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Proca action/equation describes massive spin-1 particle, but I was unable to find an equation that describes massless spin-1 particle.

Can anyone tell me what the name of this equation is?

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It's called Maxwell's equations.

A spin-1 relativistic particle has to have a 4-vector $A_\mu$ and the equation may essentially be written as $\Box A_\mu=0$, like always. However, when we quantize it, we find out that the squared norm of the states created by the time-like components has the opposite sign than the spacelike components. This would make the Hilbert space indefinite - probabilities could be negative.

So the single timelike mode has to be made unphysical. The only way to do so is to impose a gauge symmetry. So configurations related by $A'_\mu = A_\mu +\partial_\mu \lambda$ – and that's essentially the only way how the gauge invariance with 1 scalar parameter may act – must correspond to the same physical situations. Consequently, we may rewrite the equations as $\partial_\mu F^{\mu\nu}=0$, the usual equations of electromagnetism, which differ from the previous box-equation by a term that can be set to zero by a gauge choice. There can't be consistent spin-1 massless equations without a gauge invariance.

We never learn Maxwell's equations as a single-particle quantum mechanical equation because single-particle equations assume that the number of particles is approximately conserved. That's true in the non-relativistic limit. However, when the speeds approach the speed of light, we're heavily in the relativistic realm and the particle production and annihilation is important; quantum field theory is paramount. Obviously, that's also the case of the photons that move by the speed of light. After all, we know that the number of photons is changing all the time.

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