A. Zee, Quantum Field Theory in a Nutshell · Problem VI.1.2

Field Theory Redux

Physics — QFT

Problem

By thinking about mass dimensions, convince yourself that the Chern-Simons term dominates the Maxwell term at long distances. This is one reason that relativistic field theorists find anyons so appealing. As long as they are interested only in long distance physics they can ignore the Maxwell term and play with a relativistic theory (see exercise VI.1.1). Note that this picks out (2+1)(2 + 1)-dimensional spacetime as special. In (3+1)(3 + 1)-dimensional spacetime the generalization of the Chern-Simons term εμνλσfμνfλσ\varepsilon^{\mu\nu\lambda\sigma} f_{\mu\nu} f_{\lambda\sigma} has the same mass dimension as the Maxwell term f2f^2. In (4+1)(4 + 1)-dimensional spacetime the term ερμνλσaρfμνfλσ\varepsilon^{\rho\mu\nu\lambda\sigma} a_\rho f_{\mu\nu} f_{\lambda\sigma} is less important at long distances than the Maxwell term f2f^2.

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