4 research outputs found
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
QSD IV : 2+1 Euclidean Quantum Gravity as a model to test 3+1 Lorentzian Quantum Gravity
The quantization of Lorentzian or Euclidean 2+1 gravity by canonical methods
is a well-studied problem. However, the constraints of 2+1 gravity are those of
a topological field theory and therefore resemble very little those of the
corresponding Lorentzian 3+1 constraints. In this paper we canonically quantize
Euclidean 2+1 gravity for arbitrary genus of the spacelike hypersurface with
new, classically equivalent constraints that maximally probe the Lorentzian 3+1
situation. We choose the signature to be Euclidean because this implies that
the gauge group is, as in the 3+1 case, SU(2) rather than SU(1,1). We employ,
and carry out to full completion, the new quantization method introduced in
preceding papers of this series which resulted in a finite 3+1 Lorentzian
quantum field theory for gravity. The space of solutions to all constraints
turns out to be much larger than the one as obtained by traditional approaches,
however, it is fully included. Thus, by suitable restriction of the solution
space, we can recover all former results which gives confidence in the new
quantization methods. The meaning of the remaining "spurious solutions" is
discussed.Comment: 35p, LATE
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