DiracOrder

Description

  • DiracOrder brings all products of $\gamma$-matrices to alphabetical order
  • By default DiracOrder works in $D = 4$; for arbitrary $D$ one can use option DiracOrder[[Dimension: D]]
  • One can directly set trace of identity matrix (e.g. for dimensional regularisation): DiracOrder[[Dimension: D, TraceOfOne: 4]]
  • By default DiracOrder uses notation G_m for $\gamma_m$ and G5 for $\gamma_5$. DiracOrder[[Gamma: G, Gamma5: G5]] specifies the notation for $\gamma_m$ and $\gamma_5$.

Examples

Order product of $\gamma$-matrices: 

defineMatrices 'G_a', 'G5', Matrix1.matrix
def dOrder = DiracOrder
println dOrder >> 'G_b*G_a'.t

   > 2*g_{ba}-G_{a}*G_{b}

println dOrder >> 'G_d*G_c*G_b*G_a'.t

   > G_{a}*G_{b}*G_{c}*G_{d}-2*G_{c}*G_{d}*g_{ab}+2*G_{b}*G_{d}*g_{ac}-2*G_{b}*G_{c}*g_{ad}
         -2*G_{a}*G_{d}*g_{bc}+4*g_{ad}*g_{bc}+2*G_{a}*G_{c}*g_{bd}-4*g_{ac}*g_{bd}
         -2*G_{a}*G_{b}*g_{cd}+4*g_{ab}*g_{cd}

All $\gamma_5$ will be shift right: 

defineMatrices 'G_a', 'G5', Matrix1.matrix
def dOrder = DiracOrder
println dOrder >> 'G5*G_b*G_a'.t

   > 2*G5*g_{ba}-G_{a}*G_{b}*G5

Use another notation for $\gamma$-matrices: 

defineMatrices 'F_a', 'F5', Matrix1.matrix
def dOrder = DiracOrder[[Gamma: 'F_a', Gamma5: 'F5']]
println dOrder >> 'F_b*F_a'.t

   > 2*g_{ba}-F_{a}*F_{b}

See also