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| + | ====== Kronecker delta ====== | ||
| + | ---- | ||
| + | ====Basics==== | ||
| + | The notation for Kronecked delta used in Redberry is ordinary ''d^a_b'' and similarly for other index types (''d^A_B'', ''d^\\alpha_\\beta'' etc.). Kronecker delta are automatically symmetric: | ||
| + | <sxh groovy; gutter: false> | ||
| + | println 'd^a_b - d_b^a'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 0 | ||
| + | </sxh> | ||
| + | <sxh groovy; gutter: false> | ||
| + | println 'd_A^B + d^B_A'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > 2*d^B_A | ||
| + | </sxh> | ||
| + | |||
| + | Raising and lowering of Kronecker delta indices may produce [[documentation:ref:metric_tensor]]: | ||
| + | <sxh groovy; gutter: false> | ||
| + | println ('{^a -> _a}'.mapping >> 'd^a_b'.t) | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > g_ab | ||
| + | </sxh> | ||
| + | |||
| + | The transformation that simplifies contractions with Kronecker deltas is [[documentation:ref:eliminatemetrics]]: | ||
| + | <sxh groovy; gutter: false> | ||
| + | println EliminateMetrics >> 'd_a^m*F^ab*d_b^n'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > F^mn | ||
| + | </sxh> | ||
| + | |||
| + | ====Details==== | ||
| + | In addition to ''d^a_b'' notation Redberry also uses ''g^a_b'', which is the notation for [[metric_tensor]] with one upper and one lower index. | ||
| + | |||
| + | One can specify different name for Kronecker tensor by putting the following line in the beginning of the code: | ||
| + | <sxh groovy; gutter: false> | ||
| + | //change default metric name | ||
| + | CC.current().setKroneckerName('f') | ||
| + | println EliminateMetrics >> 'f_a^m*F^ab*f_b^n'.t | ||
| + | </sxh> | ||
| + | <sxh plain; gutter: false> | ||
| + | > F^mn | ||
| + | </sxh> | ||
| + | =====See also===== | ||
| + | * Related guides: [[documentation:guide:types_of_indices_and_metric]] | ||
| + | * Reference material: [[documentation:ref:metric_tensor]], [[documentation:ref:eliminatemetrics]] | ||