def test_tensorsymmetry(): Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') sym = tensorsymmetry([1] * 2) sym1 = TensorSymmetry(get_symmetric_group_sgs(2)) assert sym == sym1 sym = tensorsymmetry([2]) sym1 = TensorSymmetry(get_symmetric_group_sgs(2, 1)) assert sym == sym1 sym2 = tensorsymmetry() assert sym2.base == Tuple() and sym2.generators == Tuple(Permutation(1)) raises(NotImplementedError, lambda: tensorsymmetry([2, 1]))
def test_tensorsymmetry(): Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') sym = tensorsymmetry([1]*2) sym1 = TensorSymmetry(get_symmetric_group_sgs(2)) assert sym == sym1 sym = tensorsymmetry([2]) sym1 = TensorSymmetry(get_symmetric_group_sgs(2, 1)) assert sym == sym1 sym2 = tensorsymmetry() assert sym2.base == Tuple() and sym2.generators == Tuple(Permutation(1)) raises(NotImplementedError, lambda: tensorsymmetry([2, 1]))
def test_TensorIndexType(): D = Symbol('D') G = Metric('g', False) Lorentz = TensorIndexType('Lorentz', metric=G, dim=D, dummy_fmt='L') m0, m1, m2, m3, m4 = tensor_indices('m0:5', Lorentz) sym2 = tensorsymmetry([1]*2) sym2n = tensorsymmetry(*get_symmetric_group_sgs(2)) assert sym2 == sym2n g = Lorentz.metric assert str(g) == 'g(Lorentz,Lorentz)' assert Lorentz.eps_dim == Lorentz.dim TSpace = TensorIndexType('TSpace') i0, i1 = tensor_indices('i0 i1', TSpace) g = TSpace.metric A = tensorhead('A', [TSpace]*2, [[1]*2]) assert str(A(i0,-i0).canon_bp()) == 'A(TSpace_0, -TSpace_0)'
def test_TensorIndexType(): D = Symbol('D') G = Metric('g', False) Lorentz = TensorIndexType('Lorentz', metric=G, dim=D, dummy_fmt='L') m0, m1, m2, m3, m4 = tensor_indices('m0:5', Lorentz) sym2 = tensorsymmetry([1] * 2) sym2n = tensorsymmetry(*get_symmetric_group_sgs(2)) assert sym2 == sym2n g = Lorentz.metric assert str(g) == 'g(Lorentz,Lorentz)' assert Lorentz.eps_dim == Lorentz.dim TSpace = TensorIndexType('TSpace') i0, i1 = tensor_indices('i0 i1', TSpace) g = TSpace.metric A = tensorhead('A', [TSpace] * 2, [[1] * 2]) assert str(A(i0, -i0).canon_bp()) == 'A(TSpace_0, -TSpace_0)'
def test_TensorIndexType(): D = Symbol("D") G = Metric("g", False) Lorentz = TensorIndexType("Lorentz", metric=G, dim=D, dummy_fmt="L") m0, m1, m2, m3, m4 = tensor_indices("m0:5", Lorentz) sym2 = tensorsymmetry([1] * 2) sym2n = tensorsymmetry(*get_symmetric_group_sgs(2)) assert sym2 == sym2n g = Lorentz.metric assert str(g) == "g(Lorentz,Lorentz)" assert Lorentz.eps_dim == Lorentz.dim TSpace = TensorIndexType("TSpace") i0, i1 = tensor_indices("i0 i1", TSpace) g = TSpace.metric A = tensorhead("A", [TSpace] * 2, [[1] * 2]) assert str(A(i0, -i0).canon_bp()) == "A(TSpace_0, -TSpace_0)"
def test_canonicalize1(): Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Lorentz) sym1 = tensorsymmetry([1]) base3, gens3 = get_symmetric_group_sgs(3) sym2 = tensorsymmetry([1]*2) sym2a = tensorsymmetry([2]) sym3 = tensorsymmetry([1]*3) sym3a = tensorsymmetry([3]) # A_d0*A^d0; ord = [d0,-d0] # T_c = A^d0*A_d0 S1 = TensorType([Lorentz], sym1) A = S1('A') t = A(-d0)*A(d0) tc = t.canon_bp() assert str(tc) == 'A(L_0)*A(-L_0)' # A commuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 # T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2 t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0) tc = t.canon_bp() assert str(tc) == 'A(L_0)*A(-L_0)*A(L_1)*A(-L_1)*A(L_2)*A(-L_2)' # A anticommuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 # T_c 0 A = S1('A', 1) t = A(-d0)*A(-d1)*A(-d2)*A(d2)*A(d1)*A(d0) tc = t.canon_bp() assert tc == 0 # A commuting symmetric # A^{d0 b}*A^a_d1*A^d1_d0 # T_c = A^{a d0}*A^{b d1}*A_{d0 d1} S2 = TensorType([Lorentz]*2, sym2) A = S2('A') t = A(d0, b)*A(a, -d1)*A(d1, -d0) tc = t.canon_bp() assert str(tc) == 'A(a, L_0)*A(b, L_1)*A(-L_0, -L_1)' # A, B commuting symmetric # A^{d0 b}*A^d1_d0*B^a_d1 # T_c = A^{b d0}*A_d0^d1*B^a_d1 B = S2('B') t = A(d0, b)*A(d1, -d0)*B(a, -d1) tc = t.canon_bp() assert str(tc) == 'A(b, L_0)*A(-L_0, L_1)*B(a, -L_1)' # A commuting symmetric # A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1] # T_c = A^{a d0 d1}*A^{b}_{d0 d1} S3 = TensorType([Lorentz]*3, sym3) A = S3('A') t = A(d1, d0, b)*A(a, -d1, -d0) tc = t.canon_bp() assert str(tc) == 'A(a, L_0, L_1)*A(b, -L_0, -L_1)' # A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0 # T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3} t = A(d3, d0, d2)*A(a0, -d1, -d2)*A(d1, -d3, a1)*A(a2, a3, -d0) tc = t.canon_bp() assert str(tc) == 'A(a0, L_0, L_1)*A(a1, -L_0, L_2)*A(a2, a3, L_3)*A(-L_1, -L_2, -L_3)' # A commuting symmetric, B antisymmetric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # in this esxample and in the next three, # renaming dummy indices and using symmetry of A, # T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = 0 S2a = TensorType([Lorentz]*2, sym2a) A = S3('A') B = S2a('B') t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) tc = t.canon_bp() assert tc == 0 # A anticommuting symmetric, B anticommuting # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) tc = t.canon_bp() assert str(tc) == 'A(L_0, L_1, L_2)*A(-L_0, -L_1, L_3)*B(-L_2, -L_3)' # A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} Spinor = TensorIndexType('Spinor', metric=1, dummy_fmt='S') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Spinor) S3 = TensorType([Spinor]*3, sym3) S2a = TensorType([Spinor]*2, sym2a) A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) tc = t.canon_bp() assert str(tc) == '-A(S_0, S_1, S_2)*A(-S_0, -S_1, S_3)*B(-S_2, -S_3)' # A anticommuting symmetric, B antisymmetric anticommuting, # no metric symmetry # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 Mat = TensorIndexType('Mat', metric=None, dummy_fmt='M') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Mat) S3 = TensorType([Mat]*3, sym3) S2a = TensorType([Mat]*2, sym2a) A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2)*A(-d2, -d3, -d1)*B(-d0, d3) tc = t.canon_bp() assert str(tc) == 'A(M_0, M_1, M_2)*A(-M_0, -M_1, -M_3)*B(-M_2, M_3)' # Gamma anticommuting # Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha} # T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu} S1 = TensorType([Lorentz], sym1) S2a = TensorType([Lorentz]*2, sym2a) S3a = TensorType([Lorentz]*3, sym3a) alpha, beta, gamma, mu, nu, rho = \ tensor_indices('alpha,beta,gamma,mu,nu,rho', Lorentz) Gamma = S1('Gamma', 2) Gamma2 = S2a('Gamma', 2) Gamma3 = S3a('Gamma', 2) t = Gamma2(-mu,-nu)*Gamma(rho)*Gamma3(nu, mu, alpha) tc = t.canon_bp() assert str(tc) == '-Gamma(L_0, L_1)*Gamma(rho)*Gamma(alpha, -L_0, -L_1)' # Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha} # T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu} t = Gamma2(mu, nu)*Gamma2(beta, gamma)*Gamma(-rho)*Gamma3(alpha, -mu, -nu) tc = t.canon_bp() assert str(tc) == 'Gamma(L_0, L_1)*Gamma(beta, gamma)*Gamma(-rho)*Gamma(alpha, -L_0, -L_1)' # f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry # f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b} # g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15] # T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e} Flavor = TensorIndexType('Flavor', dummy_fmt='F') a, b, c, d, e, ff = tensor_indices('a,b,c,d,e,f', Flavor) mu, nu = tensor_indices('mu,nu', Lorentz) sym_f = tensorsymmetry([1], [2]) S_f = TensorType([Flavor]*3, sym_f) sym_A = tensorsymmetry([1], [1]) S_A = TensorType([Lorentz, Flavor], sym_A) f = S_f('f') A = S_A('A') t = f(c, -d, -a)*f(-c, -e, -b)*A(-mu, d)*A(-nu, a)*A(nu, e)*A(mu, b) tc = t.canon_bp() assert str(tc) == '-f(F_0, F_1, F_2)*f(-F_0, F_3, F_4)*A(L_0, -F_1)*A(-L_0, -F_3)*A(L_1, -F_2)*A(-L_1, -F_4)'
def test_canonicalize1(): Lorentz = TensorIndexType('Lorentz', dummy_fmt='L') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Lorentz) sym1 = tensorsymmetry([1]) base3, gens3 = get_symmetric_group_sgs(3) sym2 = tensorsymmetry([1] * 2) sym2a = tensorsymmetry([2]) sym3 = tensorsymmetry([1] * 3) sym3a = tensorsymmetry([3]) # A_d0*A^d0; ord = [d0,-d0] # T_c = A^d0*A_d0 S1 = TensorType([Lorentz], sym1) A = S1('A') t = A(-d0) * A(d0) tc = t.canon_bp() assert str(tc) == 'A(L_0)*A(-L_0)' # A commuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 # T_c = A^d0*A_d0*A^d1*A_d1*A^d2*A_d2 t = A(-d0) * A(-d1) * A(-d2) * A(d2) * A(d1) * A(d0) tc = t.canon_bp() assert str(tc) == 'A(L_0)*A(-L_0)*A(L_1)*A(-L_1)*A(L_2)*A(-L_2)' # A anticommuting # A_d0*A_d1*A_d2*A^d2*A^d1*A^d0 # T_c 0 A = S1('A', 1) t = A(-d0) * A(-d1) * A(-d2) * A(d2) * A(d1) * A(d0) tc = t.canon_bp() assert tc == 0 # A commuting symmetric # A^{d0 b}*A^a_d1*A^d1_d0 # T_c = A^{a d0}*A^{b d1}*A_{d0 d1} S2 = TensorType([Lorentz] * 2, sym2) A = S2('A') t = A(d0, b) * A(a, -d1) * A(d1, -d0) tc = t.canon_bp() assert str(tc) == 'A(a, L_0)*A(b, L_1)*A(-L_0, -L_1)' # A, B commuting symmetric # A^{d0 b}*A^d1_d0*B^a_d1 # T_c = A^{b d0}*A_d0^d1*B^a_d1 B = S2('B') t = A(d0, b) * A(d1, -d0) * B(a, -d1) tc = t.canon_bp() assert str(tc) == 'A(b, L_0)*A(-L_0, L_1)*B(a, -L_1)' # A commuting symmetric # A^{d1 d0 b}*A^{a}_{d1 d0}; ord=[a,b, d0,-d0,d1,-d1] # T_c = A^{a d0 d1}*A^{b}_{d0 d1} S3 = TensorType([Lorentz] * 3, sym3) A = S3('A') t = A(d1, d0, b) * A(a, -d1, -d0) tc = t.canon_bp() assert str(tc) == 'A(a, L_0, L_1)*A(b, -L_0, -L_1)' # A^{d3 d0 d2}*A^a0_{d1 d2}*A^d1_d3^a1*A^{a2 a3}_d0 # T_c = A^{a0 d0 d1}*A^a1_d0^d2*A^{a2 a3 d3}*A_{d1 d2 d3} t = A(d3, d0, d2) * A(a0, -d1, -d2) * A(d1, -d3, a1) * A(a2, a3, -d0) tc = t.canon_bp() assert str( tc ) == 'A(a0, L_0, L_1)*A(a1, -L_0, L_2)*A(a2, a3, L_3)*A(-L_1, -L_2, -L_3)' # A commuting symmetric, B antisymmetric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # in this esxample and in the next three, # renaming dummy indices and using symmetry of A, # T = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 # can = 0 S2a = TensorType([Lorentz] * 2, sym2a) A = S3('A') B = S2a('B') t = A(d0, d1, d2) * A(-d2, -d3, -d1) * B(-d0, d3) tc = t.canon_bp() assert tc == 0 # A anticommuting symmetric, B anticommuting # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2) * A(-d2, -d3, -d1) * B(-d0, d3) tc = t.canon_bp() assert str(tc) == 'A(L_0, L_1, L_2)*A(-L_0, -L_1, L_3)*B(-L_2, -L_3)' # A anticommuting symmetric, B antisymmetric commuting, antisymmetric metric # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = -A^{d0 d1 d2} * A_{d0 d1}^d3 * B_{d2 d3} Spinor = TensorIndexType('Spinor', metric=1, dummy_fmt='S') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Spinor) S3 = TensorType([Spinor] * 3, sym3) S2a = TensorType([Spinor] * 2, sym2a) A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2) * A(-d2, -d3, -d1) * B(-d0, d3) tc = t.canon_bp() assert str(tc) == '-A(S_0, S_1, S_2)*A(-S_0, -S_1, S_3)*B(-S_2, -S_3)' # A anticommuting symmetric, B antisymmetric anticommuting, # no metric symmetry # A^{d0 d1 d2} * A_{d2 d3 d1} * B_d0^d3 # T_c = A^{d0 d1 d2} * A_{d0 d1 d3} * B_d2^d3 Mat = TensorIndexType('Mat', metric=None, dummy_fmt='M') a, a0, a1, a2, a3, b, d0, d1, d2, d3 = \ tensor_indices('a,a0,a1,a2,a3,b,d0,d1,d2,d3', Mat) S3 = TensorType([Mat] * 3, sym3) S2a = TensorType([Mat] * 2, sym2a) A = S3('A', 1) B = S2a('B') t = A(d0, d1, d2) * A(-d2, -d3, -d1) * B(-d0, d3) tc = t.canon_bp() assert str(tc) == 'A(M_0, M_1, M_2)*A(-M_0, -M_1, -M_3)*B(-M_2, M_3)' # Gamma anticommuting # Gamma_{mu nu} * gamma^rho * Gamma^{nu mu alpha} # T_c = -Gamma^{mu nu} * gamma^rho * Gamma_{alpha mu nu} S1 = TensorType([Lorentz], sym1) S2a = TensorType([Lorentz] * 2, sym2a) S3a = TensorType([Lorentz] * 3, sym3a) alpha, beta, gamma, mu, nu, rho = \ tensor_indices('alpha,beta,gamma,mu,nu,rho', Lorentz) Gamma = S1('Gamma', 2) Gamma2 = S2a('Gamma', 2) Gamma3 = S3a('Gamma', 2) t = Gamma2(-mu, -nu) * Gamma(rho) * Gamma3(nu, mu, alpha) tc = t.canon_bp() assert str(tc) == '-Gamma(L_0, L_1)*Gamma(rho)*Gamma(alpha, -L_0, -L_1)' # Gamma_{mu nu} * Gamma^{gamma beta} * gamma_rho * Gamma^{nu mu alpha} # T_c = Gamma^{mu nu} * Gamma^{beta gamma} * gamma_rho * Gamma^alpha_{mu nu} t = Gamma2(mu, nu) * Gamma2(beta, gamma) * Gamma(-rho) * Gamma3( alpha, -mu, -nu) tc = t.canon_bp() assert str( tc ) == 'Gamma(L_0, L_1)*Gamma(beta, gamma)*Gamma(-rho)*Gamma(alpha, -L_0, -L_1)' # f^a_{b,c} antisymmetric in b,c; A_mu^a no symmetry # f^c_{d a} * f_{c e b} * A_mu^d * A_nu^a * A^{nu e} * A^{mu b} # g = [8,11,5, 9,13,7, 1,10, 3,4, 2,12, 0,6, 14,15] # T_c = -f^{a b c} * f_a^{d e} * A^mu_b * A_{mu d} * A^nu_c * A_{nu e} Flavor = TensorIndexType('Flavor', dummy_fmt='F') a, b, c, d, e, ff = tensor_indices('a,b,c,d,e,f', Flavor) mu, nu = tensor_indices('mu,nu', Lorentz) sym_f = tensorsymmetry([1], [2]) S_f = TensorType([Flavor] * 3, sym_f) sym_A = tensorsymmetry([1], [1]) S_A = TensorType([Lorentz, Flavor], sym_A) f = S_f('f') A = S_A('A') t = f(c, -d, -a) * f(-c, -e, -b) * A(-mu, d) * A(-nu, a) * A(nu, e) * A( mu, b) tc = t.canon_bp() assert str( tc ) == '-f(F_0, F_1, F_2)*f(-F_0, F_3, F_4)*A(L_0, -F_1)*A(-L_0, -F_3)*A(L_1, -F_2)*A(-L_1, -F_4)'