14 research outputs found
Regge calculus from a new angle
In Regge calculus space time is usually approximated by a triangulation with
flat simplices. We present a formulation using simplices with constant
sectional curvature adjusted to the presence of a cosmological constant. As we
will show such a formulation allows to replace the length variables by 3d or 4d
dihedral angles as basic variables. Moreover we will introduce a first order
formulation, which in contrast to using flat simplices, does not require any
constraints. These considerations could be useful for the construction of
quantum gravity models with a cosmological constant.Comment: 8 page
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
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
Polymer quantization of CGHS model- I
We present a polymer(loop) quantization of a two dimensional theory of
dilatonic gravity known as the CGHS model. We recast the theory as a
parametrized free field theory on a flat 2-dimensional spacetime and quantize
the resulting phase space using techniques of loop quantization. The resulting
(kinematical) Hilbert space admits a unitary representation of the spacetime
diffeomorphism group. We obtain the complete spectrum of the theory using a
technique known as group averaging and perform quantization of Dirac
observables on the resulting Hilbert space. We argue that the algebra of Dirac
observables gets deformed in the quantum theory. Combining the ideas from
parametrized field theory with certain relational observables, evolution is
defined in the quantum theory in the Heisenberg picture. Finally the dilaton
field is quantized on the physical Hilbert space which carries information
about quantum geometry.Comment: 50 pages, 2 figure
On (Cosmological) Singularity Avoidance in Loop Quantum Gravity
Loop Quantum Cosmology (LQC), mainly due to Bojowald, is not the cosmological
sector of Loop Quantum Gravity (LQG). Rather, LQC consists of a truncation of
the phase space of classical General Relativity to spatially homogeneous
situations which is then quantized by the methods of LQG. Thus, LQC is a
quantum mechanical toy model (finite number of degrees of freedom) for LQG(a
genuine QFT with an infinite number of degrees of freedom) which provides
important consistency checks. However, it is a non trivial question whether the
predictions of LQC are robust after switching on the inhomogeneous fluctuations
present in full LQG. Two of the most spectacular findings of LQC are that 1.
the inverse scale factor is bounded from above on zero volume eigenstates which
hints at the avoidance of the local curvature singularity and 2. that the
Quantum Einstein Equations are non -- singular which hints at the avoidance of
the global initial singularity. We display the result of a calculation for LQG
which proves that the (analogon of the) inverse scale factor, while densely
defined, is {\it not} bounded from above on zero volume eigenstates. Thus, in
full LQG, if curvature singularity avoidance is realized, then not in this
simple way. In fact, it turns out that the boundedness of the inverse scale
factor is neither necessary nor sufficient for curvature singularity avoidance
and that non -- singular evolution equations are neither necessary nor
sufficient for initial singularity avoidance because none of these criteria are
formulated in terms of observable quantities.After outlining what would be
required, we present the results of a calculation for LQG which could be a
first indication that our criteria at least for curvature singularity avoidance
are satisfied in LQG.Comment: 34 pages, 16 figure
The Phoenix Project: Master Constraint Programme for Loop Quantum Gravity
The Hamiltonian constraint remains the major unsolved problem in Loop Quantum
Gravity (LQG). Seven years ago a mathematically consistent candidate
Hamiltonian constraint has been proposed but there are still several unsettled
questions which concern the algebra of commutators among smeared Hamiltonian
constraints which must be faced in order to make progress. In this paper we
propose a solution to this set of problems based on the so-called {\bf Master
Constraint} which combines the smeared Hamiltonian constraints for all smearing
functions into a single constraint. If certain mathematical conditions, which
still have to be proved, hold, then not only the problems with the commutator
algebra could disappear, also chances are good that one can control the
solution space and the (quantum) Dirac observables of LQG. Even a decision on
whether the theory has the correct classical limit and a connection with the
path integral (or spin foam) formulation could be in reach. While these are
exciting possibilities, we should warn the reader from the outset that, since
the proposal is, to the best of our knowledge, completely new and has been
barely tested in solvable models, there might be caveats which we are presently
unaware of and render the whole {\bf Master Constraint Programme} obsolete.
Thus, this paper should really be viewed as a proposal only, rather than a
presentation of hard results, which however we intend to supply in future
submissions.Comment: LATEX, uses AMSTE
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 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 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 III. SL(2,R) Models
This is the third 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 analyze models which, despite the fact that the phase
space is finite dimensional, are much more complicated than in the second
paper: These are systems with an SL(2,\Rl) gauge symmetry and the
complications arise because non -- compact semisimple Lie groups are not
amenable (have no finite translation invariant measure). This leads to severe
obstacles in the refined algebraic quantization programme (group averaging) and
we see a trace of that in the fact that the spectrum of the Master Constraint
does not contain the point zero. However, the minimum of the spectrum is of
order which can be interpreted as a normal ordering constant arising
from first class constraints (while second class systems lead to normal
ordering constants). The physical Hilbert space can then be be obtained after
subtracting this normal ordering correction.Comment: 33 pages, no figure