Properties of the Volume Operator in Loop Quantum Gravity by Brunnemann and Rideout

This week I’ve returned more strongly to my work on the numerical spectra of quantum geometrical operators. This post looks at an analysis of the Ashtekar and Lewandowski version of the volume operator.

The authors 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. The analysis also considers generic graph vertices of valence greater than four. The authors 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.

The authors  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. it is seen that  gauge invariance connects vertex geometry and the representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum of 4-valent vertices show the presence of a volume gap.

Loop Quantum Gravity (LQG) is a candidate for a quantum theory of gravity. It is an attempt to canonically quantize General Relativity. The resulting quantum theory is formulated as an SU(2) gauge theory.

General Relativity is put into a Hamiltonian formulation by introducing a foliation of four dimensional spacetime into spatial three dimensional hypersurfaces ∑, with the orthogonal timelike
direction parametrized by t.  first class constraints which have to be imposed on the theory so 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 and the so called Hamilton constraint which generates deformations of the hypersurfaces in the t- foliation direction. In addition there are three Gauss constraints due to the introduction of additional SU(2) gauge degrees of freedom.

The 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 constructed by imposing the operator version of the constraints on the thus defined kinematical theory.

There are well defined operators in the kinematical quantum theory which correspond to differential geometric objects, such as the length of curves, the area of surfaces, and the volume of regions in the spatial foliation hypersurfaces. All these quantities have discrete spectra, which can be traced back to the compactness of the gauge group SU(2). A coherent state framework address questions on the correct semiclassical limit of the theory.

A central role in investigations is played by the area and the volume operators.The volume operator is a crucial object not only in order to analyze matter coupling to LQG, but also for evaluating the action of the Hamilton constraint operator in order to construct the physical sector of LQG. By construction the spectral properties of the constraint operators are driven by the spectrum of  the volume operator.

Loop Quantum Gravity

Hamiltonian Formulation
In order to cast General Relativity into the Hamiltonian formalism, one has to perform a foliation M ≅ Rx∑ of the four dimensional spacetime manifold M into three dimensional spacelike hypersurfaces with transverse time direction labelled by a foliation parameter t ∈ R. The four dimensional metric g can then be decomposed into the three metric q(x) on the spatial slices ∑ and its extrinsic curvature K, which serve as canonical variables. Introducing new variables due to Ashtekar  may  take
as canonical variables electric fields E and connections A as used in the canonical formulation of Yang Mills theories. Here a, b = 1 . . . 3 denote spatial (tensor) indices, i, j = 1 . . . 3 denote su(2)-

The occurrence of SU(2) as a gauge group comes from the fact that it is necessary for the coupling of spinorial matter and it is the universal covering group of SO(3) which arises naturally when one

rewrites the three metric q in terms of cotriads as


The pair (A,E) is related to (q,K) as


with Γ being the spin connection.

The pair (A,E) obeys the Poisson bracket:


The theory is subject to constraints which arise due to the background independence of General Relativity. It can be treated as suggested by Dirac . There are:

  • three vector (spatial diffeomorphism) constraints


  •  a scalar – Hamilton constraint


  • three Gauss constraints


The Volume Operator
Definition of the Volume Operator

The operator corresponding to the volume V (R) of a spatial region R ⊂ ∑


Acting on gauge invariant spin network states, is defined as


The sum in has to be taken over all vertices v of the underlying graph γ. At each vertex v one has to sum over all possible ordered triples (ei , ej , ek).

Matrix Formulation



where Q is a totally antisymmetric matrix with purely real elements. Its eigenvalues λ are purely imaginary and come in pairs λQ = ±iλ. Choosing Z = 1 the volume operator V has the same eigenstates as Q and its eigenvalues λ = |λQ|½

We are left with the task of calculating the spectra of totally antisymmetric real matrices of the form:


Analytical Results on the Gauge Invariant 4-Vertex

The spectrum of the volume operator at a given gauge invariant 4-vertex is simple, that is all its eigenvalues, except zero, come in pairs, and there are no further degeneracies or accumulation points in the spectrum. An explicit expression for the eigenstates of the volume operator in terms of polynomials of its matrix elements and its eigenvalues can be given as:




The specific matrix realization of the volume operator plays a crucial role here since it can be written as an antisymmetric purely imaginary matrix having non-zero entries only on the first off-diagonal. This makes it possible to apply  techniques from orthogonal polynomials and Jacobi-matrices. 


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