In General > s.a. canonical general relativity; hadrons [parity doubling].
* Idea: An operation defined on 3D space with a flat background, which consists of inverting all axes by mapping (x, y, z) to (–x, –y, –z) in the background.
* Remark: In higher-dimensional theories, we do not change the extra dimensions, which correspond to "internal charges".
* In field theory: One wants to have a representation of this on the space of fields; This may not always be possible (like for Dirac spinors in 5 dimensions, where one has to use a covering space, to get a faithful representation of the Clifford algebra).
* Status as symmetry: In classical physics laws are invariant under P reversal; 1957, Lee & Yang argued that P can be violated in nuclear β decay; 1957, Violation observed in β decay of polarized Co nuclei; 1957, L Landau argued that although P can be violated, CP should not be; 1982, First violation in atomic physics reported; > s.a. CP violation; CPT theorem.
* Parity anomaly: An anomaly in time-reversal or reflection symmetry in certain theories of fermions coupled to gauge fields and/or gravity in a spacetime of odd dimension.
@ General references: Rosen AJP(73)apr [form electromagnetic quantities]; Bender et al JPA(03)qp/02 [in quantum mechanics], JPA(04)qp [in PT-symmetric quantum theory], JMP(05)mp/04 [Lorentz transformation properties]; blog sa(11)aug, sa(11)sep [how to explain handedness to aliens]; blog io9(15)jul [history, Chien-Shiung Wu's 1957 experiment].
@ Related topics: Witten PRB(16)-a1605 [the parity anomaly on an unorientable manifold]; Arzano et al a1712 [at the Planck scale].

Parity Violation
* In atomic physics: The nuclear decay results are explained in the standard model by assuming that the W ± bosons that govern the weak interaction only exist in a left-handed version; The different absorption of left- and right-circularly polarized light is explained by the Z 0-boson exchange in atoms between electrons and nuclei.
* In gravity: It can arise for example in Chern-Simons modified gravity, where the Einstein-Hilbert action is modified through the addition of the gravitational parity-violating Pontryagin density coupled to a field; A manifestation would be that waves of opposite helicity travel at different speeds; A test of this modification of gravity would be to measure the moment of inertia, Love number and rotational quadrupole moment of a neutron star with gravitational-wave and electroagnetic-wave measurements, and check whether they satisfy the general relativity I-Love-Q relations or modified ones.
@ In atomic physics: Bouchiat et al PLB(82), Wood et al Sci(97)mar [cesium]; Guéna et al MPLA(05); Tsigutkin et al PRL(09) + Jungmann Phy(09) [large violation observed in ytterbium]; Darquié et al Chir(10)-a1007 [towards observation in chiral molecules].
@ In gravity: Contaldi et al PRL(08)-a0806 [and cmb polarization]; Yunes et al PRD(10)-a0912 [and neutron-star moments of inertia]; Gluscevic & Kamionkowski PRD(10)-a1002 [TB/EB correlations in cmb]; Yunes et al PRD(10)-a1005 [gravitational waves and short GRBs]; Crowder et al PLB(13)-a1212 [and the stochastic gravitational-wave background]; Contaldi AJ(17)-a1510 [imprints on the cmb]; Crisostomi et al a1710 [chiral scalar-tensor theories]; Alexander & Yunes a1712, Gupta et al CQG(18) [signatures in gravitational waves and I-Love-Q relations]; > s.a. gravitational instantons; matter phenomenology in quantum gravity.
@ In QCD: Andrianov & Espriu PLB(08)-a0709 [spontaneous, at finite baryon density]; Zhitnitsky a1411-proc [and long-range topological order].
@ In other theories: Ramsey PR(58) [and monopoles]; Wu et al PR(57) [observation in nuclear decay]; Anthony et al PRL(04) + pw(04)may [observation in e collisions]; Alexander PLB(08)ht/06 [and WMAP anomalies]; Wu et QUaD PRL(09) [bounds from cmb polarization].

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