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--- documentation:ref:spinorssimplify [2015/11/20 20:22]
poslavskysv created
+++ documentation:ref:spinorssimplify [2015/11/21 12:33] (current)
@@ Line 4: / Line 4: @@
 ====Description====
-  * ''%%DiracTrace%%'' calculates trace of Dirac matrices in $D$ dimensions
+  * ''%%SpinorsSimplify%%'' simplifies products of $\gamma$-matrices and Dirac spinors using Dirac equation
-  * By default ''%%DiracTrace%%'' works in $D = 4$; for arbitrary $D$ one can use option ''%%DiracTrace[[Dimension: D]]%%''
+  * By default ''%%SpinorsSimplify%%'' works in $D = 4$; for arbitrary $D$ one can use option ''%%SpinorsSimplify[[Dimension: D]]%%''
-  * One can directly set trace of identity matrix (e.g. for dimensional regularisation): ''%%DiracTrace[[Dimension: D, TraceOfOne: 4]]%%''
+  * One can directly set trace of identity matrix (e.g. for dimensional regularisation): ''%%SpinorsSimplify[[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.  ''%%DiracTrace[G, G5, eps]%%'' or ''%%DiracTrace[[Gamma: G, Gamma5: G5, LeviCivita: eps]]%%'' specifies the notation for $\gamma_m$, $\gamma_5$ and Levi-Civita tensor.
+  * The notation for Dirac spinors should be specified using ''%%SpinorsSimplify[[u: u, v: v, uBar: uBar, vBar: vBar, Momentum: m, Mass: m]]%%''
-  * ''%%DiracTrace[[Simplifications: rules]]%%'' will apply additional simplification ''rules'' to each processed trace
+  *  By default ''SpinorsSimplify'' uses notation ''G_m'' for $\gamma_m$ and ''G5'' for $\gamma_5$. ''%%SpinorsSimplify[[Gamma: G, Gamma5: G5]]%%'' specifies the notation for $\gamma_m$ and $\gamma_5$.
+  * ''%%SpinorsSimplify[[Simplifications: rules]]%%'' will apply additional simplification ''rules'' to each processed expression
 ====Examples====
 ----
+Simplify Dirac $\bar u$ spinors in expression:
+<sxh groovy; gutter: true>
+defineMatrices 'G_a', 'G5', Matrix1.matrix, 'cu', Matrix1.covector
+def sSimplify = SpinorsSimplify[[uBar: 'cu', Momentum: 'p_a', Mass: 'm']]
+println sSimplify >> 'cu*G^a*p_a'.t
+</sxh>
+<sxh plain; gutter: false>
+   > m*cu
+</sxh>
+<sxh groovy; gutter: true; first-line: 5>
+println sSimplify >> 'cu*G_b*G^a*p_a'.t
+</sxh>
+<sxh plain; gutter: false>
+   > -m*cu*G_{b}+2*cu*p_{b}
+</sxh>
+----
+Simplify different spinors:
+<sxh groovy; gutter: true>
+defineMatrices 'G_a', 'G5', Matrix1.matrix,
+                'cu', 'cv', Matrix1.covector,
+                'u', 'v', Matrix1.vector
+def sSimplify = SpinorsSimplify[[uBar: 'cu', vBar: 'cv', u: 'u', v: 'v',
+                                  Momentum: 'p_a', Mass: 'm']]
+println sSimplify >> 'cu*p^a*G_a*G_b*G_c*v'.t
+</sxh>
+<sxh plain; gutter: false>
+   > m*cu*G_b*G_c*v
+</sxh>
+<sxh groovy; gutter: true; first-line: 5>
+println sSimplify >> 'p^a*G5*G_a*G_b*G_cG5*u'.t
+</sxh>
+<sxh plain; gutter: false>
+   > m*G_b*G_c*u+2*G_c*u*p_b-2*G_b*u*p_c
+</sxh>
+<sxh groovy; gutter: true; first-line: 6>
+println sSimplify >> 'p^a*G_a*G_b*G_c*v'.t
+</sxh>
+<sxh plain; gutter: false>
+   > -m*G_b*G_c*v+2*G_c*v*p_b-2*G_b*v*p_c
+</sxh>
+----
+With ''DiracSimplify: true'' an additional simplification of Dirac gammas will be performed automatically:
+<sxh groovy; gutter: true; first-line: 1>
+def options = [uBar: 'cu', vBar: 'cv', u: 'u', v: 'v',
+               Momentum: 'p_a', Mass: 'm', DiracSimplify: true]
+def sSimplify = SpinorsSimplify[options]
+println sSimplify >> 'cu*G^a*G_b*G_a*p^b'.t
+</sxh>
+<sxh plain; gutter: false>
+   > -2*m*cu
+</sxh>
+Do the same in $D$ dimensions:
+<sxh groovy; gutter: true; first-line: 6>
+options['Dimension'] = 'D'
+sSimplify = SpinorsSimplify[options]
+println sSimplify >> 'cu*G^a*G_b*G_a*p^b'.t
+</sxh>
+<sxh plain; gutter: false>
+   > (-D*m+2*m)*cu
+</sxh>
 ====See also====
   * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:Setting up matrix objects]], [[documentation:guide:list_of_transformations]]
   * Related transformations: [[documentation:ref:DiracTrace]], [[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/SpinorsSimplifyTransformation.html| SpinorsSimplifyTransformation]]
+  * JavaDocs: [[http://api.redberry.cc/redberry/1.1.9/java-api//cc/redberry/physics/feyncalc/SpinorsSimplifyTransformation.html| SpinorsSimplifyTransformation]]
   * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/SpinorsSimplifyTransformation.java|SpinorsSimplifyTransformation.java]]