DiracSimplify

Description

  • DiracSimplify simplifies products of gamma matrices
  • By default DiracSimplify works in $D = 4$; for arbitrary $D$ one can use option DiracSimplify[[Dimension: D]]
  • One can directly set trace of identity matrix (e.g. for dimensional regularisation): DiracSimplify[[Dimension: D, TraceOfOne: 4]]
  • By default DiracSimplify uses notation G_m for $\gamma_m$ and G5 for $\gamma_5$. DiracSimplify[G, G5] or DiracSimplify[[Gamma: G, Gamma5: G5]] specifies the notation for $\gamma_m$ and $\gamma_5$.
  • DiracSimplify[[Simplifications: rules]] will apply additional simplification rules to each processed product of gammas

Examples

Simplify different expressions: 

defineMatrices 'G_a', 'G5', Matrix1.matrix
def dSimplify = DiracSimplify
println dSimplify >> 'G_a*G^a'.t

   > 4

println dSimplify >> 'G_a*G_b*G^a'.t

   > -2*G_{b}

println dSimplify >> 'G_a*G_b*G^a*G^b'.t

   > -8

println dSimplify >> 'G5*G_a*G_b*G^a*G^b*G5*G5'.t

   > -8*G5

println dSimplify >> 'G5*G_a*G_b*G^a*G5*G5'.t

   > 2*G_{b}

Simplify in different dimensions: 

defineMatrices 'G_a', 'G5', Matrix1.matrix
def dSimplify = DiracSimplify[[Dimension: 'D']]
println dSimplify >> 'G_a*G^a'.t

   > D

println dSimplify >> 'G_a*G_b*G^a'.t

   > -(-2+D)*G_{b}

Specify additional simplifications: 

defineMatrices 'G_a', 'G5', Matrix1.matrix
def dSimplify = DiracSimplify[[Simplifications: 'p_a*k^a = s'.t]]
println dSimplify >> 'p^b*k^c*G_a*G_b*G_c*G^a'.t

   > 4*s

See also