Kronecker delta


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:

println 'd^a_b - d_b^a'.t
   > 0
println 'd_A^B + d^B_A'.t
   > 2*d^B_A

Raising and lowering of Kronecker delta indices may produce Metric tensor:

println ('{^a -> _a}'.mapping >> 'd^a_b'.t)
   > g_ab

The transformation that simplifies contractions with Kronecker deltas is EliminateMetrics:

println EliminateMetrics >> 'd_a^m*F^ab*d_b^n'.t
   > F^mn


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:

//change default metric name
println EliminateMetrics >> 'f_a^m*F^ab*f_b^n'.t
   > F^mn

See also