# Are all quadratic terms in gauge fields necessarily mass terms?

1. Feb 2, 2010

### QuantumSkippy

It is known that for n scalar fields, any quadratic in these fields will be a mass term.
For classical fields $$\varphi_{j}$$ with the most genral possible expression being $$M^{jk}\varphi_{j}\varphi_{k}$$ , the matrix $$M^{jk}$$ is guaranteed to be symmetric and so can be diagonalised with an orthogonal similarity transformation. So there is no argument there - we get a sum of squares after diagonalisation of the form $$\sum_{j} M^{jj}\varphi_{j}\varphi_{j}$$

For the case of gauge fields, however, it does not seem (??? help me out here!!) that just any quadratic at all will necessarily be a mass term. Here is the reasoning as I see it:

No one would disagree that a term like $$M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ is definitely a mass term. Again, for classical fields the mass matrix $$M^{jk}$$ is guaranteed symmetric by the sum over symmetric terms and so is once more diagonalisable with an orthogonal similarity transformation.

Things seem different for the most general case. For example with a sum like$$M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}$$ , one would expect that a Lorentz transformation can reduce the term$$M_{\mu\nu}^{jk}$$ to something of the form $$g_{\mu\nu}m^{jk}$$.

In this way, the 'mass' term becomes $$m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ after the Lorentz transformation has been applied.

Observe however, that for the Lorentz transformation $$L_{\alpha}^{\mu}$$ which achieves this change we have

$$M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}$$ .

Multiplying both sides of this equation by $${(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}$$, we obtain the result that

$$M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}$$.

This follows from the orthogonality of the Lorentz transformations with respect to the metric $$g_{\alpha\beta}$$.

So the upshot of this appears to be that unless terms are of the form $$m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}$$ they cannot represent mass terms and are in fact, self interaction terms.