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arXiv:1108.1881

Asymptotic Limits of the Wigner $12J$-Symbol in Terms of the Ponzano-Regge Phases

Yu, Liang

Published Aug 9, 2011• 0 citations
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Abstract

There are two types of asymptotic formulas for the $12j$ symbol with one small and 11 large angular momenta. We have derived the first type of formula previously in [L. Yu, Phys. Rev. A84 022101 (2011)]. We will derive the second type in this paper. We find that this second asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram associated with two $6j$ symbols, namely, the vector diagram of two adjacent tetrahedra sharing a common face. As a result, two sets of Ponzano-Regge phases appear in the asymptotic formula. This work contributes another asymptotic formula of the Wigner $12j$ symbol to the re-coupling theory of angular momenta.

DOI
10.48550/arxiv.1108.1881

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Asymptotic Limits of the Wigner 12J12J12J-Symbol in Terms of the Ponzano-Regge Phases
There are two types of asymptotic formulas for the 12j12j12j symbol with one small and 11 large angular momenta. We have derived the first type of formula previously in [L. Yu, Phys. Rev. A84 022101 (2011)]. We will derive the second type in this paper. We find that this second asymptotic formula for the 12j12j12j symbol is expressed in terms of the vector diagram associated with two 6j6j6j symbols, namely, the vector diagram of two adjacent tetrahedra sharing a common face. As a result, two sets of Ponzano-Regge phases appear in the asymptotic formula. This work contributes another asymptotic formula of the Wigner 12j12j12j symbol to the re-coupling theory of angular momenta.
The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.
We analyze the asymptotics of the Wigner 3j3j3j-symbol as a matrix element connecting eigenfunctions of a pair of integrable systems, obtained by lifting the problem of the addition of angular momenta into the space of Schwinger's oscillators. A novel element is the appearance of compact Lagrangian manifolds that are not tori, due to the fact that the observables defining the quantum states are noncommuting. These manifolds can be quantized by generalized Bohr-Sommerfeld rules and yield all the correct quantum numbers. The geometry of the classical angular momentum vectors emerges in a clear manner. Efficient methods for computing amplitude determinants in terms of Poisson brackets are developed and illustrated.
Asymptotic Analysis of the Wigner 3j3j3j-Symbol in the Bargmann Representation
We derive the leading asymptotic limit of the Wigner 3j3j3j-symbol from a stationary phase approximation of a twelve dimensional integral, obtained from an inner product between two exact Bargmann wavefunctions. We show that, by the construction of the Bargmann inner product, the stationary phase conditions have a geometric description in terms of the Hopf fibration of C6{\mathbb C}^6C6 into R3×R3×R3{\mathbb R}^3 \times {\mathbb R}^3 \times {\mathbb R}^3R3×R3×R3. In addition, we find that, except for the usual modification of the quantum numbers by 1/2, the imaginary part of the logarithm of a Bargmann wavefunction, evaluated at the stationary points, is equal to the asymptotic phase of the 3j3j3j-symbol.
The quantum mechanical addition of an arbitrary (very large) number of identical angular momenta is considered. The resulting states jm usually occur many times. The corresponding multiplicities P j and Q m are investigated. Several explicit formulae for Q m are given. The asymptotic approximations for multiplicities P j and Q m , which are the Wigner and Gauss distributions respectively, are derived. All parameters of the distributions are explicitly obtained. A comparison of theoretical multiplicities with the exact ones is made when the number of angular moments is relatively small.
Formulas are obtained which give the matrix representation, relative to a product basis, of the projection operator for the total angular momentum of a system. If the individual angular momenta are not too large, the matrix elements depend upon a small number of parameters, independent of the number of angular momenta coupled. Recurrence relations between elements and symmetry properties of elements are derived. These results enable one to perform the vector coupling of a large number of angular momenta in a relatively simple fashion. The connection between the matrix elements and vector coupling coefficients is discussed. Several important special cases are treated in detail.