Parity |

**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];
Huang et al a1609 [2-dimensional spaces].

@ __Related topics__: Witten PRB(16)-a1605 [the parity anomaly on an unorientable manifold];
Arzano et al PLB(18)-a1712 [at the Planck scale].

**Parity Violation**

* __In atomic physics__:
The nuclear decay results are explained in the standard model by
*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 electromagnetic-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 PRD(18)-a1710 [chiral scalar-tensor theories];
Alexander & Yunes PRD(18)-a1712,
Gupta et al CQG(18) [signatures in gravitational waves and I-Love-Q relations];
Obukhov IJGMP-a2010-proc [Poincaré gauge gravity];
> 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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