54 research outputs found
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
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)
Master Constraint Operator in Loop Quantum Gravity
We introduce a master constraint operator densely defined
in the diffeomorphism invariant Hilbert space in loop quantum gravity, which
corresponds classically to the master constraint in the programme. It is shown
that is positive and symmetric, and hence has its Friedrichs
self-adjoint extension. The same conclusion is tenable for an alternative
master operator , whose quadratic form coincides with the
one proposed by Thiemann. So the master constraint programme for loop quantum
gravity can be carried out in principle by employing either of the two
operators.Comment: 11 pages, significant modification in section 2, accepted for
publication in Phys. Lett.
On the geometry of quantum constrained systems
The use of geometric methods has proved useful in the hamiltonian description
of classical constrained systems. In this note we provide the first steps
toward the description of the geometry of quantum constrained systems. We make
use of the geometric formulation of quantum theory in which unitary
transformations (including time evolution) can be seen, just as in the
classical case, as finite canonical transformations on the quantum state space.
We compare from this perspective the classical and quantum formalisms and argue
that there is an important difference between them, that suggests that the
condition on observables to become physical is through the double commutator
with the square of the constraint operator. This provides a bridge between the
standard Dirac procedure --through its geometric implementation-- and the
Master Constraint program.Comment: 14 pages, no figures. Discussion expanded. Version published in CQ
Quantum Spin Dynamics VIII. The Master Constraint
Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG)
was launched which replaces the infinite number of Hamiltonian constraints by a
single Master constraint. The MCP is designed to overcome the complications
associated with the non -- Lie -- algebra structure of the Dirac algebra of
Hamiltonian constraints and was successfully tested in various field theory
models. For the case of 3+1 gravity itself, so far only a positive quadratic
form for the Master Constraint Operator was derived. In this paper we close
this gap and prove that the quadratic form is closable and thus stems from a
unique self -- adjoint Master Constraint Operator. The proof rests on a simple
feature of the general pattern according to which Hamiltonian constraints in
LQG are constructed and thus extends to arbitrary matter coupling and holds for
any metric signature. With this result the existence of a physical Hilbert
space for LQG is established by standard spectral analysis.Comment: 19p, no figure
Spin foams with timelike surfaces
Spin foams of 4d gravity were recently extended from complexes with purely
spacelike surfaces to complexes that also contain timelike surfaces. In this
article, we express the associated partition function in terms of vertex
amplitudes and integrals over coherent states. The coherent states are
characterized by unit 3--vectors which represent normals to surfaces and lie
either in the 2--sphere or the 2d hyperboloids. In the case of timelike
surfaces, a new type of coherent state is used and the associated completeness
relation is derived. It is also shown that the quantum simplicity constraints
can be deduced by three different methods: by weak imposition of the
constraints, by restriction of coherent state bases and by the master
constraint.Comment: 22 pages, no figures; v2: remarks on operator formalism added in
discussion; correction: the spin 1/2 irrep of the discrete series does not
appear in the Plancherel decompositio
Purely geometric path integral for spin foams
Spin-foams are a proposal for defining the dynamics of loop quantum gravity
via path integral. In order for a path integral to be at least formally
equivalent to the corresponding canonical quantization, at each point in the
space of histories it is important that the integrand have not only the correct
phase -- a topic of recent focus in spin-foams -- but also the correct modulus,
usually referred to as the measure factor. The correct measure factor descends
from the Liouville measure on the reduced phase space, and its calculation is a
task of canonical analysis.
The covariant formulation of gravity from which spin-foams are derived is the
Plebanski-Holst formulation, in which the basic variables are a Lorentz
connection and a Lorentz-algebra valued two-form, called the Plebanski
two-form. However, in the final spin-foam sum, one sums over only spins and
intertwiners, which label eigenstates of the Plebanski two-form alone. The
spin-foam sum is therefore a discretized version of a Plebanski-Holst path
integral in which only the Plebanski two-form appears, and in which the
connection degrees of freedom have been integrated out. We call this a purely
geometric Plebanski-Holst path integral.
In prior work in which one of the authors was involved, the measure factor
for the Plebanski-Holst path integral with both connection and two-form
variables was calculated. Before one discretizes this measure and incorporates
it into a spin-foam sum, however, one must integrate out the connection in
order to obtain the purely geometric version of the path integral. To calculate
this purely geometric path integral is the principal task of the present paper,
and it is done in two independent ways. Gauge-fixing and the background
independence of the resulting path integral are discussed in the appendices.Comment: 21 page
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
Loop Quantum Gravity: Four Recent Advances and a Dozen Frequently Asked Questions
As per organizers' request, my talk at the 11th Marcel Grossmann Conference
consisted of two parts. In the first, I illustrated recent advances in loop
quantum gravity through examples. In the second, I presented an overall
assessment of the status of the program by addressing some frequently asked
questions. This account is addressed primarily to researchers outside the loop
quantum gravity community.Comment: 21 pages, to appear in the Proceedings of the 11th Marcel Grossmann
Conferenc
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