In this work the authors investigate the 4d path integral for Euclidean quantum gravity on a hypercubic lattice, as given by the EPRL-FK model. To tackle the problem, they restrict the path to a set of quantum geometries that reflects the lattice symmetries. In particular, the sum over intertwiners is restricted to quantum cuboids, that is, coherent intertwiners which describe a cuboidal

geometry in the large-j limit.

Using asymptotic expressions for the vertex amplitude, several interesting properties of the state sum are found.

- The value of coupling constants in the amplitude functions determines whether geometric or non-geometric configurations dominate the path integral.
- There is a critical value of the coupling constant α, which separates two phases. In one phase the main contribution

comes from very irregular and crumpled states. In the other phase, the dominant contribution comes from a highly regular configuration, which can be interpreted as flat Euclidean space, with small non-geometric perturbations around it. - States which describe boundary geometry with high

torsion have exponentially suppressed physical norm.

**The symmetry-restricted state sum**

Will work on a regular hypercubic lattice in 4d. On this lattice consider only states which conform to the lattice symmetry. This is a condition on the intertwiners, which corresponds to cuboids.

A cuboid is completely determined by its three edge lengths, or equivalently by its three areas.

All internal angles are π/2 , and the condition of regular cuboids on all dual edges of the lattice result in a high degree of symmetries on the labels: The area and hence the spin on each two parallel squares of the lattice which are translations perpendicular to the squares, have to be equal.

The high degree of symmetry will make all quantum geometries flat. The analysis carried out here is therefore not suited for describing local curvature.

**Introduction**

The plan of the paper is as follows:

- Review of the EPRL-FK spin foam model
- Semiclassical regime of the path integral
- Construction of the quantum cuboid intertwiner
- Full vertex amplitude, in particular describe its asymptotic expression for large spins
- Numerical investigation of the quantum path integral

The spin foam state sum employed is the Euclidean EPRL-FK model with Barbero-Immirzi parameter γ < 1. The EPRL-FK model is defined on an arbitrary 2-complexes. A 2-complex is determined by its vertices v, its edges e connecting two vertices, and faces f which are bounded by the edges.

The path integral is formulated as a sum over states. A state in this context is given by a collection of spins – irreducible representations

j_{f} ∈ 1/2 N of SU(2) to the faces, as well as a collection of intertwiners ι_{e} on edges.

The actual sum is given by

where A_{f} , A_{e} and A_{v} are the face-, edge- and vertex- amplitude functions, depending on the state. The sum has to be carried out over all spins, and over an orthonormal orthonormal basis in the intertwiner space at each edge.

The allowed spins j_{f} in the EPRL-FK model are such

that are both also half-integer spins.

The face amplitudes are either

The edge amplitudes A_{e} are usually taken to be equal to 1.

In Sagemath code this looks like:

**Coherent intertwiners**

In this paper, the space-time manifold used is M∼ T³×[0, 1] is the product of the 3-torus T3 and a closed interval. The space is compactified toroidally. M is covered by 4d hypercubes, which

form a regular hypercubic lattice H.There is a vertex for each hypercube, and two vertices are connected by an edge whenever two hypercubes intersect ina 3d cube. The faces of are dual to squares in H, on which four hypercubes meet.The geometry will be encoded in the state, by specification of spins j_{f}

and intertwiners ι_{e}.

Intertwiners ι_{e} can be given a geometric interpretation in terms of polyhedra in R³. Given a collection of spins j_{1}, . . . j_{n} and vectors n_{1}, . . . n_{n} which close . Can define the coherent polyhedron

The geometric interpretation is that of a polyhedron, with face areas j_{f} and face normals n_{i}. The closure condition ensures that such a polyhedron exists.

We are interested in the large j-regime of the quantum cuboids. In this limit, these become classical cuboids which are completely specified by their three areas. Therefore, a

semiclassical configuration is given by an assignment of

areas a = l_{p}² to the squares of the hypercubic lattice.

Denote the four directions in the lattice by x, y, z, t. The areas satisfy

The two constraints which reduce the twisted geometric

configurations to geometric configurations are given by:

For a non-geometric configuration, define the 4-volume of a hypercube as:

Define the four diameters to be:

then we have, V_{4} = d_{x}d_{y}d_{z}d_{t}

We also define the non- geometricity as:

as a measure of the deviation from the constraints.

In sagemath code this looks like:

**Quantum Cuboids**

We let’s look at the quantum theory. In the 2-complex, every edge has six faces attached to it, corresponding to the six faces of the cubes. So any intertwiner in the state-sum will be six-valent, and therefore can be described by a coherent polyhedron with six faces. In our setup, we restrict the state-sum to coherent cuboids, or quantum cuboids. A cuboid is characterized by areas on opposite sides of the cuboid being equal, and the respective normals being negatives of one another

The state ιj_{1},j_{2},j_{3} is given by:

The vertex amplitude for a Barbero-Immirzi parameter γ < 1 factorizes as A_{v} = A^{+}_{v}A^{–}_{v} with

with the complex action

where, a is the source node of the link l, while b is its target node.

**Large j asymptotics**

The amplitudes A^{±}_{v} possess an asymptotic expression for large j_{l}. There are two distinct stationary and critical points, satisfying the equations.

for all links ab . Using the convention shown below

having fixed g_{0} = 1, the two solutions Σ_{1} and Σ_{2} are

The amplitudes A^{±}_{v }satisfy, in the large j limit,

In the large j-limit, the norm squared of the quantum cuboid states is given by:

For the state sum, in the large-j limit on a regular hypercubic lattice:

In sagemath code this looks like:

**Related articles**

- Discrete
**harmonic functions (johndcook.com)**

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This looks great when animated:

below is the python program for these diagrams.

from scipy.special import jv, legendre, sph_harm, jacobi

from scipy.misc import factorial, comb

from numpy import floor, sqrt, sin, cos, exp, power

from math import pi

def wignerd(j,m,n=0,approx_lim=10):

”’

Wigner “small d” matrix. (Euler z-y-z convention)

example:

j = 2

m = 1

n = 0

beta = linspace(0,pi,100)

wd210 = wignerd(j,m,n)(beta)

some conditions have to be met:

j >= 0

-j <= m <= j

-j <= n <= j

The approx_lim determines at what point

bessel functions are used. Default is when:

j > m+10

and

j > n+10

for integer l and n=0, we can use the spherical harmonics. If in

addition m=0, we can use the ordinary legendre polynomials.

”’

if (j < 0) or (abs(m) > j) or (abs(n) > j):

raise ValueError(“wignerd(j = {0}, m = {1}, n = {2}) value error.”.format(j,m,n) \

+ ” Valid range for parameters: j>=0, -j<=m,n<=j.”)

if (j > (m + approx_lim)) and (j > (n + approx_lim)):

#print ‘bessel (approximation)’

return lambda beta: jv(m-n, j*beta)

if (floor(j) == j) and (n == 0):

if m == 0:

#print ‘legendre (exact)’

return lambda beta: legendre(j)(cos(beta))

elif False:

#print ‘spherical harmonics (exact)’

a = sqrt(4.*pi / (2.*j + 1.))

return lambda beta: a * conjugate(sph_harm(m,j,beta,0.))

jmn_terms = {

j+n : (m-n,m-n),

j-n : (n-m,0.),

j+m : (n-m,0.),

j-m : (m-n,m-n),

}

k = min(jmn_terms)

a, lmb = jmn_terms[k]

b = 2.*j – 2.*k – a

if (a < 0) or (b < 0):

raise ValueError(“wignerd(j = {0}, m = {1}, n = {2}) value error.”.format(j,m,n) \

+ ” Encountered negative values in (a,b) = ({0},{1})”.format(a,b))

coeff = power(-1.,lmb) * sqrt(comb(2.*j-k,k+a)) * (1./sqrt(comb(k+b,b)))

#print ‘jacobi (exact)’

return lambda beta: coeff \

* power(sin(0.5*beta),a) \

* power(cos(0.5*beta),b) \

* jacobi(k,a,b)(cos(beta))

def wignerD(j,m,n=0,approx_lim=10):

”’

Wigner D-function. (Euler z-y-z convention)

This returns a function of 2 to 3 Euler angles:

(alpha, beta, gamma)

gamma defaults to zero and does not need to be

specified.

The approx_lim determines at what point

bessel functions are used. Default is when:

j > m+10

and

j > n+10

usage:

from numpy import linspace, meshgrid

a = linspace(0, 2*pi, 100)

b = linspace(0, pi, 100)

aa,bb = meshgrid(a,b)

j,m,n = 1,1,1

zz = wignerD(j,m,n)(aa,bb)

”’

return lambda alpha,beta,gamma=0: \

exp(-1j*m*alpha) \

* wignerd(j,m,n,approx_lim)(beta) \

* exp(-1j*n*gamma)

if __name__ == ‘__main__’:

”’

just a bunch of plots in (phi,theta) for

integer and half-integer j and where m and

n take values of [-j, -j+1, …, j-1, j]

Note that all indexes can be any real number

with the conditions:

j >= 0

-j <= m <= j

-j <= n <= j

”’

from matplotlib import pyplot, cm, rc

from numpy import linspace, arange, meshgrid, real, imag, arccos

rc(‘text’, usetex=False)

ext = [0.,2.*pi,0.,pi]

phi = linspace(ext[0],ext[1],200)

theta = linspace(ext[2],ext[3],200)

pphi,ttheta = meshgrid(phi,theta)

# The maximum value of j to plot. Will plot real and imaginary

# distributions for j = 0, 0.5, … maxj

maxj = 2.0

for j in arange(0,maxj+.1,step=0.5):

fsize = (j*2+3,j*2+3)

title = ‘WignerD(j,m,n)(phi,theta)’

if j == 0:

fsize = (4,4)

else:

title += ‘, j = ‘+str(j)

figr = pyplot.figure(figsize=fsize)

figr.suptitle(r’Real Part of ‘+title)

figi = pyplot.figure(figsize=fsize)

figi.suptitle(r’Imaginary Part of ‘+title)

for fig in [figr,figi]:

fig.subplots_adjust(left=.1,bottom=.02,right=.98,top=.9,wspace=.02,hspace=.1)

if j == 0:

fig.subplots_adjust(left=.1,bottom=.1,right=.9,top=.9)

if j == 0.5:

fig.subplots_adjust(left=.2,top=.8)

if j == 1:

fig.subplots_adjust(left=.15,top=.85)

if j == 1.5:

fig.subplots_adjust(left=.15,top=.85)

if j == 2:

fig.subplots_adjust(top=.87)

if j != 0:

axtot = fig.add_subplot(1,1,1)

axtot.axesPatch.set_alpha(0.)

axtot.xaxis.set_ticks_position(‘top’)

axtot.xaxis.set_label_position(‘top’)

axtot.yaxis.set_ticks_position(‘left’)

axtot.spines[‘left’].set_position((‘outward’,10))

axtot.spines[‘top’].set_position((‘outward’,10))

axtot.spines[‘right’].set_visible(False)

axtot.spines[‘bottom’].set_visible(False)

axtot.set_xlim(-j-.5,j+.5)

axtot.set_ylim(-j-.5,j+.5)

axtot.xaxis.set_ticks(arange(-j,j+0.1,1))

axtot.yaxis.set_ticks(arange(-j,j+0.1,1))

axtot.set_xlabel(‘n’)

axtot.set_ylabel(‘m’)

nplts = 2*j+1

data_j=[]

data_zz=[]

for m in arange(-j,j+0.1,step=1):

for n in arange(-j,j+0.1,step=1):

print j,m,n

zz = wignerD(j,m,n)(pphi,ttheta)

data_j.append(j)

data_zz.append(zz)

i = n+j + nplts*(j-m)

for fig,data in zip((figr,figi), (real(zz),imag(zz))):

ax = fig.add_subplot(nplts, nplts, i+1, projection=’polar’)

plt = ax.pcolormesh(pphi,ttheta,data.copy(),

cmap=cm.jet,

#cmap=cm.RdYlBu_r,

vmin=-1., vmax=1.)

if j == 0:

ax.grid(True, alpha=0.5)

ax.set_title(r’j,m,n = (0,0,0)’, position=(0.5,1.1), size=12)

ax.set_xlabel(r’$\phi$’)

ax.set_ylabel(r’$\theta$’, rotation=’horizontal’, va=’bottom’)

ax.xaxis.set_ticks([0,.25*pi,.5*pi,.75*pi,pi,1.25*pi,1.5*pi,1.75*pi])

ax.xaxis.set_ticklabels([‘0′,r’$\frac{\pi}{4}$’,r’$\frac{\pi}{2}$’,r’$\frac{3 \pi}{4}$’,r’$\pi$’,r’$\frac{5 \pi}{4}$’,r’$\frac{3 \pi}{2}$’,r’$\frac{7 \pi}{4}$’], size=14)

ax.yaxis.set_ticks([0,.25*pi,.5*pi,.75*pi,pi])

ax.yaxis.set_ticklabels([‘0′,r’$\frac{\pi}{4}$’,r’$\frac{\pi}{2}$’,r’$\frac{3 \pi}{4}$’,r’$\pi$’], size=14)

else:

ax.xaxis.set_ticks([])

ax.yaxis.set_ticks([])

ax.set_xlim(ext[0],ext[1])

ax.set_ylim(ext[2],ext[3])

if j == 0:

fig.colorbar(plt, pad=0.07)

# uncomment the following if you want to save these to image files

#figr.savefig(‘wignerD_j’+str(j)+’_real.png’, dpi=150)

#figi.savefig(‘wignerD_j’+str(j)+’_imag.png’, dpi=150)

#pyplot.show()

#pl.plot(j_data,zz_data)

print ‘j_data=’,data_j

#print ‘zz_data=’, data_zz

**Related posts**

- The wigner d matrix
- Numerical work with sagemath 23: wigner reduced rotation matrix elements as limits of 6j symbols
- Review of wigner nj symbols
- Exact computation and asymptotic approximations of 6j symbols illustration of their semiclassical limits

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The starting point is to take the action of the area operator in the spin representation, and to consider the following normalized trace of the area operator:

If this give a well-defined operator on either the kinematical Hilbert space or on the Hilbert space of fully gauge-invariant wave functions, and also if we took the limit Λ_{fix} →∞ it is possible to read off the spectrum from the representation. There, the oscillatory behaviour of the sine function is suppressed by a factor of 1/d_{j }which leads to a discrete spectrum for sufficiently small spins j.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.1 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.3 and j=1…100.

And below the figure from the original paper, A new realization of quantum geometry.

Mathematica code for the normalized trace of the gauge invariant area operator for μ = 0.05, 0.1 and 0.3 and j=1…100

**Related posts**

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What I’m posting at the moment is some exploratory work looking at the behaviour of the Wigner D matrix elements and U(1)area operator using python and sagemath.

Below I look at the general behaviour of the Wigner D matrix elements:

Below Here I look at the behaviour of the spectrum of the area operator with U(1).

Below I look at the how the area varies with μ for the values 0.1, 0.3, 0.5:

- Exact and asymptotic computations of elementary spin networks (quantumtetrahedron.wordpress.com)
- State Sums and Geometry by Frank Hellmann (quantumtetrahedron.wordpress.com)
- The length operator in Loop Quantum Gravity by Bianchi (quantumtetrahedron.wordpress.com)
- Semiclassical analysis of Loop Quantum Gravity by Claudio Perini (quantumtetrahedron.wordpress.com)

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The quantum 6j symbol is a function of a 6-tuple jab, 1 ≤ a ≤ b ≤ 4. The 6j symbols

for q = 1 were introduced as a tool in atomic spectroscopy

by Racah, and then studied mathematically by Wigner. 6j symbols

for Uq(sl2) were introduced by Kirillov and Reshetikhin, who used them to generalize the Jones knot invariant. Turaev and Viro used them to define three manifold invariants or quantum gravity with a cosmological constant.

I have started doing some preliminary work with sagemath on 3j symbols, 6j symbols, the quantum integer and on the gram matrix.

** Related articles**

- Hamiltonian dynamics of a quantum of space (quantumtetrahedron.wordpress.com)

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The 6j symbols tend asymptotically to Wigner d^{l}_{nm} functions when some angular momenta are large where θ assumes certain discrete values.

These formulas are illustrated below:

This can be modelled using sagemath.

The routine gives some great results:

For N=320, M=320, n=0, m=0, l=20, L=0, Lmax=640

Wigner 6j vs cosθ_{L}

For N=320, M=320, n=0, m=0, l=10, L=0, Lmax=640

Wigner 6j vs cosθ_{L}

For N=320, M=320, n=0, m=0, l=5, L=0, Lmax=640

Wigner 6j vs cosθ_{L}

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- Exact and asymptotic computations of elementary spin networks
- The tetrahedron and its Regge conjugate

Fortunately sagemath can implement at number of great codes for hypergeometric functions including:

These give excellent results:

The ability to use mpmath code is very useful since it enables me to calculate a wide range of hypergeometric functions as can be seen on the reference pages. I’ll be using this over the coming posts starting with :

Exact Computation and Asymptotic Approximations of 6j Symbols:

Illustration of Their Semiclassical Limits which is in preparation.

**Related articles**

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Lets see what sagemath can do:

- Sagemath 18: Calculation of the Matrix Elements of Thiemann’s Hamiltonian Constraint in Loop Quantum Gravity (quantumtetrahedron.wordpress.com)
- A generalized Hamiltonian Constraint Operator in Loop Quantum Gravity and its simplest Euclidean Matrix Elements by Gaul and Rovelli (quantumtetrahedron.wordpress.com)

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The Rovelli-Speziale semi-classical tetrahedron is a semiclassical quantum state corresponding to the classical geometry of the tetrahedron determined by the areas A1, . . . ,A4 of its faces and

two dihedral angles θ12, θ34 between A1 and A2 respectively A3 and A4.

It is defined as a state in the intertwiner basis |j12>

with coefficients cj12 such that

in the large scale limit, for all ij. The large scale limit considered here is taken when all spins are large.

The expression of the coefficients cj12 verifying the requirements is:

where joand ko are given real numbers respectively linked to θ12 and θ34 through the following equations:

σj12 is the variance which is appropriately fixed and the phase φ(jo, ko) is the dihedral angle to join an auxiliary tetrahedron related to the asymptotic of the 6j symbol performing the change

of coupling in the intertwiner basis.

For a classical regular tetrahedron, using the expression the integrated classical curvature scales linearly in terms of the length of its hinges because the angles do not change in the equilateral configuration when the length is rescaled, which means that the

integrated classical curvature scales as a square root function of the area of a face.

In the plot above we can see that the expected values of R on coherent states and semi-classical regular tetrahedra for large spins scales as a square root function of the spin, this matches the semi-classical evolution we expect.

- Sagemath 19 Part 2: the Spectrum of the Ricci Operator for a Monochromatic 4 – Valent Node Dual to an Equilateral Tetrahedron (quantumtetrahedron.wordpress.com)
- Sagemath 19 part 1: The spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron (quantumtetrahedron.wordpress.com)

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In my last post I indicated how far I had got in porting the code for the monochromatic 4-valent node from Mathematica to sagemath. This is essentially complete now and I have just got to output a graph of eigenvalues of curvature versus spin.

*Sagemath code for the spectrum of the Ricci Operator for a monochromatic 4-valent node dual to an equilateral tetrahedron*

The curvature is written as a combination of the length of a hinge and the deficit angle around it.

As yet the code is unoptimised. Below is a sample of the output so far:

So now the eigenvalues have been found I can start to make use of them in calculations of the Hamiltonian Constraint Operator.

Update:

- Sagemath 19 part 1: The spectrum of the Ricci operator for a monochromatic 4 – valent node dual to an equilateral tetrahedron (quantumtetrahedron.wordpress.com)
- Numerical work with Sagemath 17: Exploring curvature (quantumtetrahedron.wordpress.com)

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