This week I have been following up some work which I was introduced to in Dimitri Marinelli’s PhD thesis ‘Single and collective dynamics of discretized geometries’. Essentially this involves the analysis of the volume operator. This is really exciting for me as it is in my specialist research area – the numerical analysis of Quantum geometric operators and their spectra. I’ll be following up the literature survey with numerical work in sagemath.
The paper I’ll look at this week is ‘Exact and asymptotic computations of elementary spin networks: classification of the quantum–classical boundaries’ by Bitencourt, Marzuoli, Ragni, Anderson and and Aquilanti.
There has been increasing interest to the issues of exact computations and asymptotics of spin networks. The large–entries regimes – semiclassical limits, occur in many areas of physics and in particular in discretization algorithms of applied quantum mechanics.
The authors extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient, by exploiting its self–dual properties and studying it as a function of two discrete variables. This arises from its original definition as an orthogonal angular momentum recoupling matrix. Progress comes
from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational
advances –based on traditional and new recurrence relations– which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation.
The paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines – caustics in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and highlights the key role of the Regge symmetries of the 6j.
The diagrammatic tools for spin networks were developed by the Yutsis school and in connection with applications to discretized models for quantum gravity after Penrose, Ponzano and Regge.
The basic building blocks of all spin networks are the Wigner 6j symbols or Racah coefficients, which are studied here by exploiting their self dual properties and looking at them as functions of two variables. This approach is natural in view of their origin as matrix elements describing recoupling between alternative angular momentum binary coupling schemes, or between alternative hyperspherical harmonics.
Semiclassical and asymptotic views are introduced to describe the dependence on parameters. They originated from the association due to Racah and Wigner to geometrical features, respectively a dihedral angle and the volume of an associated tetrahedron, which is the starting point of the seminal paper by Ponzano and Regge . Their results provided an impressive insight into the functional dependence of angular momentum functions showing a quantum mechanical picture in terms of formulas which describe classical and non–classical discrete wavelike regimes, as well as the transition between them.
The screen: mirror, Piero and Regge symmetries
The 6j symbol becomes the eigenfunction of the Schrodinger–like equation in the variable q, a continuous generalization of j12:
where Ψ(q) is related to
and p² is related with the square of the volume V of the associated tetrahedron.
The Cayley–Menger determinant permits to calculate the square of the volume of a generic tetrahedron in terms of squares of its edge lengths according to:
The condition for the tetrahedron with fixed edge lengths to exist as a polyhedron in Euclidean 3-space amounts to require V²> 0, while the V²= 0 and V²< 0 cases were associated by Ponzano and Regge to “flat” and nonclassical tetrahedral configurations respectively.
Major insight is provided by plotting both 6js and geometrical functions -volumes, products of face areas – of the associated tetrahedra in a 2-dimensional j12 - j23 plane , in whch the square “screen” of allowed ranges of j12 and j23 is used in all the pictures
- The mirror symmetry. The appearance of squares of tetrahedron edges entails that the invariance with respect to the exchange J ↔− J implies formally j ↔ – j -1 with respect to the entries of the 6j symbol.
- Piero line. In general, an exchange of opposite edges of a tetrahedron corresponds to different tetrahedra and different symbols. In Piero formula, there is a term due to this difference that vanishes when any pair of opposite edges are equal.
- Regge symmetries. The these arises through connection with the projective geometry of the elementary quantum of space, which
is associated to the polygonal inequalities -triangular and quadrilateral in the 6j case -, which have to be enforced in
any spin networks.
The basic Regge symmetry can be written in the following form:
The range of both J12 and J23, namely the size of the screen, is given by 2min (J1, J2, J3, J, J1 +ρ , J2 +ρ ,J3 +ρ, J + ρ).
Features of the tetrahedron volume function
Looking at the volume V as a function of x=J12 and y=J23 we get the expressions for the xVmaxand yVmax that correspond to the maximum of the volume for a fixed value of x or y:
The plots of these are called “ridge” curves on the x,y-screen. Each one marks configurations of the associated tetrahedron when two specific pairs of triangular faces are orthogonal. The corresponding values of the volume (xVmax,xand yVmax,y) are
F is the area of the triangle with sides a, b and c.Curves corresponding to V = 0, the caustic curves, obey the equations:
Symmetric and limiting cases
When some or all the j’s are equal, interesting features appear in the screen. Similarly when some are larger than others.
We can discuss the caustics of the 3j symbols as the limiting case of the corresponding 6j where three entries are larger than the other ones:
The extensive images of the exactly calculated 6j’s on the square screens illustrate how the caustic curves studied in this paper separate the classical and nonclassical regions, where they show wavelike and evanescent behaviour respectively. Limiting
cases, and in particular those referring to 3j and Wigner’s d matrix elements can be analogously depicted and discussed. Interesting also are the ridge lines, which separate the images in the screen tending to qualitatively different flattening of the quadrilateral,
namely convex in the upper right region, concave in the upper left and lower right ones, and crossed in the lower left region.