# Sage Interactions - Loop Quantum Gravity

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Contents

## Holomorphic factorization of the Quantum Tetrahedron

by David Horgan.

Moduli Space of Shapes of a Tetrahedron with Faces of Equal Area

The space of shapes of a tetrahedron with fixed face areas is naturally a symplectic manifold of real dimension two. This symplectic manifold turns out to be a Kahler manifold and can be

parametrized by a single complex coordinate Z given by the cross ratio of four complex numbers obtained by stereographically projecting the unit face normals onto the complex plane.

This Demonstration illustrates how this works in the simplest case of a tetrahedron T whose four face areas are equal. For convenience, the cross-ratio coordinate Z is shifted and rescaled

to z=(2Z-1)/Sqrt[3] so that the regular tetrahedron corresponds to z=i, in which case the upper half-plane is mapped conformally into the unit disc w=(i-z)/(i+z). The equi-area tetrahedron

T is then drawn as a function of the unit disc coordinate w.

Reference: L. Freidel, K. Krasnov, and E. R. Livine, "Holomorphic Factorization for a Quantum Tetrahedron".

## Quantum tetrahedron volume, area and angle eigenvalues

by David Horgan.

In this interact I calculate the angle, area and volume for a quantum tetrahedron The angle is found using the expression: theta = arccos((j3*(j3+1)-(j1*(j1+j1)-j2*(j2+1))/(2*sqrt(j1*(j1+j1)*j2*(j2+1)))) The area is found using the expression: A=sqrt(j1*(j1+1)) The volume is fund using the expression V^2 =M = 2/9(real antisymmetrix matrix)

Values of constants gamma is Immirzi parameter gamma =numerical_approx( ln(2)/(pi*sqrt(2))) #G = 6.63*10^-11 hbar= (1.61619926*10^-35)/(2*pi) lp is the planck length lp3=6*10^-104 Reference: Bohr-Sommerfeld Quantization of Space by Eugenio Bianchi and Hal M. Haggard. Reference: Shape in an atom of space: exploring quantum geometry phenomenology by Seth A. Major.

Research Blog: http://quantumtetrahedron.wordpress.com

Given the values of J1, J2, J3 and J4 this interact calculates the volume, area and angle eigenvalues of a quantum tetrahedron.

## Quantum tetrahedron Area Operator eigenvalues

by David Horgan.

Given the values of J1, J2, J3 and J4 this interact calculates the area eigenvalues of a quantum tetrahedron.