Relating covariant and canonical approaches to triangulated models of quantum gravity

Arnsdorf, Matthias

doi:10.1088/0264-9381/19/6/304

Cited by 10 publications

(15 citation statements)

References 34 publications

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“…A final interesting question arises: can one construct from the ideas presented here a canonical formulation of Causal Dynamical Triangulations (CDT) (see also [45])? Given that a priori all new lengths in 4D canonical Regge calculus can be freely chosen and, in particular, all be fixed equal to one while the births of baby universes are disallowed, can one construct 'classical CDT solutions' from this, such that the pre-and post-constraints determine the connectivity by rejecting or accepting certain Pachner moves?…”

Section: Discussion and Outlookmentioning

confidence: 99%

Canonical simplicial gravity

Dittrich

Hoehn

2012

Class. Quantum Grav.

254

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A general canonical formalism for discrete systems is developed which can handle varying phase space dimensions and constraints. The central ingredient is Hamilton's principal function which generates canonical time evolution and ensures that the canonical formalism reproduces the dynamics of the covariant formulation following directly from the action. We apply this formalism to simplicial gravity and (Euclidean) Regge calculus, in particular. A discrete forward/backward evolution is realized by gluing/removing single simplices step by step to/from a bulk triangulation and amounts to Pachner moves in the triangulated hypersurfaces. As a result, the hypersurfaces evolve in a discrete 'multi-fingered' time through the full Regge solution. Pachner moves are an elementary and ergodic class of homeomorphisms and generically change the number of variables, but can be implemented as canonical transformations on naturally extended phase spaces. Some moves introduce a priori free data which, however, may become fixed a posteriori by constraints arising in subsequent moves. The end result is a general and fully consistent formulation of canonical Regge calculus, thereby removing a longstanding obstacle in connecting covariant simplicial gravity models to canonical frameworks. The present scheme is, therefore, interesting in view of many approaches to quantum gravity, but may also prove useful for numerical implementations.

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Section: Discussion and Outlookmentioning

confidence: 99%

Canonical simplicial gravity

Dittrich

Hoehn

2012

Class. Quantum Grav.

254

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“…Indeed, if a unitary operator has all nonnegative matrix elements in some basis it must be rather trivial, simply acting as a permutation of the basis. However, thanks to the 'problem of time' in quantum gravity, it is more natural to use a sum over spin foams to compute, not a time evolution operator, but the projection from the space spanned by spin networks onto the physical Hilbert space of quantum gravity [4,25]. There is no reason for this to be unitary; it should be something more like a positive operator.…”

Section: Discussionmentioning

confidence: 99%

Positivity of spin foam amplitudes

Baez

Christensen

2002

Class. Quantum Grav.

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Abstract. The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols appearing for each vertex, and simpler factors for the edges and faces. We prove that these amplitudes are always nonnegative for closed spin foams. As a corollary, all open spin foams going between a fixed pair of spin networks have real amplitudes of the same sign. This means one can use the Metropolis algorithm to compute expectation values of observables in the Riemannian Barrett-Crane model, as in statistical mechanics, even though this theory is based on a real-time (e iS ) rather than imaginary-time (e −S ) path integral. Our proof uses the fact that when the Riemannian 10j symbols are nonzero, their sign is positive or negative depending on whether the sum of the ten spins is an integer or half-integer. For the product of 10j symbols appearing in the amplitude for a closed spin foam, these signs cancel. We conclude with some numerical evidence suggesting that the Lorentzian 10j symbols are always nonnegative, which would imply similar results for the Lorentzian Barrett-Crane model.

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“…In [23], in fact, a discretization of the projector operator was proposed, leading to an expression very close to that provided by spin foam models.…”

Section: The Barrett-crane Model As a Realization Of The Projector Opmentioning

confidence: 90%

“…The expression above is then interpreted as a sum over triangulations and the operators U ǫ,l , being (a sum over) local evolution operators, implement the action of the projector operator on a given spin network and are identified with the quantum operators corresponding to given Pachner moves in the spin foam case [23]. The integral over T is then the analogue of the integral over the algebraic variables in the spin foam model.…”

Section: The Barrett-crane Model As a Realization Of The Projector Opmentioning

confidence: 99%

Implementing causality in the spin foam quantum geometry

2003

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We analyse the classical and quantum geometry of the Barrett-Crane spin foam model for four dimensional quantum gravity, explaining why it has to be considering as a covariant realization of the projector operator onto physical quantum gravity states. We discuss how causality requirements can be consistently implemented in this framework, and construct causal transiton amplitudes between quantum gravity states, i.e. realising in the spin foam context the Feynman propagator between states. The resulting causal spin foam model can be seen as a path integral quantization of Lorentzian first order Regge calculus, and represents a link between several approaches to quantum gravity as canonical loop quantum gravity, sum-over-histories formulations, dynamical triangulations and causal sets. In particular, we show how the resulting model can be rephrased within the framework of quantum causal sets (or histories).2

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Relating covariant and canonical approaches to triangulated models of quantum gravity

Cited by 10 publications

References 34 publications

Canonical simplicial gravity

Canonical simplicial gravity

Positivity of spin foam amplitudes

Implementing causality in the spin foam quantum geometry

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