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Sage Interactions - Loop Quantum Gravity

Contents

Sage Interactions - Loop Quantum Gravity

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 Kähler 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.

-  ⇤ ← Revision 14 as of 2013-09-23 17:29:48 → 
  Size: 5591
  Editor: dch252
  Comment:
+   ← Revision 30 as of 2019-04-06 16:42:59 → ⇥
  Size: 10242
  Editor: chapoton
  Comment: py3 print
-Deletions are marked like this.
+Additions are marked like this.
 Line 8:
-== Quantum tetrahedron volume and angle eigenvalues ==
+== Holomorphic factorization of the Quantum Tetrahedron ==
 Line 11:
-Given the values of J1, J2, J3 and J4 this interact calculates the volume and angle eigenvalues of a quantum tetrahedron.
+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 Kähler 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".

{{{#!sagecell                

# -*- coding: utf-8 -*-
"""
Created on Mon Feb 10 20:05:11 2014

@author: David Horgan
"""

from scipy.optimize import fsolve
import math



#scene.range = 10

#cylinder(pos=(0,0,0), radius=1.0,axis=(0,0,1), color=color.white)

#scene2 = display(title='Examples of Tetrahedrons',
    # x=0, y=0, width=600, height=600,
    # center=(0,0,0), background=(1,1,1))

#scene2.visible = True
#scene2.range = 10




#unit circle plot
C = circle((0,0), 1)





#input points from unit circle

p1=0.6
p2=0.7


#face1
a1=(((1 - p2**2)/(2*(1 + p1)))**(1/4), (1/2)*(p2 + 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0)
a2=(((1 - p2**2)/(2*(1 + p1)))**(1/4),  (1/2)*(p2 - 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0 )
a3=(((1/2)*( (1 + p1)**3)* (1 - p2**2))**(1/4),p2*((2*(1 + p1))/(1 - p2**2))**(1/4), sqrt(1 - p1**2)*((1 - p2**2)/(2*(1 + p1)))**(1/4) )
a=[a1,a2,a3]


#face2

b1=(0, 0, 0)
b2=(((1 - p2**2)/(2*(1 + p1)))**(1/4), (1/2)*(p2 - 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0)
b3=(((1/2)*((1 + p1)**3)*(1 - p2**2))**(1/4),p2*((2*(1 + p1))/(1 - p2**2))**(1/4), sqrt(1-p1**2)*((1-p2**2)/(2*(1+p1)))**(1/4)  )
b=[b1,b2,b3]

#face3
c1=(0, 0, 0)
c2=( ((1 - p2**2)/(2*(1 + p1)))**(1/4), (1/2)*(p2 + 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0)
c3=(((1/2)*((1 + p1)**3)*(1 - p2**2))**(1/4), p2*((2*(1 + p1))/(1 - p2**2))**(1/4),  sqrt(1 - p1**2)*((1 - p2**2)/(2*(1 + p1)))**(1/4) )
c=[c1,c2,c3]


#face4
d1=(0, 0, 0)
d2=(((1 - p2**2)/(2*(1 + p1)))**(1/4), (1/2)*(p2 + 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0)
d3 =(((1 - p2**2)/(2*(1 + p1)))**(1/4), (1/2)*(p2 - 1)*((2*(1 + p1))/(1 - p2**2))**(1/4), 0)
d=[d1,d2,d3]

#printresults
print('a1={}'.format(a1))
print('a2={}'.format(a2))
print('a3={}'.format(a3))

print('b1={}'.format(b1))
print('b2={}'.format(b2))
print('b3={}'.format(b3))

print('c1={}'.format(c1))
print('c2={}'.format(c2))
print('c3={}'.format(c3))

print('d1={}'.format(d1))
print('d2={}'.format(d2))
print('d3={}'.format(d3))



G = Graphics()

#tetrahedron faces - coloured
P1=polygon([a1,a2, a3], color='red')
P2=polygon([b1,b2, b3],color='yellow')
P3=polygon([c1,c2, c3],color='blue')
P4=polygon([d1,d2, d3],color='green')

#polygon faces
#P1=polygon([a1,a2, a3])
#P2=polygon([b1,b2, b3])
#P3=polygon([c1,c2, c3])
#P4=polygon([d1,d2, d3])

#polygon([a1,a2, a3])
#polygon([b1,b2, b3])
#polygon([c1,c2, c3])
#polygon([d1,d2, d3])

g=G+P1+P2+P3+P4
show(g, title='tet(w)', viewer='tachyon')
show(C, title='w plane')
}}}







== 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.
-Line 15:
+Line 175:
-html('<h3>Quantum tetrahedron Volume and Angle Eigenvalues</h3>')
html('Enter the four J values into the input boxes')
html('k values k ranges from kmin to kmax in integer steps')
html('The dimension d of the Hilbert space H4,  d = kmax - kmin + 1')
html('kmin = max(|j1-j2|,|j3 -j4|) kmax = min(j1+j2,j3 +j4)')
html('The the dimension of the hilbert space is given by d = kmax -kmin + 1') 
html('V^2 =M = 2/9(real antisymmetrix matrix))')
html('Spins must satisfy (j1+j2)<= (j3+j4)')
html('Reference: Bohr-Sommerfeld Quantization of Space by Eugenio Bianchi and Hal M. Haggard ')
html('Reference: Shape in an atom of space: exploring quantum geometry phenomenology by Seth A. Major ')
-Line 53:
+Line 204:
-        print "Angle eigenvalue in radians=",(d8)
+        d8a=numerical_approx(d8, digits=4)
        angle = numerical_approx(d8*180/pi, digits=4)
        if angle != NaN:
             print('angle between faces 1 and 2 in quantum tetrahedron = {} radians'.format(d8a))
             print('angle between faces 1 and 2 in quantum tetrahedron = {} degrees'.format(angle))

        html('<h3>main sequence Area eigenvalues</h3>')
        lp=1.61619926*10^-35
        main1=numerical_approx(sqrt(j1*(j1+1)),digits=4)
        areamain1 =0.5*lp^2*main1
        print('Area of face 1=', areamain1, 'm2')
        main2=numerical_approx(sqrt(j2*(j2+1)),digits=4)
        areamain2 =0.5*lp^2*main2
        print('Area of face 2=', areamain2, 'm2')
        main3=numerical_approx(sqrt(j3*(j3+1)),digits=4)
        areamain3 =0.5*lp^2*main3
        print('Area of face 3=', areamain3, 'm2')
        main4=numerical_approx(sqrt(j4*(j4+1)),digits=4)
        areamain4 =0.5*lp^2*main4
        print('Area of face 4=', areamain4, 'm2')
        area = areamain1 + areamain3 +areamain3+areamain4
        print('Total area of quantum tetrahedron =', area, 'm2')

        html('<h3>Values of Volume Eigenvalue</h3>')
        kmin = int(max(abs(j1-j2),abs(j3 -j4)))
        kmax = int(min((j1+j2),(j3 +j4)))
        d = kmax -kmin + 1
        y=numpy.arange(kmin,kmax+1,1)
        kmatrix = matrix(CDF,int(d), int(d))
        r=list()
        for j in range(d):
            k=int(y[j])
            c1 = -i*k
            c2 = sqrt(4*k*k - 1)
            c3 = sqrt(j1*(j1+1))
            c4 = sqrt((2*j1+1))
            c5 = sqrt(j3*(j3+1))
            c6 = sqrt((2*j3+1))
            c7 = wigner_6j(j1,1,j1,k,j2,k-1)
            c8 = wigner_6j(j3,1,j3,k,j4,k-1)
            a = c1*c2*c3*c4*c5*c6*c7*c8
            r.append(a)
            q = numerical_approx(a, digits=10)
        for j in range(d-1):
            kmatrix[[j],[j+1]]=r[j+1]
            kmatrix[[j+1],[j]]=-r[j+1]
    
        M = (2/9)*kmatrix
        s = M.eigenvalues()
        lp3=6*10^-104
        for j in range(d):
            e = sqrt(s[j])
            vol = lp3*e
            volume = numerical_approx(vol, digits=2)
            if e.imag() ==0:
                print("volume eigenvalue =", e)
                print("volume of tetrahedron =", volume, 'm3')
                

}}}


== 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.

{{{#!sagecell

html('<h3>Quantum tetrahedron Volume and Angle Eigenvalues</h3>')
html('Enter the four J values into the input boxes')
html('k values k ranges from kmin to kmax in integer steps')
html('The dimension d of the Hilbert space H4,  d = kmax - kmin + 1')
html('kmin = max(|j1-j2|,|j3 -j4|) kmax = min(j1+j2,j3 +j4)')
html('The the dimension of the hilbert space is given by d = kmax -kmin + 1') 
html('V^2 =M = 2/9(real antisymmetrix matrix))')
html('Spins must satisfy (j1+j2)<= (j3+j4)')
html('Reference: Bohr-Sommerfeld Quantization of Space by Eugenio Bianchi and Hal M. Haggard ')
html('Reference: Shape in an atom of space: exploring quantum geometry phenomenology by Seth A. Major ')






import numpy



@interact




def _(j1 = input_box(6.0, 'J1'),
      j2= input_box(6.0, 'J2'),
      j3= input_box(6.0, 'J3'),
      j4= input_box(7.0, 'J1'), auto_update=False):
    if (j1+j2)<= (j3+j4):
        html('<h3>Value of Angle eigenvalue in radians</h3>')
        d2=j3*(j3+1)
        d3=j1*(j1+1)
        d4=j2*(j2+1)
        d5=d2-d3-d4
        d6=2*sqrt(d3*d4)
        d7=d5/d6
        d8=arccos(d7)
        print("Angle eigenvalue in radians=", d8)
-Line 76:
+Line 334:
-            #print r
-Line 80:
+Line 337:
-            #print kmatrix
-Line 83:
+Line 339:
-        #print M
        s=M.eigenvalues()
        #print s
+        s = M.eigenvalues()
-Line 92:
+Line 346:
-                print "volume eigenvalue =",(e)
                print "volume of tetrahedron in m3 =", volume
+                print("volume eigenvalue =", e)
                print("volume of tetrahedron in m3 =", volume)
-Line 97:
+Line 351:
-== Quantum tetrahedron Area Operator eigenvalues ==
by David Horgan. 

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

{{{#!sagecell

html('<h3>Quantum tetrahedron Volume and Angle Eigenvalues</h3>')
html('Enter the four J values into the input boxes')
html('k values k ranges from kmin to kmax in integer steps')
html('The dimension d of the Hilbert space H4,  d = kmax - kmin + 1')
html('kmin = max(|j1-j2|,|j3 -j4|) kmax = min(j1+j2,j3 +j4)')
html('The the dimension of the hilbert space is given by d = kmax -kmin + 1') 
html('V^2 =M = 2/9(real antisymmetrix matrix))')
html('Spins must satisfy (j1+j2)<= (j3+j4)')
html('Reference: Bohr-Sommerfeld Quantization of Space by Eugenio Bianchi and Hal M. Haggard ')
html('Reference: Shape in an atom of space: exploring quantum geometry phenomenology by Seth A. Major ')






import numpy



@interact




def _(j1 = input_box(6.0, 'J1'),
      j2= input_box(6.0, 'J2'),
      j3= input_box(6.0, 'J3'),
      j4= input_box(7.0, 'J1'), auto_update=False):
    if (j1+j2)<= (j3+j4):
        html('<h3>Value of Angle eigenvalue in radians</h3>')
        d2=j3*(j3+1)
        d3=j1*(j1+1)
        d4=j2*(j2+1)
        d5=d2-d3-d4
        d6=2*sqrt(d3*d4)
        d7=d5/d6
        d8=arccos(d7)
        print "Angle eigenvalue in radians=",(d8)


        html('<h3>Values of Volume Eigenvalue</h3>')
        kmin = int(max(abs(j1-j2),abs(j3 -j4)))
        kmax = int(min((j1+j2),(j3 +j4)))
        d = kmax -kmin + 1
        y=numpy.arange(kmin,kmax+1,1)
        kmatrix = matrix(CDF,int(d), int(d))
        r=list()
        for j in range(d):
            k=int(y[j])
            c1 = -i*k
            c2 = sqrt(4*k*k - 1)
            c3 = sqrt(j1*(j1+1))
            c4 = sqrt((2*j1+1))
            c5 = sqrt(j3*(j3+1))
            c6 = sqrt((2*j3+1))
            c7 = wigner_6j(j1,1,j1,k,j2,k-1)
            c8 = wigner_6j(j3,1,j3,k,j4,k-1)
            a = c1*c2*c3*c4*c5*c6*c7*c8
            r.append(a)
            q=numerical_approx(a, digits=10)
            #print r
        for j in range(d-1):
            kmatrix[[j],[j+1]]=r[j+1]
            kmatrix[[j+1],[j]]=-r[j+1]
            #print kmatrix
    
        M = (2/9)*kmatrix
        #print M
        s=M.eigenvalues()
        #print s    
        lp3=6*10^-104
        for j in range(d):
            e= sqrt(s[j])
            vol = lp3*e
            volume = numerical_approx(vol, digits=2)
            if e.imag() ==0:
                print "volume eigenvalue =",(e)
                print "volume of tetrahedron in m3 =", volume
                

}}}

Diff for "interact/Loop Quantum Gravity"

Sage Interactions - Loop Quantum Gravity

Holomorphic factorization of the Quantum Tetrahedron

Quantum tetrahedron volume, area and angle eigenvalues

Quantum tetrahedron Area Operator eigenvalues