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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>> == Quantum tetrahedron volume eigenvalues == by David Horgan. Given the values of J1, J2, J3 and J4 this interact calculates the volume eigenvalues of a quantum tetrahedron. {{{#!sagecell html('<h3>Quantum tetrahedron Volume Eigenvalue</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 ') 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>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 for j in range(d): e= sqrt(s[j]) if e.imag() ==0: print (e) }}} |
Sage Interactions - Loop Quantum Gravity
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Quantum tetrahedron volume eigenvalues
by David Horgan. Given the values of J1, J2, J3 and J4 this interact calculates the volume eigenvalues of a quantum tetrahedron.