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In general, if a Quantum Field Theory is described by a Lagrangian $\mathcal{L}$, the symmetries of $\mathcal{L}$ lead to classically conserved currents along the equations of motion and Ward identities for quantum correlation functions. However, can something be said in general about transformations which are a symmetry of the equations of motion but not the Lagrangian, i.e. dynamical symmetries? Do these symmetries manifest themselves in any way in the correlation functions for the quantum fields?

Additionally, what if we had the particular case where the equations of motion took the form

$$ C_2 \varphi(x) = 0, $$

where $C_2$ is the quadratic Casimir of some symmetry group? This of course would mean that the EoM are invariant under the action of the generators of that group, but it is not necessary that they result in a symmetry of the Lagrangian. Still, is there something to be extracted about the correlation functions for these types of transformations? Thanks in advance.

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    $\begingroup$ can you give an example of such symmetry? $\endgroup$ Nov 21, 2023 at 19:14
  • $\begingroup$ @TanmoyPati I am right now looking at the different symmetries that have been proposed for the Teukolsky equation for the Kerr black hole near its horizon. You can see a nice recent paper on it here: 2311.07933. Section 2 provides a good summary. In short, the wave operator of the Klein-Gordon equation becomes the Casimir of one or two copies of $SL(2,\mathbb{R})$ without being a spacetime isometry. $\endgroup$
    – Marcosko
    Nov 21, 2023 at 21:08

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