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Let's suppose I have a theory described by a Lagrangian as follows:

$ \mathcal{L} = A_\mu \underbrace{\left( \partial^2 g^{\mu\nu} - \partial^\mu \partial^\nu + m^2 g^{\mu \nu} \right)}_{K^{\mu \nu}} A_\nu $

(What's the actual theory does not matter.) I know that the inverse of $K^{\mu \nu}$ is called the propagator, I don't know how to call $K^{\mu \nu}$ itself. I would like to avoid something like ''operator'' because I often switch to the path integral formalism...

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I) In a free field theory

$$S[\phi]~=~ \frac{1}{2} \iint \! d^dx~ d^dy ~\phi^{\alpha}(x) K_{\alpha\beta}(x,y) \phi^{\beta}(y),$$

I would call the integral kernel

$$ K_{\alpha\beta}(x,y)~=~\frac{\delta^2 S}{\delta\phi^{\alpha}(x)\delta\phi^{\beta}(y)} $$

either the inverse propagator or the Hessian. (Here we assume that $\phi^{\alpha}$ is bosonic (Grassmann-even) in order not to clutter the formulas with sign factors. In case of gauge symmetry, the action should be gauge-fixed to remove zero-modes.)

II) In an interacting theory, the free propagator is by definition associated with the free part of the action, i.e. the part which is quadratic in the field variables $\phi$.

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