Searcharxiv⌕ Search

arXiv subjects

Madhavan Varadarajan

Publications and source records attributed to Madhavan Varadarajan.

At least 37 records · Page 2Linked to original sources
arXiv2010

The Hamiltonian constraint in Polymer Parametrized Field Theory

Recently, a generally covariant reformulation of 2 dimensional flat spacetime free scalar field theory known as Parameterised Field Theory was quantized using Loop Quantum Gravity (LQG) type `polymer' representations. Physical states were constructed, without intermediate regularization structures, by averaging over the group of gauge transformations generated by the constraints, the constraint algebra being a Lie algebra. We consider classically equivalent combinations of these constraints corresponding to a diffeomorphism and a Hamiltonian constraint, which, as in gravity, define a Dirac algebra. Our treatment of the quantum constraints parallels that of LQG and obtains the following results, expected to be of use in the construction of the quantum dynamics of LQG:(i) the (triangulated) Hamiltonian constraint acts only on vertices, its construction involves some of the same ambiguities as in LQG and its action on diffeomorphism invariant states admits a continuum limit (ii)if the regulating holonomies are in representations tailored to the edge labels of the state, all previously obtained physical states lie in the kernel of the Hamiltonian constraint, (iii) the commutator of two (density weight 1) Hamiltonian constraints as well as the operator correspondent of their classical Poisson bracket converge to zero in the continuum limit defined by diffeomorphism invariant states, and vanish on the Lewandowski- Marolf (LM) habitat (iv) the rescaled density 2 Hamiltonian constraints and their commutator are ill defined on the LM habitat despite the well defined- ness of the operator correspondent of their classical Poisson bracket there (v) there is a new habitat which supports a non-trivial representation of the Poisson- Lie algebra of density 2 constraints

gr-qc↗
arXiv2010

Polymer quantization of the free scalar field and its classical limit

Building on prior work, a generally covariant reformulation of free scalar field theory on the flat Lorentzian cylinder is quantized using Loop Quantum Gravity (LQG) type `polymer' representations. This quantization of the {\em continuum} classical theory yields a quantum theory which lives on a discrete spacetime lattice. We explicitly construct a state in the polymer Hilbert space which reproduces the standard Fock vacuum- two point functions for long wavelength modes of the scalar field. Our construction indicates that the continuum classical theory emerges under coarse graining. All our considerations are free of the "triangulation" ambiguities which plague attempts to define quantum dynamics in LQG. Our work constitutes the first complete LQG type quantization of a generally covariant field theory together with a semi-classical analysis of the true degrees of freedom and thus provides a perfect infinite dimensional toy model to study open issues in LQG, particularly those pertaining to the definition of quantum dynamics.

gr-qc↗
arXiv2008

On the resolution of the big bang singularity in isotropic Loop Quantum Cosmology

In contrast to previous work in the field, we construct the Loop Quantum Cosmology (LQC) of the flat isotropic model with a massless scalar field in the absence of higher order curvature corrections to the gravitational part of the Hamiltonian constraint. The matter part of the constraint contains the inverse triad operator which can be quantized with or without the use of a Thiemann- like procedure. With the latter choice, we show that the LQC quantization is identical to that of the standard Wheeler DeWitt theory (WDW) wherein there is no singularity resolution. We argue that the former choice leads to singularity resolution in the sense of a well defined, regular (backward) evolution through and beyond the epoch where the size of the universe vanishes. Our work along with that of the seminal work of Ashtekar, Pawlowski and Singh (APS) clarifies the role, in singularity resolution, of the three `exotic' structures in this LQC model, namely: curvature corrections, inverse triad definitions and the `polymer' nature of the kinematic representation. We also critically examine certain technical assumptions made by APS in their analysis of WDW semiclassical states and point out some problems stemming from the infrared behaviour of their wave functions

gr-qc↗
arXiv2008

Information is Not Lost in the Evaporation of 2-dimensional Black Holes

We analyze Hawking evaporation of the Callen-Giddings-Harvey-Strominger (CGHS) black holes from a quantum geometry perspective and show that information is not lost, primarily because the quantum space-time is sufficiently larger than the classical. Using suitable approximations to extract physics from quantum space-times we establish that: i)future null infinity of the quantum space-time is sufficiently long for the the past vacuum to evolve to a pure state in the future; ii) this state has a finite norm in the future Fock space; and iii) all the information comes out at future infinity; there are no remnants.

gr-qc↗
arXiv2008

Polymer Parametrised Field Theory

Free scalar field theory on 2 dimensional flat spacetime, cast in diffeomorphism invariant guise by treating the inertial coordinates of the spacetime as dynamical variables, is quantized using LQG type `polymer' representations for the matter field and the inertial variables. The quantum constraints are solved via group averaging techniques and, analogous to the case of spatial geometry in LQG, the smooth (flat) spacetime geometry is replaced by a discrete quantum structure. An overcomplete set of Dirac observables, consisting of (a) (exponentials of) the standard free scalar field creation- annihilation modes and (b) canonical transformations corresponding to conformal isometries, are represented as operators on the physical Hilbert space. None of these constructions suffer from any of the `triangulation' dependent choices which arise in treatments of LQG. In contrast to the standard Fock quantization, the non- Fock nature of the representation ensures that the algebra of conformal isometries as well as that of spacetime diffeomorphisms are represented in an anomaly free manner. Semiclassical states can be analysed at the gauge invariant level. It is shown that `physical weaves' necessarily underly such states and that such states display semiclassicality with respect to, at most, a countable subset of the (uncountably large) set of observables of type (a). The model thus offers a fertile testing ground for proposed definitions of quantum dynamics as well as semiclassical states in LQG.

gr-qc↗
arXiv2007

Towards new background independent representations for Loop Quantum Gravity

Recently, uniqueness theorems were constructed for the representation used in Loop Quantum Gravity. We explore the existence of alternate representations by weakening the assumptions of the so called LOST uniqueness theorem. The weakened assumptions seem physically reasonable and retain the key requirement of explicit background independence. For simplicity, we restrict attention to the case of gauge group U(1).

gr-qc↗
arXiv2006

Functional evolution of quantum cylindrical waves

Kucha{ř} showed that the quantum dynamics of (1 polarization) cylindrical wave solutions to vacuum general relativity is determined by that of a free axially-symmetric scalar field along arbitrary axially-symmetric foliations of a fixed flat 2+1 dimensional spacetime. We investigate if such a dynamics can be defined {\em unitarily} within the standard Fock space quantization of the scalar field. Evolution between two arbitrary slices of an arbitrary foliation of the flat spacetime can be built out of a restricted class of evolutions (and their inverses). The restricted evolution is from an initial flat slice to an arbitrary (in general, curved) slice of the flat spacetime and can be decomposed into (i) `time' evolution in which the spatial Minkowskian coordinates serve as spatial coordinates on the initial and the final slice, followed by (ii) the action of a spatial diffeomorphism of the final slice on the data obtained from (i). We show that although the functional evolution of (i) is unitarily implemented in the quantum theory, generic spatial diffeomorphisms of (ii) are not. Our results imply that a Tomanaga-Schwinger type functional evolution of quantum cylindrical waves is not a viable concept even though, remarkably, the more limited notion of functional evolution in Kucha{ř}'s `half parametrized formalism' is well-defined.

gr-qc↗
arXiv2006

Dirac Quantization of Parametrized Field Theory

Parametrized field theory (PFT) is free field theory on flat spacetime in a diffeomorphism invariant disguise. It describes field evolution on arbitrary foliations of the flat spacetime instead of only the usual flat ones, by treating the `embedding variables' which describe the foliation as dynamical variables to be varied in the action in addition to the scalar field. A formal Dirac quantization turns the constraints of PFT into functional Schrodinger equations which describe evolution of quantum states from an arbitrary Cauchy slice to an infinitesimally nearby one.This formal Schrodinger picture- based quantization is unitarily equivalent to the standard Heisenberg picture based Fock quantization of the free scalar field if scalar field evolution along arbitrary foliations is unitarily implemented on the Fock space. Torre and Varadarajan (TV) showed that for generic foliations emanating from a flat initial slice in spacetimes of dimension greater than 2, evolution is not unitarily implemented, thus implying an obstruction to Dirac quantization. We construct a Dirac quantization of PFT,unitarily equivalent to the standard Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are powerful enough to super-cede the no- go implications of the TV results. The key features of our quantization include an LQG type representation for the embedding variables, embedding dependent Fock spaces for the scalar field, an anomaly free representation of (a generalization of) the finite transformations generated by the constraints and group averaging techniques. The difference between 2 and higher dimensions is that in the latter, only finite gauge transformations are defined in the quantum theory, not the infinitesimal ones.

gr-qc↗
arXiv2006

A general solution for classical sequential growth dynamics of Causal Sets

A classical precursor to a full quantum dynamics for causal sets has been forumlated in terms of a stochastic sequential growth process in which the elements of the causal set arise in a sort of accretion process. The transition probabilities of the Markov growth process satisfy certain physical requirements of causality and general covariance, and the generic solution with all transition probabilities non-zero has been found. Here we remove the assumption of non-zero probabilities, define a reasonable extension of the physical requirements to cover the case of vanishing probabilities, and find the completely general solution to these physical conditions. The resulting family of growth processes has an interesting structure reminiscent of an ``infinite tower of turtles'' cosmology.

gr-qc↗
arXiv2004

The graviton vacuum as a distributional state in kinematic Loop Quantum Gravity

The quantum behaviour of weak gravitational fields admits an adequate, albeit approximate, description by those graviton states in which the expectation values and fluctuations of the linearised gravitational field are small. Such states must approximate corresponding states in full quantum gravity. We analyse the nature of this approximation for the graviton vacuum state in the context of kinematical Loop Quantum Gravity (LQG) wherein the constraints are ignored. We identify the graviton vacuum state with kinematically non-normalizable, distributional states in LQG by demanding that relations between linearised operator actions on the former are mirrored by those of their non-linear counterparts on the latter. We define a semi- norm on the space of kinematical distributions and show that the identification is approximate upto distributions which are small in this semi-norm. We argue that our candidate states are annihilated by the linearised constraints (expressed as operators in the full theory) to leading order in the parameter characterising the approximation. This suggests the possibility, in a scheme such as ours, of solving the full constraints order by order in this parameter. The main drawback of our considerations is that they depend on certain auxilliary constructions which, though mathematically well defined, do not arise from physical insight. Our work is an attempt to implement an earlier proposal of Iwasaki and Rovelli.

gr-qc↗
arXiv2004

Path integral quantization of parametrised field theory

Free scalar field theory on a flat spacetime can be cast into a generally covariant form known as parametrised field theory in which the action is a functional of the scalar field as well as the embedding variables which describe arbitrary, in general curved, foliations of the flat spacetime. We construct the path integral quantization of parametrised field theory in order to analyse issues at the interface of quantum field theory and general covariance in a path integral context. We show that the measure in the Lorentzian path integral is non-trivial and is the analog of the Fradkin- Vilkovisky measure for quantum gravity. We construct Euclidean functional integrals in the generally covariant setting of parametrised field theory using key ideas of Schleich and show that our constructions imply the existence of non-standard `Wick rotations' of the standard free scalar field 2 point function. We develop a framework to study the problem of time through computations of scalar field 2 point functions. We illustrate our ideas through explicit computation for a time independent 1+1 dimensional foliation. Although the problem of time seems to be absent in this simple example, the general case is still open. We discuss our results in the contexts of the path integral formulation of quantum gravity and the canonical quantization of parametrised field theory.

gr-qc↗
arXiv2002

Gravitons from a loop representation of linearised gravity

Loop quantum gravity is based on a classical formulation of 3+1 gravity in terms of a real SU(2) connection. Linearization of this classical formulation about a flat background yields a description of linearised gravity in terms of a {\em real} $U(1)\times U(1)\times U(1)$ connection. A `loop' representation, in which holonomies of this connection are unitary operators, can be constructed. These holonomies are not well defined operators in the standard graviton Fock representation. We generalise our recent work on photons and U(1) holonomies to show that Fock space gravitons are associated with distributional states in the $U(1)\times U(1)\times U(1)$ loop representation. Our results may illuminate certain aspects of the much deeper (and as yet unkown,) relation between gravitons and states in nonperturbative loop quantum gravity. This work leans heavily on earlier seminal work by Ashtekar, Rovelli and Smolin (ARS) on the loop representation of linearised gravity using {\em complex} connections. In the last part of this work, we show that the loop representation based on the {\em real} $U(1)\times U(1)\times U(1)$ connection also provides a useful kinematic arena in which it is possible to express the ARS complex connection- based results in the mathematically precise language currently used in the field.

gr-qc↗
arXiv2001

Photons from quantized electric flux representations

The quantum theory of U(1) connections admits a diffeomorphism invariant representation in which the electric flux through any surface is quantized. This representation is the analog of the representation of quantum SU(2) theory used in loop quantum gravity. We investigate the relation between this representation, in which the basic excitations are `polymer-like', and the Fock representation, in which the basic excitations are wave-like photons. We show that normalizable states in the Fock space are associated with `distributional' states in the quantized electric flux representation. This work is motivated by the question of how wave-like gravitons in linearised gravity arise from polymer-like states in non-perturbative loop quantum gravity.

gr-qc↗
arXiv2000

Kruskal coordinates as canonical variables for Schwarzschild black holes

We derive a transformation from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes, to new canonical variables which have the interpretation of Kruskal coordinates. We explicitly show that this transformation is non-singular, even at the horizon. The constraints of the theory simplify in terms of the new canonical variables and are equivalent to the vanishing of the canonical momenta. Our work is based on earlier seminal work by Kuchar in which he reconstructed curvature coordinates and a mass function from spherically symmetric canonical data. The key feature in our construction of a nonsingular canonical transformation to Kruskal variables, is the scaling of the curvature coordinate variables by the mass function rather than by the mass at left spatial infinity.

gr-qc↗
arXiv2000

A proposal for analyzing the classical limit of kinematic loop gravity

We analyze the classical limit of kinematic loop quantum gravity in which the diffeomorphism and hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along {\em all} possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasi-classical states themselves. We construct ``macroscopic'' operators based on such lattices and propose that these operators be used to analyze the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations. Although, in principle, the holonomies of `large' loops on these lattices could be used to analyze the classical limit, we argue that it may be simpler to base the analysis on an alternate set of ``flux'' based operators. We explicitly construct candidate quasi-classical states in 2 spatial dimensions and indicate how these constructions may generalize to 3d. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism invariant Rovelli model which couples a matter reference system to the Hussain Kucha{\v r} model.

gr-qc↗
arXiv2000

Fock representations from U(1) holonomy algebras

We revisit the quantization of U(1) holonomy algebras using the abelian C* algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of ``smeared loops'' and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering efforts to construct Fock space representations of linearised gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity.

gr-qc↗
arXiv1999

On the metric operator for quantum cylindrical waves

Every (1 polarization) cylindrical wave solution of vacuum general relativity is completely determined by a corresponding axisymmetric solution to the free scalar wave equation on an auxilliary 2+1 dimensional flat spacetime. The physical metric at radius R is determined by the energy, $γ(R)$, of the scalar field in a box (in the flat spacetime) of radius R. In a recent work, among other important results, Ashtekar and Pierri have introduced a strategy to study the quantum geometry in this system, through a regularized quantum counterpart of $γ(R)$. We show that this regularized object is a densely defined symmetric operator, thereby correcting an error in their proof of this result. We argue that it admits a self adjoint extension and show that the operator, unlike its classical counterpart, is not positive.

gr-qc↗