A. Zee, Quantum Field Theory in a Nutshell · Problem VI.1.1

Field Theory Redux

Physics — QFT

Problem

In a nonrelativistic theory you might think that there are two separate Chern-Simons terms,

\varepsilon^{ij} a_i \partial_j a_0

and

\varepsilon^{ij} a_0 \partial_i a_j

. Show that gauge invariance forces the two terms to combine into a single Chern-Simons term

\varepsilon^{\mu\nu\lambda} a_\mu \partial_\nu a_\lambda

. For the Chern-Simons term, gauge invariance implies Lorentz invariance. In contrast, the Maxwell term would in general be nonrelativistic, consisting of two terms,

f_{0i}^2

and

f_{ij}^2

, with an arbitrary relative coefficient between them (with

f_{\mu\nu} = \partial_\mu a_\nu - \partial_\nu a_\mu

as usual).

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