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In a Hamiltonian approach to anomalies parity and time reversal symmetriescan be restored by introducing suitable impure (or mixed) states. However, theexpectation values of observables such as the Hamiltonian diverges in suchimpure states. Here we show that such divergent expectation values can betreated within a renormalization group framework, leading to a set of$\beta$-functions in the moduli space of the operators representing theobservables. This leads to well defined expectation values of the Hamiltonianin a phase where the impure state restores the $P$ and $T$ symmetry. We alsoshow that this RG procedure leads to a mass gap in the spectrum. Such aframework may be relevant for long wavelength descriptions of condensed mattersystems such as the quantum spin Hall effect.

Anomalies and Renormalization of Impure States in Quantum Theories