Calculating the propagator of a Spin-2 field

[avnl]

TL;DR 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.

 

[AndyW]

The final relation you are referring to is known as the "functional derivative" or "variational derivative" of the field $h_{\mu\nu}$ with respect to the source term $\tau_{\rho\sigma}$. It is a way of calculating how the field $h_{\mu\nu}$ changes in response to small changes in the source term $\tau_{\rho\sigma}$.

To understand where this relation comes from, we first need to understand what is meant by a "functional derivative". In ordinary calculus, we are familiar with the concept of a derivative of a function with respect to a variable. For example, if we have a function $f(x)$, the derivative of this function with respect to the variable $x$ is given by $\frac{df}{dx}$. This tells us how the function changes as we vary the variable $x$.

In the case of a functional derivative, we are dealing with functions that depend on other functions. In other words, our "variables" are actually functions themselves. So instead of taking the derivative with respect to a variable $x$, we take the functional derivative with respect to another function $g(x)$, denoted by $\frac{\delta f}{\delta g}$. This tells us how the function $f$ changes as we vary the function $g$.

Now, let's apply this concept to the relation given in the forum post. We have a field $h_{\mu\nu}$ that depends on the source term $\tau_{\rho\sigma}$. The functional derivative of $h_{\mu\nu}$ with respect to $\tau_{\rho\sigma}$ is written as $\frac{\delta h_{\mu\nu}}{\delta \tau_{\rho\sigma}}$. This tells us how the field $h_{\mu\nu}$ changes as we vary the source term $\tau_{\rho\sigma}$.

The reason this relation is used to calculate the propagator is because the propagator is defined as the response of the field to a point source. In other words, it is the functional derivative of the field with respect to a point source. This is why we set the source term $\tau_{\rho\sigma}$ to be zero in the final relation - it represents the point source.