Diffeomorphism symmetry in simplicial quantum gravity

Lehto, M.; Nielsen, Holger Bech; Ninomiya, M.

doi:10.1016/0550-3213(86)90349-4

Cited by 24 publications

(16 citation statements)

References 10 publications

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“…Thus one reason to take the refinement limit is actually to restore the diffeomorphism symmetry [61,23], as is also used in the perfect action program [56] for lattice QCD with regard to Lorentz symmetry. There are a number of arguments for such a restoration: one is that the pseudo gauge modes should have a small lattice correlation lengths and decouple in the continuum limit [61]. Another is that for instance for Regge calculus with flat building blocks the eigenvalues of the Hessian of the action associated to the pseudo gauge modes scale with the curvature per building block of the solution [15].…”

Section: (Decorated) Tensor Network Renormalization For Spin Nets Andmentioning

confidence: 99%

The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory

Dittrich¹

2017

100 Years of General Relativity

173

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The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We discuss iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme constructs a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit. Solving the dynamics of loop quantum gravityLoop quantum gravity led to a rigorous non-perturbative framework, in which to formulate the dynamics of quantum gravity. It allowed fascinating insights into quantum geometry and a possible structure of quantum space time. To get a complete picture of the dynamics of the theory -in the form of constructing the so-called physical Hilbert space of wave functions satisfying the Wheeler deWitt equation -we need to construct the continuum limit. In the framework presented here physical states, i.e. solutions of the Wheeler deWitt equations which give the equations of motions of the theory, are constructed by taking the refinement limit via a coarse graining procedure.The conceptual underpinnings of this framework rely on the inductive limit Hilbert space construction used in loop quantum gravity to define the continuum (so far kinematical) Hilbert space. We point out the powerful concept of this inductive limit construction if one allows for a generalization of the refinement maps that define the inductive limit Hilbert spaces. It leads to a framework in which physical states are computed in a truncation scheme, where the type of truncation is determined by the dynamics itself.This procedure allows for an understanding of the dynamics of quantum gravity on all scaleswhere a notion of scale is given by the coarseness or fineness of configurations. The different scales of the theory are connected via the cylindrical consistency condition inherent in the inductive limit construction. This replaces the notion of renormalization flow in theories with a background scale.We start our considerations with a short explanation of the inductive limit construction in section 2 and discuss the difference between kinematical and dynamical understanding of the continuum limit. In section 3 we start with the task to construct the physical Hilbert space of the theory and explain that it necessitates the construction of the refinement limit for the dynamics of the theory. This results in an iterative coarse graining scheme, in which physical states -or amplitude maps -are constructed in a certain truncation, labelled by the coarseness or fineness of the discrete structures involved. The relation of this scheme with a renormalization flow is clarified in section 4. Concrete realizations of this scheme in the form of (decorated) tensor network methods are shortly explained in section 5. We then point out the powerful notion of diffeomorphism symmetry for discrete systems ...

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Section: (Decorated) Tensor Network Renormalization For Spin Nets Andmentioning

confidence: 99%

The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory

Dittrich¹

2017

100 Years of General Relativity

173

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“…But without a realization of diffeomorphism invariance in the path integral (1.1), it cannot act as a projector onto the physical states. To deal with this issue, one attempts to restore diffeomorphism invariance via refining the building blocks and finding effective amplitudes for the coarser building blocks by integrating out the finer degrees of freedom [35,36]. We usually refer to this as coarse graining flow (although the initial step is a refining).…”

Section: Overviewmentioning

confidence: 99%

Time evolution as refining, coarse graining and entangling

2014

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We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows construction of a (cylindrically) consistent continuum limit of the theory.In this paper we point out that time evolution maps, which appear in simplicial discretizations [13,14], can also be interpreted as refining and coarse graining maps. As we will argue here, this applies in particular to gravitational dynamics, e.g. spin foams [15][16][17][18].One reason why the appearance of time evolution as coarse graining or refining maps applies in particular to gravitational or other diffeomorphism invariant systems is the following: As argued in [19][20][21][22][23], diffeomorphism symmetry in discrete systems translates to a symmetry, which can be interpreted as moving vertices in the discrete space time described by the dynamical variables of the theory. These vertex translations can also be understood as time evolution. Now, vertices can be even moved on top of each other, which gives a coarse graining of the underlying state. Alternatively, vertices can split into two and in this way give a refinement. Indeed, this argument was used in [23] to show that diffeomorphism symmetry implies discretization independence.More generally, diffeomorphism invariant systems are totally constrained, i.e. the Hamiltonian is given by a combination of constraints. In the case of a totally constrained system, the time evolution operator should be a projection operator [24,25], projecting onto socalled physical states. Thus physical states should not evolve 1 .For discrete time evolutions that change the number of degrees of freedom, this leads to the puzzle of how to identify states from Hilbert spaces of 'different size' 2 . We will argue that such states describe indeed the same physical state, expressed, however on two different discretizations. The equivalence relation is provided by the refining time evolution operator. We will explain how this notion can be formalized into the construction of an inductive limit Hilbert space. Such an inductive limit construction is also used for the (kinematical) Hilbert space of loop quantum gravity [27,28].The inductive limit Hilbert spaces, however, which are defined via an equivalence relation between states from Hilbert spaces based on different discretizations, require (so-called cylindrical) consistency conditions: Physical observables should not depend on which representative they have been determined on. Indeed, we will connect these consistency conditions with a notion of path independence for (refining) time evolution. This relates, then, to the requirement of diffeomorphism invariance.Discrete (non-topological) theories typically break the diffeomorphism symmetry [22]. The hope, however, is that diffeomorphism symmetry can be recovered in the continuum limit. We will ex...

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“…These variations in edge-lengths that preserve geometry are the simplicial analogs of diffeomorphisms [13,14]. Further, for large triangulations where the local geometry is near to flat we expect there to be approximate simplicial diffeomorphisms -small changes in the edge-lengths which approximately preserve the geometry [13,15,16]. Thus, the continuum limit of T (M) is not the superspace of three-geometries but the space of three-metrics.…”

Section: A Definitionmentioning

confidence: 99%

Signature of the simplicial supermetric

Hartle

Miller

Williams³

1997

Class. Quantum Grav.

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We investigate the signature of the Lund-Regge metric on spaces of simplicial three-geometries which are important in some formulations of quantum gravity. Tetrahedra can be joined together to make a three-dimensional piecewise linear manifold. A metric on this manifold is specified by assigning a flat metric to the interior of the tetrahedra and values to their squared edge-lengths. The subset of the space of squared edge-lengths obeying triangle and analogous inequalities is simplicial configuration space. We derive the Lund-Regge metric on simplicial configuration space and show how it provides the shortest distance between simplicial three-geometries among all choices of gauge inside the simplices for defining this metric (Regge gauge freedom). We show analytically that there is always at least one physical timelike direction in simplicial configuration space and provide a lower bound on the number of spacelike directions. We show that in the neighborhood of points in this space corresponding to flat metrics there are spacelike directions corresponding to gauge * Permanent address:

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Diffeomorphism symmetry in simplicial quantum gravity

Cited by 24 publications

References 10 publications

The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory

The Continuum Limit of Loop Quantum Gravity: A Framework for Solving the Theory

Time evolution as refining, coarse graining and entangling

Signature of the simplicial supermetric

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