Dirac quantization of parametrized field theory

Varadarajan, Madhavan

doi:10.1103/physrevd.75.044018

Cited by 19 publications

(23 citation statements)

References 12 publications

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“…We showed in this paper in particular that a full sector of the theory is completely equivalent to a free scalar field, the gravity field only being there to allow for a diffeomorphism covariant formulation. This sector is actually fairly similar to what was already developed with parametrized field theories [24][25][26], although in higher dimensions and with a different language.…”

Section: Discussionsupporting

confidence: 74%

Abelian 2+1D loop quantum gravity coupled to a scalar field

Charles

2019

Gen Relativ Gravit

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In order to study 3d loop quantum gravity coupled to matter, we consider a simplified model of abelian quantum gravity, the so-called U(1) 3 model. Abelian gravity coupled to a scalar field shares a lot of commonalities with parameterized field theories. We use this to develop an exact quantization of the model. This is used to discuss solutions to various problems that plague even the 4d theory, namely the definition of an inverse metric and the role of the choice of representation for the holonomy-flux algebra.

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Section: Discussionsupporting

confidence: 74%

Abelian 2+1D loop quantum gravity coupled to a scalar field

Charles

2019

Gen Relativ Gravit

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“…One motivation for this example is to provide an interpretation for graph changing Hamiltonians appearing in loop quantum gravity [34,64] or for the parametrized scalar field [65]. This example will illustrate that refining evolution does in fact split into an embedding map and a proper evolution.…”

Section: Massless Scalar Field In a 2d Lorentzian Space Timementioning

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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“…Subsequently, he constructed a Dirac quantization of this model and used it as a toy system for canonical quantum gravity [3,4]. More recently, through Laddha and Varadarajan's analysis of PFT within the loop (polymer) quantization framework [5][6][7], this system has emerged as a robust testing ground for the quantization techniques employed in loop quantum gravity (LQG).…”

Section: Introductionmentioning

confidence: 99%

Quantum geometry with a nondegenerate vacuum: A toy model

Sengupta¹

2013

Phys. Rev. D

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Motivated by a recent proposal (by Koslowski-Sahlmann) of a kinematical representation in loop quantum gravity (LQG) with a nondegenerate vacuum metric, we construct a polymer quantization of the parametrized massless scalar field theory on a Minkowskian cylinder. The diffeomorphism covariant kinematics is based on states that carry a continuous label corresponding to smooth embedding geometries, in addition to the discrete embedding and matter labels. The physical state space, obtained through the group averaging procedure, is nonseparable. A physical state in this theory can be interpreted as a quantum spacetime, which is composed of discrete strips and supersedes the classical continuum. We find that the conformal group is broken in the quantum theory and consists of all Poincaré translations. These features are remarkably different compared to the case without a smooth embedding. Finally, we analyze the length operator whose spectrum is shown to be a sum of contributions from the continuous and discrete embedding geometries, being in perfect analogy with the spectra of geometrical operators in LQG with a nondegenerate vacuum geometry.

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Dirac quantization of parametrized field theory

Cited by 19 publications

References 12 publications

Abelian 2+1D loop quantum gravity coupled to a scalar field

Abelian 2+1D loop quantum gravity coupled to a scalar field

Time evolution as refining, coarse graining and entangling

Quantum geometry with a nondegenerate vacuum: A toy model

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