# An equation that describes massless spin-1 particle

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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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.