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)'
