992 research outputs found
Reality conditions inducing transforms for quantum gauge field theory and quantum gravity
For various theories, in particular gauge field theories, the algebraic form
of the Hamiltonian simplifies considerably if one writes it in terms of certain
complex variables. Also general relativity when written in the new canonical
variables introduced by Ashtekar belongs to that category, the Hamiltonian
being replaced by the so-called scalar (or Wheeler-DeWitt) constraint. In order
to ensure that one is dealing with the correct physical theory one has to
impose certain reality conditions on the classical phase space which generally
are algebraically quite complicated and render the task of finding an
appropriate inner product into a difficult one. This article shows, for a
general theory, that if we prescribe first a {\em canonical} complexification
and second a representation of the canonical commutation relations in
which the real connection is diagonal, then there is only one choice of a
holomorphic representation which incorporates the correct reality conditions
{\em and} keeps the Hamiltonian (constraint) algebraically simple ! We derive a
canonical algorithm to obtain this holomorphic representation and in particular
explicitly compute it for quantum gravity in terms of a {\em Wick rotation
transform}.Comment: Latex, 23 page
Volume and Quantizations
The aim of this letter is to indicate the differences between the
Rovelli-Smolin quantum volume operator and other quantum volume operators
existing in the literature. The formulas for the operators are written in a
unifying notation of the graph projective framework. It is clarified whose
results apply to which operators and why.Comment: 8 page
Quantum Theory of Gravity I: Area Operators
A new functional calculus, developed recently for a fully non-perturbative
treatment of quantum gravity, is used to begin a systematic construction of a
quantum theory of geometry. Regulated operators corresponding to areas of
2-surfaces are introduced and shown to be self-adjoint on the underlying
(kinematical) Hilbert space of states. It is shown that their spectra are {\it
purely} discrete indicating that the underlying quantum geometry is far from
what the continuum picture might suggest. Indeed, the fundamental excitations
of quantum geometry are 1-dimensional, rather like polymers, and the
3-dimensional continuum geometry emerges only on coarse graining. The full
Hilbert space admits an orthonormal decomposition into finite dimensional
sub-spaces which can be interpreted as the spaces of states of spin systems.
Using this property, the complete spectrum of the area operators is evaluated.
The general framework constructed here will be used in a subsequent paper to
discuss 3-dimensional geometric operators, e.g., the ones corresponding to
volumes of regions.Comment: 33 pages, ReVTeX, Section 4 Revised: New results on the effect of
topology of a surface on the eigenvalues and eigenfunctions of its area
operator included. The proof of the bound on the level spacing of eigenvalues
(for large areas) simplified and its ramification to the Bekenstein-Mukhanov
analysis of black-hole evaporation made more explicit. To appear in CQ
Quantum Spin Dynamics (QSD) II
We continue here the analysis of the previous paper of the Wheeler-DeWitt
constraint operator for four-dimensional, Lorentzian, non-perturbative,
canonical vacuum quantum gravity in the continuum. In this paper we derive the
complete kernel, as well as a physical inner product on it, for a non-symmetric
version of the Wheeler-DeWitt operator. We then define a symmetric version of
the Wheeler-DeWitt operator. For the Euclidean Wheeler-DeWitt operator as well
as for the generator of the Wick transform from the Euclidean to the Lorentzian
regime we prove existence of self-adjoint extensions and based on these we
present a method of proof of self-adjoint extensions for the Lorentzian
operator. Finally we comment on the status of the Wick rotation transform in
the light of the present results.Comment: 27 pages, Latex, preceded by a companion paper before this on
An Invitation to Higher Gauge Theory
In this easy introduction to higher gauge theory, we describe parallel
transport for particles and strings in terms of 2-connections on 2-bundles.
Just as ordinary gauge theory involves a gauge group, this generalization
involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie
group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes,
which play an important role in string theory and multisymplectic geometry.
Second, every group representation gives a Lie 2-group; the representation of
the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which
leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint
representation of any Lie group on its own Lie algebra gives a 'tangent
2-group', which serves as a gauge 2-group in 4d BF theory, which has
topological gravity as a special case. Fourth, every Lie group has an 'inner
automorphism 2-group', which serves as the gauge group in 4d BF theory with
cosmological constant term. Fifth, every Lie group has an 'automorphism
2-group', which plays an important role in the theory of nonabelian gerbes. And
sixth, every compact simple Lie group gives a 'string 2-group'. We also touch
upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra
that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum
Gravity and Quantum Geometry at the 2009 Corfu Summer Institut
Quantum Geometry and Black Hole Entropy
A `black hole sector' of non-perturbative canonical quantum gravity is
introduced. The quantum black hole degrees of freedom are shown to be described
by a Chern-Simons field theory on the horizon. It is shown that the entropy of
a large non-rotating black hole is proportional to its horizon area. The
constant of proportionality depends upon the Immirzi parameter, which fixes the
spectrum of the area operator in loop quantum gravity; an appropriate choice of
this parameter gives the Bekenstein-Hawking formula S = A/4*l_p^2. With the
same choice of the Immirzi parameter, this result also holds for black holes
carrying electric or dilatonic charge, which are not necessarily near extremal.Comment: Revtex, 8 pages, 1 figur
On the relation between the connection and the loop representation of quantum gravity
Using Penrose binor calculus for () tensor expressions, a
graphical method for the connection representation of Euclidean Quantum Gravity
(real connection) is constructed. It is explicitly shown that: {\it (i)} the
recently proposed scalar product in the loop-representation coincide with the
Ashtekar-Lewandoski cylindrical measure in the space of connections; {\it (ii)}
it is possible to establish a correspondence between the operators in the
connection representation and those in the loop representation. The
construction is based on embedded spin network, the Penrose graphical method of
calculus, and the existence of a generalized measure on the space of
connections modulo gauge transformations.Comment: 19 pages, ioplppt.sty and epsfig.st
Matrix Elements of Thiemann's Hamiltonian Constraint in Loop Quantum Gravity
We present an explicit computation of matrix elements of the hamiltonian
constraint operator in non-perturbative quantum gravity. In particular, we
consider the euclidean term of Thiemann's version of the constraint and compute
its action on trivalent states, for all its natural orderings. The calculation
is performed using graphical techniques from the recoupling theory of colored
knots and links. We exhibit the matrix elements of the hamiltonian constraint
operator in the spin network basis in compact algebraic form.Comment: 32 pages, 22 eps figures. LaTeX (Using epsfig.sty,ioplppt.sty and
bezier.sty). Submited to Classical and Quantum Gravit
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