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| This page features interactive sage calculations concerned with aspects of loop Quantum Gravity | = Sage Interactions - Loop Quantum Gravity = goto [[interact|interact main page]] <<TableOfContents>> {{attachment:5-cell.gif}} == 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,angle eigenvalues of a quantum tetrahedron. {{{#!sagecell 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) d8a=numerical_approx(d8, digits=4) angle = numerical_approx(d8*180/pi, digits=4) print 'angle between faces 1 and 2 in quantum tetrahedron = ',d8a, 'radians' print 'angle between faces 1 and 2 in quantum tetrahedron = ',angle,'degrees' 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) #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 }}} == 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) 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 }}} |
Sage Interactions - Loop Quantum Gravity
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Contents
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,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.