical regime of Regge calculus and spin foams

Bianchi

Satz

2009

Nuclear Physics B

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“…The equation is non-linear, and we do not know the analitic form of the solution. However, as we see from (14), only the linear dependence of A η on η matters. Let us therefore expand A η in a power series in η,…”

Section: Free Propagator: Non-abelian Casementioning

confidence: 77%

“…In particular, it has been recently argued that the definition of the theory on a compact spacetime region with boundaries is more suitable to extract physical information from background independent theories [2,13,14,15]. This is perhaps the situation where the proposal of [3] appears to be more interesting.…”

Section: Consequences For Gravitymentioning

confidence: 99%

On the expansion of a quantum field theory around a topological sector

Rovelli

Speziale

2006

Gen Relativ Gravit

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The idea of treating general relativistic theories in a perturbative expansion around a topological theory has been recently put forward in the quantum gravity literature. Here we investigate the viability of this idea, by applying it to conventional Yang-Mills theory on flat spacetime. We find that the expansion around the topological theory coincides with the usual expansion around the abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. Some implications for gravity are discussed.

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“…Before going on to the specific case of gravity, we take the opportunity to first discuss field theory, and introduce what is known as the general boundary formulation of quantum mechanics [6,7,2]. In the case of field theory, instead of integrating over possible paths x(t) of a particle from time…”

Section: Field Theory and The General Boundary Formulation Of Quantummentioning

confidence: 99%

Spin Foams

Engle¹

2014

Springer Handbook of Spacetime

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Ever since special relativity, space and time have become seamlessly merged into a single entity, and space-time symmetries, such as Lorentz invariance, have played a key role in our fundamental understanding of nature. Quantum mechanics, however, did not originally conform to this new way of thinking. The original formulation of quantum mechanics, called 'canonical', involves wavefunctions, operators, Hamiltonians, and time evolution in a way that treats time very differently from space. This situation was improved by Feynman, who formulated quantum mechanics in terms of probabilities calculated by summing over amplitudes associated to classical histories -the path integral formulation of quantum mechanics. As histories are naturally space-time objects in which space and time can be viewed 'on equal footing', the path integral formulation allowed, for the first time, space-time symmetries to be manifest in a general quantum theory.The key insight of Einstein's theory of gravity, general relativity, is that gravity is spacetime geometry. Space-time geometry, the one 'background structure' -i.e., non-dynamical space-time structure -remaining after special relativity, was discovered to be dynamical and to describe the gravitational field, revealing nature to be 'background independent.' Background independence can equivalently be expressed in terms of a profound enlargement of the basic spacetime symmetry group of physics: invariance under Lorentz transformations and translations is replaced by invariance under the much larger group of space-time diffeomorphisms.We have already seen in the chapter by Sahlmann on gravity, geometry and the quantum, a canonical quantization of Einstein's gravity, and hence of geometry, in which geometric operators are derived with discrete eigenvalues [1, 2, 3]. Instead of space being a smooth continuum, we see that it comes in discrete quanta -minimal 'chunks of space.' Furthermore, as discussed in the chapter by Agullo and Corichi, when applied to cosmology, this quantum theory of gravity leads to a new understanding of the Big Bang in which usually problematic infinities are resolved, and one can actually ask what happened before the Big Bang. In spite of these successes, because it is a canonical theory, it has as a drawback that space-time symmetries, in particular space-time diffeomorphism symmetry, are not manifest. Equivalently, the preferred separation between space and time prevents full background independence from being manifest.One can ask: Is there a way to construct a path-integral formulation of quantum gravity, in which the most radical discovery of general relativity, background independence, or equivalently, space-time diffeomorphism invariance, can be fully manifest, which nevertheless retains the successes of the canonical theory? This is the question leading to the spin foam program. In answering it one must understand more carefully the relationship between the canonical and path integral formulations of quantum mechanics, and in particular how these apply to g...

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A “general boundary” formulation for quantum mechanics and quantum gravity

Cited by 153 publications

References 9 publications

Semiclassical regime of Regge calculus and spin foams

Semiclassical regime of Regge calculus and spin foams

On the expansion of a quantum field theory around a topological sector

Spin Foams

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