Generating Functionals for Spin Foam Amplitudes by Hnybida

This week I have been reading the PhD thesis ‘Generating Functionals for Spin Foam Amplitudes’ by Jeff Hnybida. This is a few useful topic because the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

In the various approaches to Quantum Gravity such as Loop Quantum Gravity, Spin Foam Models and Tensor-Group Field theories use invariant tensors on a group, called intertwiners, as the basic building block of transition amplitudes. For the group SU(2) the contraction of these intertwiners in the pattern of a graph produces spin network amplitudes.

In this paper a generating functional for the exact evaluation of a coherent representation of these spin network amplitudes is constructed. This generating functional is defined for arbitrary graphs and depends only on a pair of spinors for each edge. The generating functional is a meromorphic polynomial in the spinor invariants which is determined by the cycle structure of the graph.
The expansion of the spin network generating function is given in terms of a basis of SU(2) intertwiners consisting of the monomials of the holomorphic spinor invariants. This basis, the discrete-coherent basis, is labelled by the degrees of the monomials and is  discrete. It also contains the precise amount of data needed to specify points in the classical space of closed polyhedra.

The focus the paper is on the 4-valent basis, which is the case of interest for Quantum Gravity. Simple relations between the discrete-coherent basis, the orthonormal basis, and the coherent basis are found.

The 4-simplex amplitude in this basis depends on 20 spins and is referred to as the 20j symbol. The 20j symbol is the exact evaluation of the coherent 4-simplex amplitude. 

20j

The asymptotic limit of the 20j symbol is found to give a generalization of the Regge action to Twisted Geometry.

3d quantum gravity

A triple of edge vectors meeting at a node must be invariant under the local rotational gauge transformations

hamiltonianequ328a

There is only one invariant rank three tensor on SU(2) up to normalization: The Wigner 3j symbol or Clebsch-Gordan coefficient. The 3j symbol has the interpretation as a quantum triangle and its three spins correspond to the lengths of its three edges, which close to form a triangle due to the SU(2) invariance. Contracting four 3j symbols in the pattern of a tetrahedron gives the  well-known 6j symbol which is the amplitude for each tetrahedron.

Coherent BF Theory
The coherent intertwiners are  a coherent state representation of the space of invariant tensors on SU(2). The exact evaluations computed later are a result of a special exponentiating property of
coherent states. Each SU(2) coherent state is labelled by a spinor |z 〉,  |z] denotes its contragradient version. Using a bra-ket notation for the spinors

generatinhgequ1.11

such that given two spinors z and w the two invariants which can be formed by contracting with either epsilon or delta are denoted

generatinhgequ1.12

The exponentiating property of the coherent states corresponds to the fact that the spin j representation is simply the tensor product of 2j copies of the spinor |z〉⊗2j   . A coherent rank n tensor on SU(2) is therefore the tensor product of n exponentiated spinors.

To make the coherent tensor invariant we group average using the Haar measure

generatinhgequ1.13

which is the denition of the Livine-Speziale intertwiner.

The coherent 6j symbol is constructed by contracting 4 coherent intertwiners in the pattern of a tetrahedron. Labeling each vertex by i = 1,..,4 and edges by pairs (ij) this amplitude depends on 6 spins jij = jji and 12 spinors |zij 〉≠ |zji〉 where the upper index denotes the vertex and the lower index the connected vertex. The coherent
amplitude in 3d is given by

generatinhgequ1.14

The asymptotics of the coherent amplitude have been studied extensively, however the actual evaluation of these amplitudes was not known. While the asymptotic analysis is important to check the semi-classical limit, the exact evaluation could be useful to study
recursion relations, coarse graining moves, or to perform numerical calculations.

To obtain the exact evaluation we use a special property of the Haar measure on SU(2) to express the group integrals above as Gaussian integrals. The generating functional is defined as

generatinhgequ1.15

we are able to compute the Gaussian integrals in above, not just for the tetrahedral graph but for any arbitrary graph. Performing the Gaussian integrals produces a determinant depending purely on the spinors. The determinant can be evaluated in general and can be expressed in terms of loops of the spin network graph.

For example, after integration and evaluating the determinant, the generating functional of the 3-simplex takes the form:

generatinhgequ1.16

4d quantum gravity

General Relativity in four dimensions is not topological, but it can  be formulated by a constrained four dimensional BF theory. That is if B is constrained to be of the form

generatinhgequ1.21

for a real tetrad 1-form e then the BF action becomes the Hilbert Palatini action for General Relativity. The aim of the spin foam program is to formulate a discretized version of these constraints that can break the topological invariance of BF theory and give rise
to the local degrees of freedom of gravity.

The advantage of formulating GR as a constrained BF theory is that, instead of quantizing Plebanski’s action, we can instead use the topological nature of BF theory to quantize  the discretized BF action and impose the  discretized constraints at the quantum level.
The first model of this type was proposed by Barret and Crane.
While this is not a quantization of a constrained system in the sense of Dirac it is a quantization of the Gupta-Bleuler type which was realised by Livine and Speziale  and led to corrected versions of the Barret-Crane model by Engle, Livine, Pereira, Rovelli  and by Freidel, Krasnov.

The behaviour of our spin network generating functional under
general coarse graining moves is a simple transformation of the coarse grained action in terms of lattice paths. For a square lattice, the generating functional expressed as sums over loops similar to gives precisely the partition function for the 2d Ising model.

ising

Since the Ising model and its renormalization are very well understood this example could provide a toy model for which one could base a study of the more complicated spin foam renormalization.

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Hamiltonian dynamics of a quantum of space

In this post I follow up on some of the work reviewed in the post:

The action of the quantum mechanical volume operator plays a fundamental role in discrete quantum gravity models, can be given as a second order difference equation. By a complex phase change this can be  turned into a discrete Schrödinger-like equation.

The introduction of discrete potential–like functions reveals the role of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols.

I’ll look at the underlying geometric features. When the spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary quantum of space, and an asymptotic picture emerges of the semiclassical and classical regimes.

The definition of coordinates adapted to Regge symmetry is used to construct a set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.

Introduction

For an elementary spin network as shown below

hamiltonianfig1

A quadrilateral and its Regge “conjugate”illustrating the elementary spin network representation of the symmetric coupling scheme: each quadrilateral is dissected into two triangles sharing, as a common side, the diagonal l.

 

the volume operator K = J1 .J2 × J3,  acts democratically on vectors J1; J2 and J3 plus a fourth one, J4, which closes a not necessarily planar quadrilateral vector diagram J1 + J2 + J3 + J4 = 0.

The matrix elements of K are computed to provide a Hermitian representation, whose features have been studied by many see posts:

By a suitable complex change of phase we can transform the imaginary antisymmetric representation into a real, time-independent Schrödinger equation which governs the Hamiltonian dynamics as a function of a discrete variable denoted l. The Hilbert space spanned by the eigenfunctions of the volume operator can be constructed combinatorially and geometrically, applying polygonal relationships to the two quadrilateral vector diagram which are conjugated by a hidden symmetry discovered by Regge .

Discrete schrodinger equation and Regge symmetry

Eigenvalues k and eigenfunctions Ψl(k) of the volume operator are obtained through the three–terms recursion relationship – see post:

Applying a change of phaseΨl (k)=(-i)lΦl(k) to obtain a real, finite difference Schrodinger–like equation

hamiltonianequ1The Ψl are the eigenfunctions of the volume operator expanded in the J12 = J1 +J2 basis. The matrix elements αl in  are given in terms of geometric quantities, namely

hamiltonianequ2αl is proportional to the product of the areas of the two triangles sharing the side of length l and forming a quadrilateral of sides J1 +½, J2 +½, J3 +½ and J4 +½.

The requirement that the four vectors form a (not necessarily planar) quadrilateral leads to identify the range of l with

hamiltonianequ3

which is also the dimension of the Hilbert space where the volume operator acts.

Hamiltonian Dynamics

The Hamiltonian operator for the discrete Schrödinger equation

hamiltonianequ1

can be written, in terms of the shift operator

hamiltonianequ6

The two-dimensional phase space (l, φ) supports the corresponding classical Hamiltonian function given by

hamiltonianequ7This is  illustrated in below for the two Regge conjugate quadrilaterals of the diagram above.

hamiltonianfig2

 

The quadrilaterals are now allowed to fold along l with φ seen as a torsion angle.

The classical regime occurs when quantum numbers j are large and l can be considered as a continuous variable. This limit for l permits us to draw the closed curves in the (l, k) plane when φ = 0 or φ=π . These curves have the physical meaning of torsional-like potential functions

hamiltonianequ8

viewing the quadrilaterals as mechanical systems.

hamiltonianfig4Potential functions U+ and U in  hamiltonianequ8are shown for two cases where the conjugated tetrahedra coincide.

  • left panel:   j1,j2, j3, j4=100,110,130,140 the tangential quadrilateral
  • right panel: j1= j2=j3=j4 =120 the ex-tangential quadrilateral.

During the classical motion, the diagonal l changes its value preserving the energy of the system. The result is a geometric configuration  a tetrahedron changing continuously its shape but preserving its volume as a constant of motion.

Quantum mechanics extends the domain of the canonical variables to regions of phase space classically not allowed. Boundaries of these regions are the so-called potential-energy curves particularly important in applications. They are defined as turning points, namely the points where for each value of energy the classical  changes sign. This happens when the momentum φ is either 0 or π.

hamiltonianequ328

The above conditions define closed curved in the l-energy plane. These curves have the physical meaning of torsional like potential functions.

At each value of the possible values E’ of the Hamiltonian are bounded by

hamiltonianequ328a

and the eigenvalues λk of the quantum system are bounded by

hamiltonianequ329

 

The sagemath code and output for this work is shown below:

hamiltonianfig5

hamiltonianfig5a

hamiltonianfig5b