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].

@ __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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