In general relativity, on-shell spin 2 graviton has 2 degree of freedom in 3+1D due to gauge symmetry, which is because of einstein equation, however, for off-shell graviton, does it still have 2 degree of freedom?
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The off-shell graviton has 6 degrees of freedom, while the on-shell graviton has 2 dynamical degrees of freedom. The off-shell graviton can be decomposed into a spin-2, a spin-1 and a spin-0 component, each with 2 degrees of freedom. The spin-1 and spin-0 components are eliminated by the gauge symmetry on-shell, leaving only the spin-2 component as the physical degree of freedom. However, off-shell, these components can mix due to quantum effects2, and their couplings to matter can be different from the classical theory.
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$\begingroup$ Thanks! On shell it only has two dof means that there are 4 equation of motion, so it subtract 4 more dof? $\endgroup$– feng linCommented May 28, 2023 at 17:46