Properties of the volume operator in loop quantum gravity: I. Results

Abstract

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a 'volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5-7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices are included, for which the presence of a volume gap is shown. This paper presents our main results; details are provided by a companion paper [52]. IntroductionLoop Quantum Gravity (LQG) [1,2,3] has become a promising candidate for a quantum theory of gravity over the last 15 years. It is an attempt to canonically quantize General Relativity (GR) while preserving its key principle: background independence 1 . The resulting quantum theory is formulated as an SU(2) gauge theory. * brunnemann@math.uni-hamburg.de † drideout@perimeterinstitute.ca 1 With the term 'background independence' we imply independence on a choice of fixed background geometry. In order to allow for the possibility of topology change in quantum gravity, it would be desirable to have the theory in its final formulation be independent of the topology of the underlying manifold, however to our knowledge this has not been achieved so far within LQG. 1Upon casting GR into a Hamiltonian formulation by introducing a foliation of four dimensional spacetime (M, g µν ) 2 into spatial three dimensional hypersurfaces Σ, with the orthogonal timelike direction parametrized by t ∈ Ê, one obtains first class constraints which have to be imposed on the reformulated theory such that it obeys the dynamics of Einstein's equations and is independent of the particular choice of foliation. These constraints are the three spatial diffeomorphism or vector constraints which generate diffeomorphisms inside Σ, and the so called Hamilton constraint which generates deformations of the hypersurfaces Σ in the t-(foliation) direction. In addition one obtains three Gauss constraints due to the introduction of additional SU(2) gauge degrees of freedom.(There is also a more recent reformulation of the constraints, as proposed in the master constraint programme. See [9] and references therein.) The resulting theory is then quantized on the kinematical level in terms of holonomies h and electric fluxes E. Kinematical states are defined over collection of edges of embedded graphs. The physical states have to be constructe...

The length operator in Loop Quantum Gravity

“…The construction of the operator L I for finite ∆s and given cylindrical function is discussed in section 3.3.5. It requires two ingredients: the two-hand operatorŶ i Iαβ corresponding to the quantization of the quantity introduced in (30), and the inverse of the local volume operatorV I . These two ingredients enter in the definition of L I in the way specified by equations (29) and (31).…”

Canonical quantum gravity on noncommutative space–time

“…The area operator and the volume operator in the context of loop quantum gravity have been considered in [44] for the first time. Further studies concerning the area operator can be found in [45], [46], [47] and concerning the volume operator in [48], [49], [50], [51], [52]. In this paper the considerations will remain restricted to the area operator.…”

Matrix elements of Lorentzian Hamiltonian constraint in loop quantum gravity

“…The action of the remaining holonomies in (34) can be simplified by Theorem (loop trick) (36) where ði; j; kÞ ¼ 1 if the edges e i , e j , e k are ordered anticlockwise, otherwise it equals À1.…”

“…Inserting this in the first equality proves the theorem. As in the example on page 11, adding h s k to (35) transforms the solid line c 1 into a dashed line so that with (36) and…”

“…The volume operator in LQG has been studied intensively under both the canonical [32][33][34] and the covariant perspective [35]. Yet, the matrix elements are getting very complicated with the more edges involved so that one has to apply numerical methods [36,37] or semiclassical analysis [38,39] to evaluate them. The second difficulty appears due to the regularization of the connections by holonomies.…”