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DiracTrace calculates trace of Dirac matrices in $D$ dimensionsDiracTrace works in $D = 4$; for arbitrary $D$ one can use option DiracTrace[[Dimension: D]]DiracTrace[[Dimension: D, TraceOfOne: 4]]DiracTrace uses notation G_m for $\gamma_m$, G5 for $\gamma_5$ and e_abcd for Levi-Civita tensor. or DiracTrace[[Gamma: G, Gamma5: G5, LeviCivita: eps]] specifies the notation for $\gamma_m$, $\gamma_5$ and Levi-Civita tensor.
* DiracTrace[[Simplifications: rules]] will apply additional simplification rules to each processed trace
====Examples====
—-
Calculate trace of $\gamma$-matrices:
<sxh groovy; gutter: false>
defineMatrices 'G_a', 'G5', Matrix1.matrix
println DiracTrace['G_a'] » 'Tr[G_a*G_b]'.t
</sxh>
<sxh plain; gutter: false>
> 4*g_ab
</sxh>
Another example:
<sxh groovy; gutter: false>
set up matrix objects
defineMatrices 'G_a', 'G5', Matrix1.matrix
DiracTrace transformation
def dTrace = DiracTrace 'G_a'
calculate trace
println dTrace » 'Tr[(p_a*G^a + m)*G_m*(q_a*G^a-m)*G_n]'.t
</sxh>
<sxh plain; gutter: false>
> 4*p_{m}*q_{n}+4*p_{n}*q_{m}-4*m2*g_{mn}-4*p^{a}*g_{mn}*q_{a}
</sxh>
—-
Calculate trace involving $\gamma_5$:
<sxh groovy; gutter: true>
set up matrix objects
defineMatrices 'G_a', 'G5', Matrix1.matrix
DiracTrace transformation
def dTrace = DiracTrace 'e_abcd'
calculate trace
println dTrace » 'Tr[G_a*G_b*G_c*G_d*G5]'.t
</sxh>
<sxh plain; gutter: false>
> -4*I*e_{abcd}
</sxh>
<sxh groovy; gutter: true; first-line: 7>
println dTrace » 'Tr[(p_a*G^a + m)*G_m*G5*(q_a*G^a-m)*G_n]'.t
</sxh>
<sxh plain; gutter: false>
> -4*I*p_{b}*q_{a}*e^{a}_{n}^{b}_{m}
</sxh>
—-
Calculate trace in different dimensions:
<sxh groovy; gutter: false>
defineMatrices 'G_a', 'G5', Matrix1.matrix
println DiracTrace 6 » 'Tr[G_c*G_a*G_b*G^c]'.t
</sxh>
<sxh plain; gutter: false>
> 48*g_ab
</sxh>
By default, Tr[1] is equal to $2^{\frac{D-1}{2}}$ for odd $D$ and $2^{\frac{D}{2}}$ for even. For symbolic $D$ it will be assumed that it is even:
<sxh groovy; gutter: false>
defineMatrices 'G_a', 'G5', Matrix1.matrix
println DiracTrace 'D'.t » 'Tr[G_c*G_a*G_b*G^c]'.t
</sxh>
<sxh plain; gutter: false>
> D*2**(D/2)*g_ab
</sxh>
One can directly overcome predefined value of Tr[1] by using additional option (required for dimensional regularisation):
<sxh groovy; gutter: false>
defineMatrices 'G_a', 'G5', Matrix1.matrix
def dTrace = DiracTrace 4
println dTrace » 'Tr[G_c*G_a*G_b*G^c]'.t
</sxh>
<sxh plain; gutter: false>
> 4*D*g_ab
</sxh>
—-
For traces involving $\gamma_5$ in $D$ dimensions, all $\gamma_5$-related calculations will be performed as in 4 dimensions ($Tr[\gamma_a \gamma_b \gamma_c \gamma_d \gamma_5] = -4 i e_{abcd}$ and Chiholm-Kahane identitie: $\gamma_a \gamma_b \gamma_c = g_{ab} \gamma_c-g_{ac} \gamma_b+g_{bc} \gamma_a-i e_{abcd} \gamma_5 \gamma^d$):
<sxh groovy; gutter: true>
defineMatrices 'G_a', 'G5', Matrix1.matrix
def dTrace = DiracTrace 4
println dTrace » 'Tr[G_a*G_b*G_c*G_d*G5]'.t
</sxh>
<sxh plain; gutter: false>
> -4*I*e_{abcd}
</sxh>
<sxh groovy; gutter: true; first-line: 4>
println dTrace » 'Tr[G_a*G_b*G_c*G_d*G_e*G^a*G5]'.t
</sxh>
<sxh plain; gutter: false>
> 4*I*e_{debc}-4*I*e_{decb}+4*I*e_{dbce}-4*I*e_{ebcd}
</sxh>
—-
DiracTrace expands out products of sums containing traces of $\gamma$-matrices and leaves unexpanded other parts of expressions:
<sxh groovy; gutter: false>
defineMatrices 'G_a', 'G5', Matrix1.matrix
def expr = '(k_c+p_c)*(k_d+p_d) + Tr[(p^a + k^a)*(p^b + k^b)*G_a*G_b*G_c*G_d]'.t
println dTrace » expr
</sxh>
<sxh plain; gutter: false>
> (k_c+p_c)*(k_d+p_d)+4*g_cd*p_b*p^b+4*k_b*k^b*g_cd+8*k^b*g_cd*p_b
</sxh>
====Options====
====See also====
* Related guides: Applying and manipulating transformations, Setting up matrix objects, List of common transformations
* Related tutorials: Compton scattering in QED
* Related transformations: LeviCivitaSimplify, UnitaryTrace
* JavaDocs: DiracTraceTransformation
* Source code: DiracTraceTransformation.java