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Parity anomaly
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<p>In <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a> a <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a> is said to have a parity anomaly if its classical <a href="/facts/Action_(physics)/NhWS2Mts">action</a> is invariant under a change of <a href="/facts/Parity_(physics)/emSbJ4hD">parity</a> of the universe, but the quantum theory is not invariant. </p><p>This kind of <a href="/facts/Anomaly_(physics)/RbeExSmM">anomaly</a> can occur in odd-dimensional <a href="/facts/Gauge_theories/fSW3gggv">gauge theories</a> with <a href="/facts/Fermions/LEckdDsR">fermions</a> whose gauge groups have odd <a href="/facts/Dual_Coxeter_number/4B80wH8V">dual Coxeter numbers</a>. They were first introduced by Antti J. Niemi and <a href="/facts/Gordon_Walter_Semenoff/UwNQSmWA">Gordon Walter Semenoff</a> in the letter <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.51.2077">Axial-Anomaly-Induced Fermion Fractionization and Effective Gauge-Theory Actions in Odd-Dimensional Space-Times</a> and by A. Norman Redlich in the letter <a href="https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.52.18">Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions</a> and the article <a href="https://journals.aps.org/prd/abstract/10.1103/PhysRevD.29.2366">Parity violation and gauge noninvariance of the effective gauge field action in three dimensions</a>. It is in some sense an odd-dimensional version of <a href="/facts/Edward_Witten/lPUylPmj">Edward Witten</a>'s <a href="/facts/Anomaly_(physics)/RbeExSmM">SU(2) anomaly</a> in 4-dimensions, and in fact Redlich writes that his demonstration follows Witten's. </p>