4,982 research outputs found
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
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.Comment: 56 pages, No figure
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 functionsComment: 26 pages, no figure
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 constraintsComment: 53 page
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