1,046 research outputs found
Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
This is the fourth paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. We now move on to free field theories with constraints, namely Maxwell
theory and linearized gravity. Since the Master constraint involves squares of
constraint operator valued distributions, one has to be very careful in doing
that and we will see that the full flexibility of the Master Constraint
Programme must be exploited in order to arrive at sensible results.Comment: 23 pages, no figure
Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
This is the second paper in our series of five in which we test the Master
Constraint Programme for solving the Hamiltonian constraint in Loop Quantum
Gravity. In this work we begin with the simplest examples: Finite dimensional
models with a finite number of first or second class constraints, Abelean or
non -- Abelean, with or without structure functions.Comment: 23 pages, no figure
Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories
This is the final fifth paper in our series of five in which we test the
Master Constraint Programme for solving the Hamiltonian constraint in Loop
Quantum Gravity. Here we consider interacting quantum field theories,
specificlly we consider the non -- Abelean Gauss constraints of Einstein --
Yang -- Mills theory and 2+1 gravity. Interestingly, while Yang -- Mills theory
in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field
theory on Minkowski space, in background independent quantum field theories
such as Loop Quantum Gravity (LQG) this might become possible by working in a
new, background independent representation.Comment: 20 pages, no figure
Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was
proposed as a classically equivalent way to impose the infinite number of
Wheeler -- DeWitt constraint equations in terms of a single Master Equation.
While the proposal has some promising abstract features, it was until now
barely tested in known models. In this series of five papers we fill this gap,
thereby adding confidence to the proposal. We consider a wide range of models
with increasingly more complicated constraint algebras, beginning with a finite
dimensional, Abelean algebra of constraint operators which are linear in the
momenta and ending with an infinite dimensional, non-Abelean algebra of
constraint operators which closes with structure functions only and which are
not even polynomial in the momenta. In all these models we apply the Master
Constraint Programme successfully, however, the full flexibility of the method
must be exploited in order to complete our task. This shows that the Master
Constraint Programme has a wide range of applicability but that there are many,
physically interesting subtleties that must be taken care of in doing so. In
this first paper we prepare the analysis of our test models by outlining the
general framework of the Master Constraint Programme. The models themselves
will be studied in the remaining four papers. As a side result we develop the
Direct Integral Decomposition (DID) for solving quantum constraints as an
alternative to Refined Algebraic Quantization (RAQ).Comment: 42 pages, no figure
Gauge Field Theory Coherent States (GCS) : I. General Properties
In this article we outline a rather general construction of diffeomorphism
covariant coherent states for quantum gauge theories.
By this we mean states , labelled by a point (A,E) in the
classical phase space, consisting of canonically conjugate pairs of connections
A and electric fields E respectively, such that (a) they are eigenstates of a
corresponding annihilation operator which is a generalization of A-iE smeared
in a suitable way, (b) normal ordered polynomials of generalized annihilation
and creation operators have the correct expectation value, (c) they saturate
the Heisenberg uncertainty bound for the fluctuations of and
(d) they do not use any background structure for their definition, that is,
they are diffeomorphism covariant.
This is the first paper in a series of articles entitled ``Gauge Field Theory
Coherent States (GCS)'' which aim at connecting non-perturbative quantum
general relativity with the low energy physics of the standard model. In
particular, coherent states enable us for the first time to take into account
quantum metrics which are excited {\it everywhere} in an asymptotically flat
spacetime manifold. The formalism introduced in this paper is immediately
applicable also to lattice gauge theory in the presence of a (Minkowski)
background structure on a possibly {\it infinite lattice}.Comment: 40 pages, LATEX, no figure
QSD VI : Quantum Poincar\'e Algebra and a Quantum Positivity of Energy Theorem for Canonical Quantum Gravity
We quantize the generators of the little subgroup of the asymptotic
Poincar\'e group of Lorentzian four-dimensional canonical quantum gravity in
the continuum. In particular, the resulting ADM energy operator is densely
defined on an appropriate Hilbert space, symmetric and essentially
self-adjoint. Moreover, we prove a quantum analogue of the classical positivity
of energy theorem due to Schoen and Yau. The proof uses a certain technical
restriction on the space of states at spatial infinity which is suggested to us
given the asymptotically flat structure available. The theorem demonstrates
that several of the speculations regarding the stability of the theory,
recently spelled out by Smolin, are false once a quantum version of the
pre-assumptions underlying the classical positivity of energy theorem is
imposed in the quantum theory as well. The quantum symmetry algebra
corresponding to the generators of the little group faithfully represents the
classical algebra.Comment: 24p, LATE
Gauge Field Theory Coherent States (GCS) : II. Peakedness Properties
In this article we apply the methods outlined in the previous paper of this
series to the particular set of states obtained by choosing the complexifier to
be a Laplace operator for each edge of a graph. The corresponding coherent
state transform was introduced by Hall for one edge and generalized by
Ashtekar, Lewandowski, Marolf, Mour\~ao and Thiemann to arbitrary, finite,
piecewise analytic graphs. However, both of these works were incomplete with
respect to the following two issues : (a) The focus was on the unitarity of the
transform and left the properties of the corresponding coherent states
themselves untouched. (b) While these states depend in some sense on
complexified connections, it remained unclear what the complexification was in
terms of the coordinates of the underlying real phase space. In this paper we
resolve these issues, in particular, we prove that this family of states
satisfies all the usual properties : i) Peakedness in the configuration,
momentum and phase space (or Bargmann-Segal) representation, ii) Saturation of
the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These
states therefore comprise a candidate family for the semi-classical analysis of
canonical quantum gravity and quantum gauge theory coupled to quantum gravity,
enable error-controlled approximations and set a new starting point for {\it
numerical canonical quantum general relativity and gauge theory}. The text is
supplemented by an appendix which contains extensive graphics in order to give
a feeling for the so far unknown peakedness properties of the states
constructed.Comment: 70 pages, LATEX, 29 figure
Regularized Hamiltonians and Spinfoams
We review a recent proposal for the regularization of the scalar constraint
of General Relativity in the context of LQG. The resulting constraint presents
strengths and weaknesses compared to Thiemann's prescription. The main
improvement is that it can generate the 1-4 Pachner moves and its matrix
elements contain 15j Wigner symbols, it is therefore compatible with the
spinfoam formalism: the drawback is that Thiemann anomaly free proof is spoiled
because the nodes that the constraint creates have volume.Comment: 4 pages, based on a talk given at Loops '11 in Madrid, to appear in
Journal of Physics: Conference Series (JPCS
On the Relation between Operator Constraint --, Master Constraint --, Reduced Phase Space --, and Path Integral Quantisation
Path integral formulations for gauge theories must start from the canonical
formulation in order to obtain the correct measure. A possible avenue to derive
it is to start from the reduced phase space formulation. In this article we
review this rather involved procedure in full generality. Moreover, we
demonstrate that the reduced phase space path integral formulation formally
agrees with the Dirac's operator constraint quantisation and, more
specifically, with the Master constraint quantisation for first class
constraints. For first class constraints with non trivial structure functions
the equivalence can only be established by passing to Abelian(ised) constraints
which is always possible locally in phase space. Generically, the correct
configuration space path integral measure deviates from the exponential of the
Lagrangian action. The corrections are especially severe if the theory suffers
from second class secondary constraints. In a companion paper we compute these
corrections for the Holst and Plebanski formulations of GR on which current
spin foam models are based.Comment: 43 page
Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order
In our companion paper we identified a complete set of manifestly
gauge-invariant observables for general relativity. This was possible by
coupling the system of gravity and matter to pressureless dust which plays the
role of a dynamically coupled observer. The evolution of those observables is
governed by a physical Hamiltonian and we derived the corresponding equations
of motion. Linear perturbation theory of those equations of motion around a
general exact solution in terms of manifestly gauge invariant perturbations was
then developed. In this paper we specialise our previous results to an FRW
background which is also a solution of our modified equations of motion. We
then compare the resulting equations with those derived in standard
cosmological perturbation theory (SCPT). We exhibit the precise relation
between our manifestly gauge-invariant perturbations and the linearly
gauge-invariant variables in SCPT. We find that our equations of motion can be
cast into SCPT form plus corrections. These corrections are the trace that the
dust leaves on the system in terms of a conserved energy momentum current
density. It turns out that these corrections decay, in fact, in the late
universe they are negligible whatever the value of the conserved current. We
conclude that the addition of dust which serves as a test observer medium,
while implying modifications of Einstein's equations without dust, leads to
acceptable agreement with known results, while having the advantage that one
now talks about manifestly gauge-invariant, that is measurable, quantities,
which can be used even in perturbation theory at higher orders.Comment: 51 pages, no figure
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