arXiv subjects

Madhavan Varadarajan

Publications and source records attributed to Madhavan Varadarajan.

At least 19 recordsLinked to original sources

arXiv2026

Area discreteness, Lorentz covariance and Hilbert space non-separability

We show how quantum discreteness of spatial area is consistent with a unitary implementation of Lorentz boosts in an LQG type quantization of a diffeomorphism invariant reformulation of free scalar field theory on 2d flat spacetime known as Parameterized Field Theory (PFT). This consistency is a result of Hilbert space non-separability which is a characteristic feature of LQG representations. Our results suggest a possible interpretation of Hilbert space non-separability in terms of observer perspectives.

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arXiv2024

Spherical collapse and black hole evaporation

We consider spherically symmetric gravity coupled to a spherically symmetric scalar field with a specific coupling which depends on the Areal Radius. Classical collapse is described by the Vaidya solution. The semiclassical Einstein equations are well defined. Their solution describes an evaporating black hole. A balance law at future null infinity relates the rate of change of a back reaction-corrected Bondi mass to a manifestly positive flux. Its detailed form suggests a quantum extension of future null infinity along which all information including that in the collapsing matter is expected to emerge.

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arXiv2022

Anomaly free quantum dynamics for Euclidean LQG

Classical gravitational evolution admits an elegant and compact re-expression in terms of gauge covariant generalizations of Lie derivatives with respect to a spatial phase space dependent $su(2)$ valued vector field called the Electric Shift [1]. A quantum dynamics for Euclidean Loop Quantum Gravity which ascribes a central role to the Electric Shift operator is derived in [2]. $Here\; we\; show\; that\; this \;quantum\; dynamics\; is\; nontrivially\; anomaly$ $free$. Specifically, we show that on a suitable space of off shell states (a) the (non-vanishing) commutator between a pair of Hamiltonian constraint operators mirrors the Poisson bracket between their classical correspondents, (b) the group of finite spatial diffeomorphisms is faithfully represented and (c) the action of the Hamiltonian constraint operator is diffeomorphism covariant with respect to the action of spatial diffeomorphisms.

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arXiv2021

On propagation in Loop Quantum Gravity

A rigorous implementation of the Wheeler-Dewitt equations was derived in the context of Loop Quantum Gravity (LQG) and was coined Quantum Spin Dynamics (QSD). The Hamiltonian constraint of QSD was criticised as being too local and to prevent "propagation" in canonical LQG. That criticism was based on an algorithm developed for QSD for generating solutions to the Wheeler-DeWitt equations. The fine details of that algorithm could not be worked out because the QSD Hamiltonian constraint makes crucial use of the volume operator which cannot be diagonalised analytically. In this paper, we consider the U(1)$^3$ model for Euclidean vacuum LQG which consists in replacing the structure group SU(2) by U(1)$^3$ and otherwise keeps all properties of the SU(2) theory intact. This enables analytical calculations and the fine details of the algorithm {\it can} be worked out. By considering one of the simplest possible non-trivial class of solutions based on very small graphs, we show that 1. an infinite number of solutions {\it exist} which are 2. generically {\it not normalisable} with respect to the inner product on the space of spatially diffeomorphism invariant distributions and 3. generically display {\it propagation}. Due to the closeness of the U(1)$^3$ model to Euclidean LQG, it is extremely likely that all three properties hold also in the SU(2) case and even more so in physical Lorentzian LQG. These arguments can in principle be made water tight using modern numerical (e.g. ML or QC) methods combined with the techniques developed in this paper which we reserve for future work.

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arXiv2021

Gravitational dynamics: A novel shift in the Hamiltonian paradigm

It is well known that Einstein's equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that {\em time} evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in {\em spatial} directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomorphism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the `integrable' (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the `double copy' ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.

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arXiv2021

Euclidean LQG Dynamics: An Electric Shift in Perspective

Loop Quantum Gravity (LQG) is a non-perturbative attempt at quantization of a classical phase space description of gravity in terms of $SU(2)$ connections and electric fields. As emphasized recently [1], on this phase space, classical gravitational evolution in $time$ can be understood in terms of certain gauge covariant generalizations of Lie derivatives with respect to a $spatial$ $SU(2)$ Lie algebra valued vector field called the Electric Shift. We present a derivation of a quantum dynamics for Euclidean LQG which is informed by this understanding. In addition to the physically motivated nature of the action of the Euclidean Hamiltonian constraint so derived, the derivation implies that the spin labels of regulating holonomies are determined by corresponding labels of the spin network state being acted upon thus eliminating the `spin $j$-ambiguity' pointed out by Perez. By virtue of Thiemann's seminal work, the Euclidean quantum dynamics plays a crucial role in the construction of the Lorentzian quantum dynamics so that our considerations also have application to Lorentzian LQG.

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arXiv2019

On quantum propagation in Smolin's weak coupling limit of 4d Euclidean Gravity

Two desireable properties of a quantum dynamics for Loop Quantum Gravity (LQG) are that its generators provide an anomaly free representation of the classical constraint algebra and that physical states which lie in the kernel of these generators encode propagation. A physical state in LQG is expected to be a sum over graphical $SU(2)$ spin network states. By propagation we mean that a quantum perturbation at one vertex of a spin network state propagates to vertices which are `many links away' thus yielding a new spin network state which is related to the old one by this propagation. A physical state encodes propagation if its spin network summands are related by propagation. Here we study propagation in an LQG quantization of Smolin's weak coupling limit of Euclidean Gravity based on graphical $U(1)^3$ `charge' network states. Building on our earlier work on anomaly free quantum constraint actions for this system, we analyse the extent to which physical states encode propagation. In particular, we show that a slight modification of the constraint actions constructed in our previous work leads to physical states which encode robust propagation. Under appropriate conditions, this propagation merges, seperates and entangles vertices of charge network states. The `electric' diffeomorphism constraints introduced in prevous work play a key role in our considerations. The main import of our work is that there are choices of quantum constraint constructions through LQG methods which are consistent with vigorous propagation thus providing a counterpoint to Smolin's early observations on the difficulties of propagation in the context of LQG type operator constructions. Whether the choices considered in this work are physically appropriate is an open question worthy of further study.

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arXiv2018

From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier

We construct a positive complexifier, differentiable almost everywhere on the classical phase space of real triads and $SU(2)$ connections, which generates a Wick Transform from Euclidean to Lorentzian gravity everywhere except on a phase space set of measure zero. This Wick transform assigns an equal role to the self dual and anti-self dual Ashtekar variables in quantum theory. We argue that the appropriate quantum arena for an analysis of the properties of the Wick rotation is the diffeomorphism invariant Hilbert space of Loop Quantum Gravity (LQG) rather than its kinematic Hilbert space. We examine issues related to the construction, in quantum theory, of the positive complexifier as a positive operator on this diffeomorphism invariant Hilbert space. Assuming the existence of such an operator, we explore the possibility of identifying physical states in Lorentzian LQG as Wick rotated images of physical states in the Euclidean theory. Our considerations derive from Thiemann's remarkable proposal to define Lorentzian LQG from Euclidean LQG via the implementation in quantum theory of a phase space `Wick rotation' which maps real Ashtekar-Barbero variables to Ashtekar's complex, self dual variables.

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arXiv2018

The constraint algebra in Smolins' $G\rightarrow 0$ limit of 4d Euclidean Gravity

Smolin's generally covariant $G_{\mathrm{Newton}}\rightarrow0$ limit of 4d Euclidean gravity is a useful toy model for the study of the constraint algebra in Loop Quantum Gravity. In particular, the commutator between its Hamiltonian constraints has a metric dependent structure function. While a prior LQG like construction of non-trivial anomaly free constraint commutators for the model exists, that work suffers from two defects. First, Smolin's remarks on the inability of the quantum dynamics to generate propagation effects apply. Second, the construction only yields the action of a single Hamiltonian constraint together with the action of its commutator through a continuum limit of corresponding discrete approximants; the continuum limit of a product of 2 or more constraints does not exist. Here, we incorporate changes in the quantum dynamics through structural modifications in the choice of discrete approximants to the quantum Hamiltonian constraint. The new structure is motivated by that responsible for propagation in an LQG like quantization of Paramaterized Field Theory and significantly alters the space of physical states. We study the off shell constraint algebra of the model in the context of these structural changes and show that the continuum limit action of multiple products of Hamiltonian constraints is (a) supported on an appropriate domain of states (b) yields anomaly free commutators between pairs of Hamiltonian constraints and (c) is diffeomorphism covariant. Many of our considerations seem robust enough to be applied to the setting of 4d Euclidean gravity.

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arXiv2016

Propagation in Polymer Parameterised Field Theory

The Hamiltonian constraint operator in Loop Quantum Gravity acts ultralocally. Smolin has argued that this ultralocality seems incompatible with the existence of a quantum dynamics which propagates perturbations between macroscopically seperated regions of quantum geometry. We present evidence to the contrary within an LQG type `polymer' quantization of two dimensional Parameterised Field Theory (PFT). PFT is a generally covariant reformulation of free field propagation on flat spacetime. We show explicitly that while, as in LQG, the Hamiltonian constraint operator in PFT acts ultralocally, states in the joint kernel of the Hamiltonian and diffeomorphism constraints of PFT necessarily describe propagation effects. The particular structure of the finite triangulation Hamiltonian constraint operator plays a crucial role, as does the necessity of imposing (the continuum limit of) its kinematic adjoint as a constraint. Propagation is seen as a property encoded by physical states in the kernel of the constraints rather than that of repeated actions of the finite triangulation Hamiltonian constraint on kinematic states. The analysis yields robust structural lessons for putative constructions of the Hamiltonian constraint in LQG for which ultralocal action co-exists with a description of propagation effects by physical states.

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arXiv2016

A Note on Entanglement Entropy, Coherent States and Gravity

The entanglement entropy of a free quantum field in a coherent state is independent of its stress energy content. We use this result to highlight the fact that while the Einstein equations for first order variations about a locally maximally symmetric vacuum state of geometry and quantum fields seem to follow from Jacobson's principle of maximal vacuum entanglement entropy, their possible derivation from this principle for the physically relevant case of finite but small variations remains an open issue. We also apply this result to the context of Bianchi's identification, independent of unknown Planck scale physics, of the first order variation of Bekenstein Hawking area with that of vacuum entanglement entropy. We argue that under certain technical assumptions this identification seems not to be extendible to the context of finite but small variations to coherent states. Our particular method of estimation of entanglement entropy variation reveals the existence of certain contributions over and above those appearing in Jacobson's and Bianchi's works. We discuss the sense in which these contributions may be subleading to those already present in these works.

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arXiv2014

A quantum kinematics for asymptotically flat spacetimes

We construct a quantum kinematics for asymptotically flat spacetimes based on the Koslowski-Sahlmann (KS) representation. The KS representation is a generalization of the representation underlying Loop Quantum Gravity (LQG) which supports, in addition to the usual LQG operators, the action of `background exponential operators' which are connection dependent operators labelled by `background' $su(2)$ electric fields. KS states have, in addition to the LQG state label corresponding to 1 dimensional excitations of the triad, a label corresponding to a `background' electric field which describes 3 dimensional excitations of the triad. Asymptotic behaviour in quantum theory is controlled through asymptotic conditions on the background electric fields which label the {\em states} and the background electric fields which label the {\em operators}. Asymptotic conditions on the triad are imposed as conditions on the background electric field state label while confining the LQG spin net graph labels to compact sets. We show that KS states can be realised as wave functions on a quantum configuration space of generalized connections and that the asymptotic behaviour of each such generalized connection is determined by that of the background electric fields which label the background exponential operators. Similar to the spatially compact case, the Gauss Law and diffeomorphism constraints are then imposed through Group Averaging techniques to obtain a large sector of gauge invariant states. It is shown that this sector supports a unitary action of the group of asymptotic rotations and translations and that, as anticipated by Friedman and Sorkin, for appropriate spatial topology, this sector contains states which display fermionic behaviour under $2π$ rotations.

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arXiv2014

The Koslowski-Sahlmann representation: Quantum Configuration Space

The Koslowski-Sahlmann (KS) representation is a generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity (LQG), to accommodate states labelled by smooth spatial geometries. As shown recently, the KS representation supports, in addition to the action of the holonomy and flux operators, the action of operators which are the quantum counterparts of certain connection dependent functions known as "background exponentials". Here we show that the KS representation displays the following properties which are the exact counterparts of LQG ones: (i) the abelian $*$ algebra of $SU(2)$ holonomies and `$U(1)$' background exponentials can be completed to a $C^*$ algebra (ii) the space of semianalytic $SU(2)$ connections is topologically dense in the spectrum of this algebra (iii) there exists a measure on this spectrum for which the KS Hilbert space is realised as the space of square integrable functions on the spectrum (iv) the spectrum admits a characterization as a projective limit of finite numbers of copies of $SU(2)$ and $U(1)$ (v) the algebra underlying the KS representation is constructed from cylindrical functions and their derivations in exactly the same way as the LQG (holonomy-flux) algebra except that the KS cylindrical functions depend on the holonomies and the background exponentials, this extra dependence being responsible for the differences between the KS and LQG algebras. While these results are obtained for compact spaces, they are expected to be of use for the construction of the KS representation in the asymptotically flat case.

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arXiv2013

The Koslowski- Sahlmann representation: Gauge and diffeomorphism invariance

The discrete spatial geometry underlying Loop Quantum Gravity (LQG) is degenerate almost everywhere. This is at apparent odds with the non- degeneracy of asymptotically flat metrics near spatial infinity. Koslowski generalised the LQG representation so as to describe states labelled by smooth non- degenerate triad fields. His representation was further studied by Sahlmann with a view to imposing gauge and spatial diffeomorphism invariance through group averaging methods. Motivated by the desire to model asymptotically flat quantum geometry by states with triad labels which are non- degenerate at infinity but not necessarily so in the interior, we initiate a generalisation of Sahlmann's considerations to triads of varying degeneracy. In doing so, we include delicate phase contributions to the averaging procedure which are crucial for the correct implementation of the gauge and diffeomorphism constraints, and whose existence can be traced to the background exponential functions recently constructed by one of us. Our treatment emphasizes the role of symmetries of quantum states in the averaging procedure. Semianalyticity, influential in the proofs of the beautiful uniqueness results for LQG, plays a key role in our considerations. As a by product, we re- derive the group averaging map for standard LQG, highlighting the role of state symmetries and explicitly exhibiting the essential uniqueness of its specification.

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arXiv2013

The generator of spatial diffeomorphisms in the Koslowski- Sahlmann representation

A generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity, to accomodate states labelled by smooth spatial geometries, was discovered by Koslowski and further studied by Sahlmann. We show how to construct the diffeomorphism constraint operator in this Koslowski- Sahlmann (KS) representation from suitable connection and triad dependent operators. We show that the KS representation supports the action of hitherto unnoticed "background exponential" operators which, in contrast to the holonomy- flux operators, change the smooth spatial geometry label of the states. These operators are shown to be quantizations of certain connection dependent functions and play a key role in the construction of the diffeomorphism constraint operator.

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arXiv2012

Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity

The G -->0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)xU(1)xU(1) gauge theory. The Poisson bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. Motivated by recent results in Parameterized Field Theory and by the search for an anomaly-free quantum dynamics for Loop Quantum Gravity (LQG), the quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1) theory is constructed so as to produce a non-trivial LQG-type representation of its Poisson brackets through the following steps. First, the constraint at finite triangulation, as well as the commutator between a pair of such constraints, are constructed as operators on the `charge' network basis. Next, the continuum limit of the commutator is evaluated with respect to an operator topology defined by a certain space of `vertex smooth' distributions. Finally, the operator corresponding to the Poisson bracket between a pair of Hamiltonian constraints is constructed at finite triangulation in such a way as to generate a `generalised' diffeomorphism and its continuum limit is shown to agree with that of the commutator between a pair of finite triangulation Hamiltonian constraints. Our results in conjunction with the recent work of Henderson, Laddha and Tomlin in a 2+1-dimensional context, constitute the necessary first steps toward a satisfactory treatment of the quantum dynamics of this model.

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arXiv2012

Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity: Diffeomorphism Covariance

The G-->0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)xU(1)xU(1) gauge theory. In an earlier paper, Tomlin and Varadarajan constructed the quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1) theory so as to produce a non-trivial anomaly free LQG-type representation of the Poisson bracket between a pair of Hamiltonian constraints. These constructions involved a choice of regulating coordinate patches. The use of these coordinate patches is in apparent conflict with spatial diffeomorphism covariance. In this work we show how an appropriate choice of coordinate patches together with suitable modifications of these constructions results in the diffeomorphism covariance of the continuum limit action of the Hamiltonian constraint operator, while preserving the anomaly free property of the continuum limit action of its commutator.

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arXiv2011

The Diffeomorphism Constraint Operator in Loop Quantum Gravity

We construct the smeared diffeomorphism constraint operator at finite triangulation from the basic holonomy- flux operators of Loop Quantum Gravity, evaluate its continuum limit on the Lewandowski- Marolf habitat and show that the action of the continuum operator provides an anomaly free representation of the Lie algebra of diffeomorphisms of the 3- manifold. Key features of our analysis include: (i) finite triangulation approximants to the curvature, $F_{ab}^i$ of the Ashtekar- Barbero connection which involve not only small loop holonomies but also small surface fluxes as well as an explicit dependence on the edge labels of the spin network being acted on (ii) the dependence of the small loop underlying the holonomy on both the direction and magnitude of the shift vector field (iii) continuum constraint operators which do {\em not} have finite action on the kinematic Hilbert space, thus implementing a key lesson from recent studies of parameterised field theory by the authors. Features (i) and (ii) provide the first hints in LQG of a conceptual similarity with the so called "mu- bar" scheme of Loop Quantum Cosmology. We expect our work to be of use in the construction of an anomaly free quantum dynamics for LQG.

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