View Single Post
Nov5-11, 12:29 PM
P: 375
1. The problem statement, all variables and given/known data

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)


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]
Phys.Org News Partner Science news on
World's largest solar boat on Greek prehistoric mission
Google searches hold key to future market crashes
Mineral magic? Common mineral capable of making and breaking bonds