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| - | ====== SpinorsSimplify ====== | ||
| - | ---- | ||
| - | ====Description==== | ||
| - | |||
| - | * ''%%SpinorsSimplify%%'' simplifies products of $\gamma$-matrices and Dirac spinors using Dirac equation | ||
| - | |||
| - | * 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): ''%%SpinorsSimplify[[Dimension: D, TraceOfOne: 4]]%%'' | ||
| - | |||
| - | * The notation for Dirac spinors should be specified using ''%%SpinorsSimplify[[u: u, v: v, uBar: uBar, vBar: vBar, Momentum: m, Mass: m]]%%'' | ||
| - | |||
| - | * 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]] | ||
| - | * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/SpinorsSimplifyTransformation.java|SpinorsSimplifyTransformation.java]] | ||