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| - | ====== LeviCivitaSimplify ====== | ||
| - | ---- | ||
| - | ====Description==== | ||
| - | * ''LeviCivitaSimplify'' simplifies combinations of Levi-Civita tensors. | ||
| - | |||
| - | * ''LeviCivitaSimplify.euclidean[eps]'' simplifies combinations of Levi-Civita tensors (denoted as ''eps'') assuming that space is Euclidean. | ||
| - | |||
| - | |||
| - | * ''LeviCivitaSimplify.minkowski[eps]'' simplifies combinations of Levi-Civita tensors (denoted as ''eps'') assuming that metric has signature {+, -, -, ...}. | ||
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| - | |||
| - | * ''LeviCivitaSimplify'' works in arbitrary dimensions: if one specified e.g. tensor ''eps_{abcd}'' as Levi-Civita tensor (using e.g. ''LeviCivitaSimplify.euclidean['eps_{abcd}'.t]'' ) then space dimension will be considered equal to 4 (number of Levi-Civita indices) and Kronecker trace will be substituted (i.e. '''d^n_n = 4'.t'' will be applied). | ||
| - | |||
| - | |||
| - | * When using ''LeviCivitaSimplify'' one should be ensured that symmetries of Levi-Civita tensor are already set up. | ||
| - | ====Examples==== | ||
| - | |||
| - | ---- | ||
| - | Simplify combinations of Levi-Civita tensors in dimension 3 in Euclidean space: | ||
| - | <sxh groovy; gutter: true> | ||
| - | setAntiSymmetric 'e_abc' | ||
| - | println LeviCivitaSimplify.euclidean['e_abc'] >> 'e_abc*e^abd'.t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 2*d^d_c | ||
| - | </sxh> | ||
| - | <sxh groovy; gutter: true; first-line: 3> | ||
| - | println LeviCivitaSimplify.euclidean['e_abc'] >> 'e_abc*e^abc'.t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 6 | ||
| - | </sxh> | ||
| - | <sxh groovy; gutter: true; first-line: 4> | ||
| - | def t = 'e_abc*e^amd*e_mnk*e^bnk'.t | ||
| - | println LeviCivitaSimplify.euclidean['e_abc'] >> t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 4*d_{c}^{d} | ||
| - | </sxh> | ||
| - | |||
| - | ---- | ||
| - | Simplify combinations of Levi-Civita tensors in dimension 4 in Euclidean space: | ||
| - | <sxh groovy; gutter: true> | ||
| - | setAntiSymmetric 'e_abcd' | ||
| - | def t = '4*e^h_d^fb*e_abch*e_e^d_gf'.t | ||
| - | println LeviCivitaSimplify.euclidean['e_abcd'] >> t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 16*e_{eagc} | ||
| - | </sxh> | ||
| - | Simplify same expression in Minkowski space: | ||
| - | <sxh groovy; gutter: true> | ||
| - | setAntiSymmetric 'e_abcd' | ||
| - | def t = '4*e^h_d^fb*e_abch*e_e^d_gf'.t | ||
| - | println LeviCivitaSimplify.minkowski['e_abcd'] >> t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > -16*e_{eagc} | ||
| - | </sxh> | ||
| - | |||
| - | ---- | ||
| - | Simplify combinations of Levi-Civita tensors in dimension 5 in Minkowski space: | ||
| - | <sxh groovy; gutter: true> | ||
| - | setAntiSymmetric 'e_abcde' | ||
| - | def t = '''e^{m}_{g}^{kci}*e_{pdj}^{l}_{o}*e_{c}^{n}_{mi}^{p} | ||
| - | *e_{khnef}*e^{g}_{a}^{efd}*e_{l}^{hj}_{b}^{o}'''.t | ||
| - | println LeviCivitaSimplify.minkowski['e_abcde'] >> t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 864*g_{ab} | ||
| - | </sxh> | ||
| - | ---- | ||
| - | Simplify expression where Levi-Civita is contracted with symmetric tensor: | ||
| - | <sxh groovy; gutter: true> | ||
| - | setAntiSymmetric 'e_abcd' | ||
| - | def t = 'e_abcd*(A^a + C^a)*(A^b + C^b)'.t | ||
| - | println LeviCivitaSimplify.minkowski['e_abcd'] >> t | ||
| - | </sxh> | ||
| - | <sxh plain; gutter: false> | ||
| - | > 0 | ||
| - | </sxh> | ||
| - | ---- | ||
| - | ====See also==== | ||
| - | * Related guides: [[documentation:guide:applying_and_manipulating_transformations]], [[documentation:guide:list_of_transformations]] | ||
| - | * Related transformations: [[documentation:ref:DiracTrace]], [[documentation:ref:UnitaryTrace]] | ||
| - | * JavaDocs: [[http://api.redberry.cc/redberry/1.1.8/java-api//cc/redberry/physics/feyncalc/LeviCivitaSimplifyTransformation.html| LeviCivitaSimplifyTransformation]] | ||
| - | * Source code: [[https://bitbucket.org/redberry/redberry/src/tip/physics/src/main/java/cc/redberry/physics/feyncalc/LeviCivitaSimplifyTransformation.java|LeviCivitaSimplifyTransformation.java]] | ||