Quantum states of elementary three–geometry by Carbone, Carfora , and Marzuoli

This is a relatively old paper but its so clearly written that it is well worth reviewing and understanding the material.

In this paper the authors introduce a quantum volume operator K in three–dimensional Quantum Gravity by taking into account a symmetrical coupling scheme of three SU(2) angular momenta. The spectrum of K is discrete and defines a complete set of eigenvectors
which is an alternative with respect to the complete sets employed when the usual binary coupling schemes of angular momenta are considered. Each of these states,  called quantum bubbles, represents an interference of extended configurations which provides a rigorous meaning to the heuristic notion of quantum tetrahedron. They study the generalized recoupling coefficients connecting the symmetrical and the binary basis vectors, and provide an explicit recursive solution for such coefficients by analyzing also its asymptotic limit.


Ponzano–Regge gravity is based on an asymptotic formula for theSU(2) 6j symbols,


The physical interpretation follows if we recognize that the exponential includes the classical Regge action – the discretized version of the Hilbert– Einstein action for the tetrahedron T. The
presence of the slowly varying volume term in front of the phase factor means that {6j}as can be interpreted as a probability amplitude in the approach to the classical limit. The probability amplitude for an elementary block of Euclidean three–geometry to emerge from the recoupling of quantum angular momenta or from a spin network.

Physical reality can only be ascribed to the tetrahedron T if and only if its volume –written in terms of the squares of the edges through the Cayley determinant– satisfies the condition (V (T))² > 0. The triangle inequalities on the four triples of spin variables  – associated with the four faces of the tetrahedron – which ensure the existence of the 6j symbol are weaker than the condition that T exists as a realizable solid.

The authors try to answer the following questions:

  • Why does a classical three–geometry emerge from a recoupling of angular momenta?
  • What does it mean to give the quantum state of an elementary block of geometry?
  • What are the degrees of freedom of a quantum tetrahedron?

Quantum bubbles

The theory of the coupling of states of three SU(2) angular momenta operators J₁, J₂,  J₃ to states of sharp total angular momentum J is usually developed in the framework of binary couplings. Starting from the ordered triple,  denote symbolically the admissible schemes according to



The corresponding state vectors may be written as


In this framework the Wigner 6j symbol is just the recoupling coefficient relating the two sets  and , namely


where Φ ≡ J₁+J₂+J₃

In spite of the tetrahedral symmetry of the 6j symbol, at the quantum level we cannot recognize something like a quantum tetrahedron: it is only in the approach to the classical limit  that both the coupling schemes coexist and the tetrahedron may take shape.

This situation changes  if we bring into play the symmetrical coupling
scheme for the addition of three SU(2) angular momenta to give a fourth definite angular momentum J with projection J₀ symbolically

3gequ7 such a coupling is characterized by the simultaneous diagonalization of the six Hermitian operators

J²₁,J²₂,J²₃,  K, J² and J₀, where K is a scalar operator built from the irreducible tensor operators J₁, J₂,  J₃ and defined according to


Since K is the mixed product of three angular momentum vectors we can call it a volume.

Each of the eigenvectors of this new set of operators is denoted by 3gequ8

the set of eigenvalues in each subspace with (jm) fixed is discrete and
consists of pairs (k,−k), with at most one zero eigenvalue.

From a geometrical point of view a state |kjm >, for k ≠ 0, represents a quantum volume of size kħ³ and k2 > 0 is the quantum counterpart of the condition (V (T))² > 0.

According to the Correspondence Principle in the region of large quantum numbers the states state |kjm >characterize angular momentum vectors confined to narrow ranges
around specific values: the narrower are the ranges, the closer we approach the classical regime. This implies that the classical limit of the operator coincides with the expression of the volume of the classical tetrahedron T spanned by the counterparts of the operators  J₁, J₂,  J₃.

The state |kjm > for fixed (jm) provides a proper description of the quantum state of an elementary block of Euclidean three–geometry. However |kjm > cannot be directly interpreted as a quantum tetrahedron, but rather as a quantum interference of extendend configurations from which information about all significant geometrical quantities can be extracted.

 Generalized recoupling theory

The 6j symbol can be interpreted as a propagator between two states belonging to alternative binary bases. The asymptotic expression:  3gequ1is actually the generating functional of a path–sum evaluated at the semiclassical level and the physical information about the underlying classical theory is encoded in the form of the action.

For the quantitative analysis of the relations between one of the sets of eigenvectors arising from a binary coupling parametrized by the eigenvalue of the intermediate momentum J₁₂ and the symmetrical basis  we introduce a unitary transformation defined as


The generalized recoupling coefficient


and its inverse defined as


A three–term recursion relation for the generalized recoupling coefficient is


The functions α (l =  J₁₂, J₁₂ + 1) can be cast in the form


Since 3gequ21the conditions at the extrema fixed by  are


The formal solution of 3gequ19



Asymptotic limit

To explore the generalized recoupling coefficient we use a semiclassical (WKB) approximation to the recursion relation


Which  gives an ordinary second order difference equation
of the type


we search for a solution of this for each k, in the form


with ρ and A to be determined. By substitution we get a pair of differential equations corresponding respectively to the imaginary and real parts;


Finally, since the classical counterpart of the operator K is the volume V ≡ V (T) of the Euclidean tetrahedron T we see that the generalized recoupling coefficient in the asymptotic limit behaves as



where θ₁₂ is the dihedral angle between the faces of the tetrahedron T which share J₁₂.

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