# How do you interpret quadratic terms in the gauge field in a Lagrangian?

1. Feb 9, 2010

### QuantumSkippy

Consider a one dimensional gauge theory where the field has mass. The term,

$$m^{2}A^{\mu}A_{\mu}$$

is the conventional mass term. What if you find terms in your Unified Field Theory lagrangian of the form

$$M_{\mu\nu}A^{\mu}A^{\nu}$$ ?

In this case $$M_{\mu\nu}$$ is constant.

When it is not the case that

$$M_{\mu\nu}$$

is of the form

$$m^{2}g_{\mu\nu}$$ ,

are these to be interpreted as self-interaction terms, or self-interaction terms somehow related to mass for the gauge field, or as bona fide mass terms?

2. Feb 9, 2010

### chrispb

As far as my understanding goes, it's the mass eigenstates that would propagate. This is what we (I, at least) believe happens with neutrinos (with a spinor instead of a vector field, of course). You'd diagonalize your M matrix and find that the mass eigenstates are not identical to flavor eigenstates, but mass eigenstates are ones that diagonalize the Hamiltonian and therefore evolve under an e^(-iHt). Writing down the mass eigenstate at t=0, time-evolving it, then rewriting it in the flavor basis allows you to see the now-popular neutrino oscillations. What IS peculiar, as far as I know, is the fact that the neutrino masses are so small.

3. Feb 9, 2010

### chrispb

I should also point out that in QED, a mass term like that breaks local gauge invariance and is therefore generally disallowed.