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--- documentation:ref:diractrace [2015/10/07 09:38]
poslavskysv [Examples]
+++ documentation:ref:diractrace [2015/11/21 12:33] (current)
@@ Line 3: / Line 3: @@
 ====Description====
   * ''%%DiracTrace%%'' calculates trace of Dirac matrices in $D$ dimensions
   * By default ''%%DiracTrace%%'' works in $D = 4$; for arbitrary $D$ one can use option ''%%DiracTrace[[Dimension: D]]%%''
-  * Sometimes (e.g. for dimensional regularisation) it is convenient to directly set trace of identity matrix: ''%%DiracTrace[[Dimension: D, TraceOfOne: 4]]%%''
+  * One can directly set trace of identity matrix (e.g. for dimensional regularisation): ''%%DiracTrace[[Dimension: D, TraceOfOne: 4]]%%''
-  * By default ''DiracTrace'' uses notation ''G_m'' for $\gamma_m$, ''G5'' for $\gamma_5$ and ''e_abcd'' for Levi-Civita tensor.
+  * By default ''DiracTrace'' uses notation ''G_m'' for $\gamma_m$, ''G5'' for $\gamma_5$ and ''e_abcd'' for Levi-Civita tensor.  ''%%DiracTrace[G, G5, eps]%%'' or ''%%DiracTrace[[Gamma: G, Gamma5: G5, LeviCivita: eps]]%%'' specifies the notation for $\gamma_m$, $\gamma_5$ and Levi-Civita tensor.
-  * ''%%DiracTrace[G]%%'' or ''%%DiracTrace[[Gamma: G]]%%'' allows to specify the notation used for $\gamma_\mu$.
+  * ''%%DiracTrace[[Simplifications: rules]]%%'' will apply additional simplification ''rules'' to each processed trace
-  * ''%%DiracTrace[G, G5, eps]%%'' or ''%%DiracTrace[[Gamma: G, Gamma5: G5, LeviCivita: eps]]%%'' specifies the notation for $\gamma_m$, $\gamma_5$ and Levi-Civita tensor
-  * ''%%DiracTrace%%'' first applies [[Expand]] to all parts of expressions containing traces of gammas
-  * ''%%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', Matrix1.matrix
+defineMatrices 'G_a', 'G5', Matrix1.matrix
-println DiracTrace['G_a'] >> 'Tr[G_a*G_b]'.t
+println DiracTrace >> 'Tr[G_a*G_b]'.t
 </sxh>
 <sxh plain; gutter: false>
@@ Line 32: / Line 29: @@
 <sxh groovy; gutter: false>
 //set up matrix objects
-defineMatrices 'G_a', Matrix1.matrix
+defineMatrices 'G_a', 'G5', Matrix1.matrix
-//DiracTrace transformation
-def dTrace = DiracTrace[[Gamma: 'G_a']]
 //calculate trace
-println  dTrace >> 'Tr[(p_a*G^a + m)*G_m*(q_a*G^a-m)*G_n]'.t
+println  DiracTrace >> 'Tr[(p_a*G^a + m)*G_m*(q_a*G^a-m)*G_n]'.t
 </sxh>
 <sxh plain; gutter: false>
@@ Line 47: / Line 42: @@
 //set up matrix objects
 defineMatrices 'G_a', 'G5', Matrix1.matrix
-//DiracTrace transformation
-def dTrace = DiracTrace[[Gamma: 'G_a', Gamma5: 'G5', LeviCivita: 'e_abcd']]
 //calculate trace
-println dTrace >> 'Tr[G_a*G_b*G_c*G_d*G5]'.t
+println DiracTrace >> 'Tr[G_a*G_b*G_c*G_d*G5]'.t
 </sxh>
 <sxh plain; gutter: false>
@@ Line 56: / Line 49: @@
 </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
+println DiracTrace >> 'Tr[(p_a*G^a + m)*G_m*G5*(q_a*G^a-m)*G_n]'.t
 </sxh>
 <sxh plain; gutter: false>
@@ Line 65: / Line 58: @@
 Calculate trace in different dimensions:
-<sxh groovy; gutter: true>
+<sxh groovy; gutter: false>
-defineMatrices 'G_a', Matrix1.matrix
+defineMatrices 'G_a', 'G5', Matrix1.matrix
 println DiracTrace[[Dimension: 6]] >> 'Tr[G_c*G_a*G_b*G^c]'.t
 </sxh>
@@ Line 72: / Line 65: @@
    > 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[[Dimension: '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[[Dimension: 'D'.t, TraceOfOne: 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[[Dimension: 'D'.t, TraceOfOne: 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>
+----
+Use another notation for gamma matrices:
+<sxh groovy; gutter: false>
+defineMatrices 'F_\\mu', 'F5', Matrix2.matrix
+def dTrace = DiracTrace[[Gamma: 'F_\\mu', Gamma5: 'F5', LeviCivita: 'Eps_{\\mu\\nu\\alpha\\beta}']]
+println dTrace >> 'Tr[F_\\mu*F_\\nu*F_\\alpha*F_\\beta * F5]'.t
+</sxh>
+<sxh plain; gutter: false>
+   > -4*I*Eps_{\mu\nu\alpha\beta}
+</sxh>
+====Options====
+  * ''Simplifications'': one can specify additional simplifications that will be applied to each evaluated trace:<sxh groovy; gutter: false>
+defineMatrices 'G_a', 'G5', Matrix1.matrix
+def expr = 'Tr[(p^a + k^a)*(p^b + k^b)*G_a*G_b*G_c*G_d]'.t
+def mandelstam = setMandelstam([k_a: '0', p_a: '0', q_a: 'm', r_a: 'm'], 's', 't', 'u')
+println DiracTrace[[Simplifications: mandelstam]] >> expr
+</sxh><sxh plain; gutter: false>
+   > 4*s*g_{cd}
+</sxh>which is same as<sxh groovy; gutter: false>
+println( (DiracTrace & mandelstam) >> expr )
+</sxh><sxh plain; gutter: false>
+   > 4*s*g_{cd}
+</sxh>
+  * ''ExpandAndEliminate'': ''DiracTrace'' expands out products of sums containing traces of $\gamma$-matrices using [[ExpandAndEliminate]] transformation. One can replace it with another instance using ''%%DiractTrace[[ExpandAndEliminate: tr]]%%''.
+  * ''LeviCivitaSimplify'': When traces involve $\gamma_5$, ''DiracTrace'' uses [[LeviCivitaSimplify]] in for simplifying resulting expressions with Levi-Civita tensors. One can replace the default instance of [[LeviCivitaSimplify]] with another one using ''%%DiractTrace[[ExpandAndEliminate: tr]]%%''.
 ====See also====
   * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:Setting up matrix objects]], [[documentation:guide:list_of_transformations]]
   * Related tutorials: [[documentation:tutorials:Compton scattering in QED]]
-  * Related transformations: [[documentation:ref:LeviCivitaSimplify]], [[documentation:ref:UnitaryTrace]]
+  * Related transformations: [[documentation:ref:DiracSimplify]], [[documentation:ref:DiracOrder]],  [[documentation:ref:LeviCivitaSimplify]], [[documentation:ref:UnitaryTrace]]
-  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.8/java-api//cc/redberry/physics/feyncalc/DiracTraceTransformation.html| DiracTraceTransformation]]
+  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api//cc/redberry/physics/feyncalc/DiracTraceTransformation.html| DiracTraceTransformation]]
   * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/DiracTraceTransformation.java|DiracTraceTransformation.java]]