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## Summary:

- Right now I'm working on spin-2 theories and have to calculate the propagator for a general Lagrangian, I'm trying to use the same method use by P. van Nieuwenhuizen in his paper about ghost free tensor Lagrangians https://doi.org/10.1016/0550-3213(73)90194-6.

Nieuwenhuizen uses a method for calculating the propagator by decomposing the field ## h_{\mu\nu}, ## first into symmetric part ## \varphi_{\mu\nu} ## and antisymmetric part ## \psi_{\mu\nu} ##, and then by a spin decomposition using projector operators. Using this he writes the dynamical equation

## M^{\rho\sigma}_{\mu\nu} h_{\rho\sigma} = 2 \kappa \tau_{\mu\nu}, ##

as follows

## M^S h = \left( \sum_i C_i^S P_i^S \right) h = 2 \kappa \left( \sum_{i} \hat{P}^S_i \right) \tau ,##

## M^A h = \left( \sum_i C_i^A P_i^A \right) h = 2 \kappa \left( \sum_{i} \hat{P}^A_i \right) \tau ,##

Where ## M^{(A),S} ## is the (anti)symmetric part of the dynamical operator, ## C^{(A),S}_i ## are the (anti)symmetric coefficients which in momentum space are linear to the momentum squared ##k^2## and the squared mass ##m^2##, and the ## P^{(A),S}_i ## are the (anti)symmetric projector operators, he then solves the dynamical equation for each projection and uses those solutions to calculate the propagator ## \Pi_{\mu\nu\rho\sigma} ## using the following relation

## \Pi_{\mu\nu\rho\sigma} = \left. \frac{\delta h_{\mu\nu}}{\delta \tau^{\rho\sigma}} \right|_{\tau = 0}. ##

what I don't understand is this final relation and where does it come from. If someone can explain it and give me somewhere to read about it I would be grateful.

## M^{\rho\sigma}_{\mu\nu} h_{\rho\sigma} = 2 \kappa \tau_{\mu\nu}, ##

as follows

## M^S h = \left( \sum_i C_i^S P_i^S \right) h = 2 \kappa \left( \sum_{i} \hat{P}^S_i \right) \tau ,##

## M^A h = \left( \sum_i C_i^A P_i^A \right) h = 2 \kappa \left( \sum_{i} \hat{P}^A_i \right) \tau ,##

Where ## M^{(A),S} ## is the (anti)symmetric part of the dynamical operator, ## C^{(A),S}_i ## are the (anti)symmetric coefficients which in momentum space are linear to the momentum squared ##k^2## and the squared mass ##m^2##, and the ## P^{(A),S}_i ## are the (anti)symmetric projector operators, he then solves the dynamical equation for each projection and uses those solutions to calculate the propagator ## \Pi_{\mu\nu\rho\sigma} ## using the following relation

## \Pi_{\mu\nu\rho\sigma} = \left. \frac{\delta h_{\mu\nu}}{\delta \tau^{\rho\sigma}} \right|_{\tau = 0}. ##

what I don't understand is this final relation and where does it come from. If someone can explain it and give me somewhere to read about it I would be grateful.