- #1
QuantumSkippy
- 18
- 1
Can someone please help me out with mass terms in the general case for a lagrangian?
It is known that for n scalar fields, any quadratic in these fields will be a mass term.
For classical fields [tex]\varphi_{j}[/tex] with the most general possible expression being [tex]M^{jk}\varphi_{j}\varphi_{k}[/tex] , the matrix [tex]M^{jk}[/tex] 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 [tex]\sum_{j} M^{jj}\varphi_{j}\varphi_{j}[/tex]
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 [tex] M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu} [/tex] is definitely a mass term. Again, for classical fields the mass matrix [tex]M^{jk}[/tex] 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[tex] M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}[/tex] , one would expect that a Lorentz transformation can reduce the term[tex] M_{\mu\nu}^{jk}[/tex] to something of the form [tex]g_{\mu\nu}m^{jk}[/tex].
In this way, the 'mass' term becomes [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] after the Lorentz transformation has been applied.
Observe however, that for the Lorentz transformation [tex]L_{\alpha}^{\mu}[/tex] which achieves this change we have
[tex]M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}[/tex] .
Multiplying both sides of this equation by [tex]{(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}[/tex], we obtain the result that
[tex]M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}[/tex].
This follows from the orthogonality of the Lorentz transformations with respect to the metric [tex]g_{\alpha\beta}[/tex].
So the upshot of this appears to be that unless terms are of the form [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] they cannot represent mass terms and are in fact, self interaction terms.
Please help me out here, as this is the only way I can interpret quadratic terms in the gauge fields at present.
It is known that for n scalar fields, any quadratic in these fields will be a mass term.
For classical fields [tex]\varphi_{j}[/tex] with the most general possible expression being [tex]M^{jk}\varphi_{j}\varphi_{k}[/tex] , the matrix [tex]M^{jk}[/tex] 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 [tex]\sum_{j} M^{jj}\varphi_{j}\varphi_{j}[/tex]
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 [tex] M^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu} [/tex] is definitely a mass term. Again, for classical fields the mass matrix [tex]M^{jk}[/tex] 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[tex] M_{\mu\nu}^{jk}A^{\mu}_{j}A^{\nu}_{k}[/tex] , one would expect that a Lorentz transformation can reduce the term[tex] M_{\mu\nu}^{jk}[/tex] to something of the form [tex]g_{\mu\nu}m^{jk}[/tex].
In this way, the 'mass' term becomes [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] after the Lorentz transformation has been applied.
Observe however, that for the Lorentz transformation [tex]L_{\alpha}^{\mu}[/tex] which achieves this change we have
[tex]M_{\mu\nu}^{jk}L_{\alpha}^{\mu}L_{\beta}^{\nu} = m^{jk}g_{\alpha\beta}[/tex] .
Multiplying both sides of this equation by [tex]{(L^{-1})}^{\alpha}_{\mu}{(L^{-1})}^{\beta}_{\nu}[/tex], we obtain the result that
[tex]M_{\alpha\beta}^{jk} = m^{jk}g_{\alpha\beta}[/tex].
This follows from the orthogonality of the Lorentz transformations with respect to the metric [tex]g_{\alpha\beta}[/tex].
So the upshot of this appears to be that unless terms are of the form [tex]m^{jk}A^{\mu}_{j}A^{\nu}_{k}g_{\mu\nu}[/tex] they cannot represent mass terms and are in fact, self interaction terms.
Please help me out here, as this is the only way I can interpret quadratic terms in the gauge fields at present.