DiracSimplify

• 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

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

• Related guides: Applying and manipulating transformations, Setting up matrix objects, List of common transformations
• Related tutorials: Compton scattering in QED
• Related transformations: DiracTrace, SpinorsSimplify, DiracOrder, LeviCivitaSimplify, UnitarySimplify, UnitaryTrace
• JavaDocs: DiracSimplifyTransformation
• Source code: DiracSimplifyTransformation.java