- #1
creepypasta13
- 375
- 0
Homework Statement
So I'm having some difficulty with my QFT assignment. I have to solve the following problem.
In three spacetime dimensions (two space plus one time) an antisymmetric Lorentz tensor
F[itex]^{\mu\nu}[/itex] = -F[itex]^{\nu\mu}[/itex] is equivalent to an axial Lorentz vector, F[itex]^{\mu\nu}[/itex] = e[itex]^{\mu\nu\lambda}[/itex]F[itex]_{\lambda}[/itex]. Consequently, in 3D
one can have a massive photon despite unbroken gauge invariance of the electromagnetic
field A[itex]_{\mu}[/itex]. Indeed, consider the following Lagrangian:
L = -(1/2)*F[itex]_{\lambda}[/itex]F[itex]^{\lambda}[/itex] + (m/2)*F[itex]_{\lambda}[/itex]A[itex]^{\lambda}[/itex] (6)
where
F[itex]_{\lambda}[/itex](x) = (1/2)*[itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex]F[itex]^{\mu\nu}[/itex] = [itex]\epsilon[/itex][itex]_{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]^{\mu}[/itex]A[itex]^{\nu}[/itex],
or in components, F[itex]_{0}[/itex] = -B, F1 = +E[itex]^{2}[/itex], F[itex]_{2}[/itex] = -E[itex]^{1}[/itex].
(a) Show that the action S = [itex]\int[/itex]d[itex]^{3}[/itex]x*L is gauge invariant (although the Lagrangian (6) is not invariant).
So I tried substituting A[itex]^{\lambda}[/itex] -> A[itex]^{\lambda'}[/itex] = A[itex]^{\lambda}[/itex] + [itex]\partial[/itex][itex]^{\lambda}[/itex][itex]\Lambda[/itex]
and F[itex]^{\lambda}[/itex] -> F[itex]^{\lambda'}[/itex] = [itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex]A[itex]_{\nu}[/itex]'
then I obtained L' = L + (1/2)*[ F[itex]_{\lambda}[/itex] [itex]\epsilon^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex] [itex]\partial[/itex][itex]_{\nu}[/itex] [itex]\Lambda[/itex] + some other terms]
What I don't understand is how these leftover terms would vanish after being integrated (to obtain S'), but they don't all vanish if they are not integrated (since L is not invariant). Is there some kind of special mathematical trick I have to use? I just don't see how I can integrate terms like [itex]\int[/itex]d[itex]^{3}[/itex]x F[itex]_{\lambda}[/itex][itex]\epsilon[/itex][itex]^{\lambda\mu\nu}[/itex][itex]\partial[/itex][itex]_{\mu}[/itex][itex]\partial[/itex][itex]_{\nu}[/itex][itex]\Lambda[/itex]