[xAct] Help with correct index factorization in Lagrangian with symmetric rank-3 tensor
João Vitor Santos Perles
Dear colleagues, how are you?
We are working with a symmetric rank-3 tensor field, h_{\mu\nu\rho}, and trying to write a Lagrangian in the form (h^{\alpha\mu\nu}{O_{\alpha\mu\nu}}^{\gamma\beta\delta}h_{\gamma\beta\delta}). That is, a quadratic form with the kinetic operator acting between two identical fields, with all indices properly contracted.
We followed the sequence shown in the attached images, trying to apply commands such as MakeRule, SeparateMetric, and a manual substitution like PD[-μ]@PD[-ν] -> ω[-μ, -ν] □, with the goal of rewriting the terms and reorganizing the indices. However, the final result does not show the fields properly factorized with matching free indices. The structure looks correct, but the index arrangement does not reflect the desired symmetry with the fields at the ends and the operator in the middle.
Has anyone faced this kind of issue? Is there any reliable approach in xAct/xTensor that allows obtaining directly the form (h^{\alpha\mu\nu}{O_{\alpha\mu\nu\gamma\beta\delta}}h^{\gamma\beta\delta}), with organized indices and clearly separated fields?
