Semiclassical Mechanics of the Wigner 6j-Symbol by Aquilanti et al

This week I have been continuing my research on the the expectation values of the Ricci curvature on Carlo-Speziale semi-classical states, for different values of the spins in the case of a simple graph : a monochromatic 4 – valent node dual to an equilateral tetrahedron. I will be posting about this later.

As part of this work I am studying a paper on the semiclassical mechanics of the wigner 6j symbol.  In this paper the semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the methods of Roberts and others for deriving the Ponzano- Regge formula are discussed, and a new approach, based on the recoupling of four angular momenta, is presented. A generalization of the Yutsis-type of spin network is developed for this purpose. Special attention is given to symplectic reduction, the reduced phase space of the 6j-symbol and the reduction of Poisson bracket expressions for semiclassical amplitudes. 

Introduction
The Wigner 6j-symbol or Racah W-coefficient)is a central object in angular momentum theory, with many applications in atomic, molecular and nuclear physics. These usually involve the recoupling of three angular momenta, that is, the 6j symbol contains the unitary matrix elements of the transformation connecting the two bases that arise when three angular momenta are added in two different ways.

More recently the 6j- and other 3nj-symbols have found applications
in quantum computing and in algorithms for molecular
scattering calculations , which make use of their connection with discrete orthogonal polynomials.
The 6j-symbol is an example of a spin network, a graphical representation for contractions between tensors that occur in angular momentum theory. The graphical notation has been developed since the ’60s.

The 6j symbol is the simplest, nontrivial, closed spin network -one that represents a rotational invariant. Spin networks are important in lattice QCD and in loop quantum gravity where they provide a gauge-invariant basis for the field. Applications in quantum gravity are described by Rovelli and Smolin , Baez, Carlip , Barrett and Crane, Regge and Williams , Rovelli and Thiemann .

There are three approaches to the evaluation of rotational SU(2)invariants.

  • The Yutsis school of graphical notation
  • The Clebsch-Gordan school of algebraic manipulation
  •  The chromatic evaluation – which grew out of Penrose’s doctoral work on the graphical representation of tensors and is closely related to knot theory. 

The asymptotics of spin networks and especially the 6j-symbol has played an important role in many areas. Aymptotics refer to the asymptotic expansion for the spin network when all j’s are large, equivalent to a semiclassical approximation since large j is equivalent to small h/2π. The asymptotic expression for the 6j-symbol the leading term in the asymptotic series)was first obtained by Ponzano and Regge. In the  same paper those authors gave the first spin foam model for quantum gravity. The formula of Ponzano and Regge is notable  for its high symmetry and the manner in which it is related to the geometry of a tetrahedron in three-dimensional space. It is also remarkable because the phase of the asymptotic  expression is identical to the Einstein-Hilbert action for three-dimensional gravity integrated over a tetrahedron, in Regge’s simplicial approximation to general relativity. The semiclassical limit of the 6j-symbol thus plays a crucial role in simplicial approaches to the quantization of the gravitational field.

 Spin network notation

The 3j-symbol and Wigner intertwiner

semifig1

Bras, kets and scalar products

semifig2

 

semifig3

 

semifig4

semifig5

Intertwiners

An SU(2) intertwiner is a linear map between two vector spaces that commutes with the action of SU(2) on the two spaces

semifig6

Tensor products and resolution of identity

The outer product of a ket with a bra is represented in spin network language simply by placing the spin networks for the ket and the bra on the same page as shown below

semifig7

 

semifig8

 

semifig9

The 2j symbol and intertwiner

|Ki> is  expressed in terms of the Clebsch-Gordan coefficients by

semiequ1

This vector can also be expressed in terms of the 2j-symbol,which is defined in terms of the usual 3j-symbol by

semiequ2

The invariant vector |Ki> can also be written,

semiequ3

semifig10

semifig11

 

semifig13

semiequ4

Kets to bras

semifig14

semifig15

semifig16

 

Bras to kets

semifig17

semifig18

semifig19

semifig20

 

semifig21

semifig22

 

 

semifig23

semifig24

 

semifig25

Raising and lowering indices

semifig26

semifig27

 

semifig28semifig29

Models for the 6j-symbol

For given values of the six j’s, the 6j-symbol is just a number, but to study its semiclassical limit it is useful to write it as a scalar product <B|A>of wave functions in some Hilbert space. This can be done in many different ways, corresponding to different models of the 6j-symbol.

The 12j-model of the 6j-symbol

semifig30

 

semifig31

 

semifig32

The 6j-symbol is represented as a product of six copies of a 2j-symbol and four of a 3j-symbol. Using the definition for the
2j-symbol, the result is

semiequ7

 

 

semifig33

semiequ8

 

semifig34

semiequ9

 

semifig35

semiequ10

semiequ11

 

The Triangle and Polygon Inequalities

semiequ12

This is equivalent to the triangle inequalities when n = 3. In general, it represents the necessary and sufficient condition that line segments of the given, nonnegative lengths can be fitted together to form a polygon with n sides

The 4j-model of the 6j-symbol

semiequ13

 

semifig36

semifig37

The 6j-symbol is proportional to
a scalar product

semiequ19

 

The Ponzano-Regge Formula

semiequ93

 

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