71 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
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
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.Comment: 57 pages, 9 figure
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,
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.Comment: 37 pages, 6 figure
A Comment on the Degrees of Freedom in the Ashtekar Formulation for 2+1 Gravity
We show that the recent claim that the 2+1 dimensional Ashtekar formulation
for General Relativity has a finite number of physical degrees of freedom is
not correct.Comment: 6 pages LaTex, to appear in Classical and Quantum Gravit
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