SpinorsSimplify simplifies products of $\gamma$-matrices and Dirac spinors using Dirac equationSpinorsSimplify works in $D = 4$; for arbitrary $D$ one can use option SpinorsSimplify[[Dimension: D]]SpinorsSimplify[[Dimension: D, TraceOfOne: 4]]SpinorsSimplify[[u: u, v: v, uBar: uBar, vBar: vBar, Momentum: m, Mass: m]]SpinorsSimplify uses notation G_m for $\gamma_m$ and G5 for $\gamma_5$. SpinorsSimplify[[Gamma: G, Gamma5: G5]] specifies the notation for $\gamma_m$ and $\gamma_5$.SpinorsSimplify[[Simplifications: rules]] will apply additional simplification rules to each processed expressionSimplify Dirac $\bar u$ spinors in expression:
defineMatrices 'G_a', 'G5', Matrix1.matrix, 'cu', Matrix1.covector def sSimplify = SpinorsSimplify[[uBar: 'cu', Momentum: 'p_a', Mass: 'm']] println sSimplify >> 'cu*G^a*p_a'.t
> m*cu
println sSimplify >> 'cu*G_b*G^a*p_a'.t
> -m*cu*G_{b}+2*cu*p_{b}
Simplify different spinors:
defineMatrices 'G_a', 'G5', Matrix1.matrix,
'cu', 'cv', Matrix1.covector,
'u', 'v', Matrix1.vector
def sSimplify = SpinorsSimplify[[uBar: 'cu', vBar: 'cv', u: 'u', v: 'v',
Momentum: 'p_a', Mass: 'm']]
println sSimplify >> 'cu*p^a*G_a*G_b*G_c*v'.t
> m*cu*G_b*G_c*v
println sSimplify >> 'p^a*G5*G_a*G_b*G_cG5*u'.t
> m*G_b*G_c*u+2*G_c*u*p_b-2*G_b*u*p_c
println sSimplify >> 'p^a*G_a*G_b*G_c*v'.t
> -m*G_b*G_c*v+2*G_c*v*p_b-2*G_b*v*p_c
With DiracSimplify: true an additional simplification of Dirac gammas will be performed automatically:
def options = [uBar: 'cu', vBar: 'cv', u: 'u', v: 'v',
Momentum: 'p_a', Mass: 'm', DiracSimplify: true]
def sSimplify = SpinorsSimplify[options]
println sSimplify >> 'cu*G^a*G_b*G_a*p^b'.t
> -2*m*cuDo the same in $D$ dimensions:
options['Dimension'] = 'D' sSimplify = SpinorsSimplify[options] println sSimplify >> 'cu*G^a*G_b*G_a*p^b'.t
> (-D*m+2*m)*cu