The equation for a massive spin 1 particle is given by the Proca equation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \partial_\mu(\partial^\mu A^\nu - \partial^\nu A^\mu)+ \left(\frac{mc}{\hbar}\right)^2 A^\nu=0 [/tex]

My question is why this equation? In particular, why can't it be like this instead:

[tex] \partial_\mu(\partial^\mu A^\nu +2 \partial^\nu A^\mu)+ \left(\frac{mc}{\hbar}\right)^2 A^\nu=0 [/tex]

The Lagrangian is given by:

[tex] \mathcal{L}=-\frac{1}{16\pi}(\partial^\mu A^\nu-\partial^\nu A^\mu)(\partial_\mu A_\nu-\partial_\nu A_\mu)+\frac{m^2 c^2}{8\pi \hbar^2}A^\nu A_\nu [/tex]

and again, why can't the derivative terms instead be:

[tex]X\partial_\nu A_\mu \partial^\nu A^\mu+Y\partial_\nu A^\nu \partial^\mu A_\mu +Z\partial_\nu A^\mu \partial_\mu A^\nu[/tex]

for arbitrary real numbers X,Y, and Z?

The answer can't be that there is a U(1) symmetry, because the mass terms don't obey the U(1) symmetry.

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# Form of the spin 1 equation

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