applies mapping of indices to tensors:
println '{m -> b, n -> a}'.mapping >> 't_mn'.t |
> t_ba |
collects terms by patterns:
def t = 'A_m*B_n + A_n*C_m'.tprintln Collect['A_m'.t] >> t |
> A_i*(d^i_m*B_n + d^i_n*C_m) |
collects similar scalar factors in products:
println CollectScalars >> 'A_m*A^m*A_n*A^n'.t |
> (A_{m}*A^{m})**2 |
collects terms in sums with same tensorial parts:
println CollectNonScalars >> 'A_m*A^m*A_n + A_n'.t |
> (1+A_{m}*A^{m})*A_{n} |
replaces complex numbers in the expression with their complex conjugations:
println Conjugate >> 'a + I*b'.t |
> a - I*b |
gives the denominator of expression:
println Denominator >> '(a + b)/(c + d)'.t |
> c + d |
differentiates expressions with respect to specified variables:
println Differentiate['x_a'] >> 'x_a*x^a - Sin[x_a*x^a]'.t |
> (2 - 2*Cos[x_{b}*x^{b}])*x^{a} |
simplifies products of gamma matrices:
defineMatrices 'G_a', 'G5', Matrix1.matrixprintln DiracSimplify >> 'G5*G_a*G5*G_b*G^a*G^b'.t |
> -8 |
order products of gamma matrices:
defineMatrices 'G_a', 'G5', Matrix1.matrixprintln DiracOrder >> 'G5*G_c*G_b*G_a'.t |
> G_{a}*G_{b}*G_{c}*G5+2*G_{b}*G5*g_{ca}-2*G_{c}*G5*g_{ba}-2*G_{a}*G5*g_{cb} |
evaluates trace of gamma matrices:
defineMatrices 'G_m', 'G5', Matrix1.matrixprintln DiracTrace[[Gamma: 'G_m']] >> 'Tr[G_m*G_n]'.t |
> 4*g_{mn} |
removes parts of expressions, which are zero because of the symmetries (symmetric and antisymmetric at the same time):
println EliminateDueSymmetries >> '(A_mn - A_nm)*(A^mn + A^nm)'.t |
> 0 |
eliminates metric tensors and kronecker deltas:
println EliminateMetrics >> 'g_mn*A^m + d_n^a*B_a'.t |
> A_n + B_n |
expands out product of sums and positive integer powers and permanently eliminates metric and kronecker deltas:
println ExpandAndEliminate >> '(g_mn - A_nm)*(A^mn + g^nm)'.t |
> -A^{mn}*A_{nm}+d^{n}_{n} |
expands out products and positive integer powers:
println Expand >> '(g_mn - A_nm)*(A^mn + g^nm)'.t |
> -A^{mn}*A_{nm}+g^{nm}*g_{mn} |
expands out all products and integer powers in any part of expression:
println ExpandAll >> '1/((g_mn - A_nm)*(A^mn + g^nm)').t |
> 1/(-A^{mn}*A_{nm}+g^{nm}*g_{mn}) |
expands out products and powers that appear in the numerator:
println ExpandDenominator >> '(a + b)**2/(c + d)**2'.t |
> (a+b)**2/(c**2+d**2+2*d*c) |
expands out products and powers that appear as numerators:
println ExpandNumerator >> '(a + b)**2/(c + d)**2'.t |
> (a**2+2*a*b+b**2)/(c+d)**2 |
expands only indexed parts:
println ExpandTensors >> '(a + b)**2*(A_a + B_a)*(A^a + B^a)'.t |
> (a+b)**2*A^{a}*A_{a}+2*(a+b)**2*A^{a}*B_{a}+(a+b)**2*B^{a}*B_{a} |
symmetrizes expression with respect to all free indices
println FullyAntiSymmetrize >> 'f_abc'.t |
> -(1/6)*f_{bac}+(1/6)*f_{bca}+(1/6)*f_{cab}-(1/6)*f_{acb}+(1/6)*f_{abc}-(1/6)*f_{cba} |
symmetrizes expression with respect to all free indices
println FullySymmetrize >> 'f_abc'.t |
> (1/6)*f_{cba}+(1/6)*f_{bca}+(1/6)*f_{abc}+(1/6)*f_{acb}+(1/6)*f_{bac}+(1/6)*f_{cab} |
factors a polynomial over the integers:
println Factor >> 'x**2 - 2*x*y + y**2'.t |
> (x - y)**2 |
just identity:
def expr = 'A_mn*(p^m + q^m) + T_n'.tassert Identity >> expr == expr |
inverts indices of expression:
println InvertIndices >> 'A_mn*(p^m + q^m) + T_n'.t |
> A_m^n*(p^m + q^m) + T^n |
simplifies combinations with Levi-Civita tensors:
println LeviCivitaSimplify.minkowski['e_abcd'.t] >> 'e_abcm*e^abcn'.t |
> -6*d_{m}^{n} |
gives the numerical value of expression:
println Numeric >> 'Sin[2]'.t |
> 0.9092974268256817 |
expands all powers of products and powers with respect to specified variables:
println PowerExpand >> '(a*b*c)**d'.t |
> a**d*b**d*c**d |
expands all powers of products and powers with respect to specified variables and unfolds powers of indexed arguments into products:
println PowerUnfold >> '(A_m*A^m)**2'.t |
> A_{m}*A^{m}*A_{a}*A^{a} |
reverses the order of matrices of specified matrix type:
defineMatrices 'A', 'B', 'C', Matrix1.matrixprintln Reverse[Matrix1] >> 'A*B*C'.t |
> C*B*A |
simplifies Dirac spinors:
defineMatrices 'G_a', 'G5', Matrix1.matrix, 'cu', Matrix1.covectordef sSimplify = SpinorsSimplify[[uBar: 'cu', Momentum: 'p_a', Mass: 'm']]println sSimplify >> 'cu*G^a*p_a'.t |
> -m*cu*G_{b}+2*cu*p_{b} |
gives a symmetrization of tensor with respect to specified indices under the specified symmetries:
def indices = '_abc'.siindices.symmetries.setSymmetric()println Symmetrize[indices] >> 't_abc'.t |
> (1/6)*(t_{cab} + t_{acb} + t_{bca} + t_{cba} + t_{abc} + t_{bac}) |
puts terms in a sum over a common denominator, and cancels factors in the result:
println Together >> '1/a + 1/b'.t |
> (a + b)/(a*b) |
simplifies combinations of unitary matrices and SU(N) structural and d-constants
defineMatrices 'T_A', Matrix2.matrixprintln UnitarySimplify[[Matrix: 'T_A']] >> 'T_A*T^A'.t |
> (1/2)*N**(-1)*(N**2-1) |
evaluates trace of unitary matrices:
defineMatrices 'T_A', Matrix2.matrixprintln UnitaryTrace[[Matrix: 'T_A']] >> 'Tr[T_A*T_B]'.t |
> (1/2)*g_{BA} |