CPT Symmetry and Theorem

In General > s.a. anomalies; Charge Conjugation; M-theory; particle statistics; QED; quantum klein-gordon field.
* Remark: It is a symmetry of any Lorentz-invariant, local quantum field theory, whether or not the individual factors are.
* Operator: The operator θ = PCT is anti-unitary; for a charged scalar field, θ⁻¹ φ(x) θ = φ(−x)*.
* Proofs: Lüders and Pauli proved it based on Lagrangian quantum field theory; Jost gave a more general axiomatic quantum field theory proof.
@ Proofs of theorem: Lüders AP(57), reprinted AP(00); Streater & Wightman 64; in Feynman in(87) [topological]; Greenberg FP(06)hp/03; Greaves & Thomas SHPMP(14)-a1204; Selover & Sudarshan a1308 [more general, using action principles].
@ Related topics: Norbury EJP(90) [for electromagnetism]; Greenberg PLB(98)ht/97 [and locality]; Duck & Sudarshan 98; Borchers & Yngvason mp/00; Kostelecký ed-02; Socolovsky IJTP(04)mp [Dirac field]; Scurek AJP(04)may [group and representations]; Carballo & Socolovsky IJTP(09)-a0811 [CPT from P and T subgroups of Lorentz group]; Carballo & Socolovsky a0906 [irreducible representations of CPT group for QED]; Greaves BJPS(10)#1 [geometrical understanding, using classical analog]; > s.a. spin-3/2 fields.
@ Generalized: Grimus & Rebelo PRP(97) [P and CP, in gauge theories]; Soloviev JMP(98) [non-local quantum field theory]; Dastidar & Dastidar MPLA(99) [non-local gauge theories]; Chaichian et al PLB(03)ht/02, Franco & Polito JMP(05) [non-commutative theories]; Jannussis et al NCB(05) [non-Hermitian H]; Mannheim PLB(16)-a1512 [non-Hermitian H and unstable states]; Betzios et al a2004 [gauging the CPT symmetry, and the Riemann hypothesis].

Violation and Tests > s.a. CP violation; early-universe baryogenesis; matter; parity.
* Possible reasons: Extra spacetime dimensions; Violation of Lorentz invariance; Quantum-gravity effects.
* For electrons: The relative difference for g of e⁺ and e⁻ is less than 2 × 10⁻¹².
* For neutral kaons: 2009, No deviation from CPT symmetry and quantum mechanics observed at the KLOE experiment at the DAΦNE \(e^+\)-\(e^-\) collider.
* For baryons: The relative mass difference of p and \(\bar p\) is less than \(10^{-10}\) [@ news pn(98)may], and the magnitudes of the antiproton and proton magnetic moments differ by less than 5 parts per million [> see hadrons].
@ General references: Colladay AIP(03)hp [rev]; Kostelecký a1010-conf [introduction]; Bailey a1906-conf [status].
@ And Lorentz invariance: Greenberg PRL(02)hp, objection Chaichian et al PLB(11)-a1103, response Greenberg a1105 [implies violation]; Dütsch & Gracia-Bondía PLB(12) [not so clear].
@ Theoretical models: Bertolami et al PLB(97)hp/96 [and baryogenesis]; Adam & Klinkhamer NPB(01)hp, PLB(01)ht [abelian Chern-Simons theory]; Klinkhamer ht/01-conf, PRD(02)ht/01, hp/05-conf [chiral fermions, non-trivial spacetime topology]; Chaichian et al EPJC(13)-a1205 [Lorentz-invariant]; > s.a. modified lorentz symmetry and QED.
@ Cosmological, cmb: Feng et al PRL(06); Cabella et al PRD(07)-a0705 [WMAP 3-year data]; Auriemma a0711-conf [rev]; Xia et al PLB(10)-a0908 [polarization]; Li et al ApJ(15)-a1405; Zhao et al JCAP(15)-a1504 [efficient probe].
@ Neutrinos: Minataka & Uchinami PRD(05)hp [supernova neutrinos]; Hooper et al PRD(05)hp [high-energy]; Tsukerman a1006; Wang & Pan a1512 [oscillations]; Huang et al a1906 [IceCube data, and Lorentz symmetry].
@ Other tests: Commins AJP(93)sep [RL]; Colladay & Kostelecký PLB(95)hp, PRD(95)hp, hp/96-conf [high-energy physics]; Bluhm et al PRL(99); Geer et APEX PRL(00); Hughes et al PRL(01) [muonium spectroscopy]; Murayama PLB(04) [Ks vs neutrinos]; Canè et al PRL(04) [neutron, bound on boost effects]; Bernabeu et al hp/06-conf; Di Domenico et KLOE FP(10) [neutral kaons]; Toma et al PRL(12) [bound from GRB polarization]; Friedman et al a1809 [active galaxies]; Arzano a1904 [from muon lifetime, Planck-scale deformations]; Wang & Zhao a2002 [binary black hole gravitational waves].

Time Reversal > s.a. arrow of time; electromagnetism; entropy; finsler geometry and physics; PT Symmetry.
* Remarks: Distinguish between irreversibility and T or CPT violation (Lüders: T should be called "motion reversal"); Finding T violation would be equivalent to finding CP violation, because of the CPT theorem.
* And experiment: One of the ways of looking for T violation is to look for a neutron electric dipole moment; The Hamiltonian would contain a term d σ · E, and, under T, σ \(\mapsto\) −σ and E \(\mapsto\) E, so H would not be invariant; > for the current best bound, see neutrons.
@ General references: Schrödinger SPAW(31) [time reversal of a diffusion process]; Ramsey PR(58) [and magnetic poles]; Rosen AJP(73)apr [for electromagnetic quantities]; Sachs Sci(72)may, 87; Domingos IJTP(79) [rev]; Price qp/96-conf [interactions and boundary conditions]; Kuenzi et al PRA(02)cm [in solid-state physics]; North PhSc(08)apr [new view]; Arntzenius & Greaves BJPS(09) [in classical electromagnetism]; Oreshkov & Cerf nPhys(15)-a1507 [in quantum theory, operational formulation]; Roberts a1607 [comments on the definition]; Ardakani a1802-MS; Huang et al a1609 [2-dimensional spaces]; Roberts psa(18).
@ Violation: Ryder CP(94); news PT(99)feb; Gutkin JPA(07) [dynamical, and chaos]; Greentree & Martin Phy(10) [in photon lattices]; Vaccaro FP(11)-a0911 [and unidirectionality of time]; Polonyi PRD(11)-a1109 [dynamical breakdown, and causality]; de Vries et al PRL(11) + news sd(11)oct [and deuteron electric-dipole and magnetic-quadrupole form factors]; news PT(12)nov, cbs(12)nov + Zeller Phy(12) [first clear, direct evidence of T violation in BaBar observations of transition rates between B-meson states]; Roberts a1306, a1306-conf [three mechanisms]; Ashtekar SHPMP-a1307-conf [new perspective]; Polonyi Symm-a1503 [explicit]; Roberts psa(15).
@ Experiment: news pn(98)nov, Mavromatos pw(98)dec [violation observed in kaons]; Mumm et al PRL(11) + news po(11)nov [in beta decay, limit].
> And gravity: see canonical general relativity; modified theories.

In Curved Spacetime > s.a. quantum field theory effects in curved spacetime [Unruh effect].
* Remark: It does not hold for a black hole background metric, in the sense that there is no operator θ such that $⁻¹ = θ $ θ⁻¹, where $ is the superscattering matrix for pure and mixed states.
@ References: Brout & Englert NPB(81), Anandan PRL(98) [and cosmology]; Buchholz et al CQG(00) [AdS]; Hollands CMP(04)gq/02 [and operator product expansion]; Arntzenius SHPMP(04) [Lorentz group representations, and t-orientation]; Varlamov PLB(05)mp [de Sitter space, spinor fields].

In Quantum Gravity > s.a. non-commutative field theory; time in quantum gravity.
* Idea: The CPT theorem is a consequence of usual quantum field theory; It uses special relativity (in that it assumes the symmetries of Minkowski spacetime) and quantum mechanics, and therefore it need not hold in quantum gravity.
* Scenarios: It may arise from modified uncertainty relations [@ Amelino-Camelia MPLA(97)gq], or from geons.
@ General references: Penrose in(79), in(81); Page PRL(80); Wald PRD(80); Wald in(81); Hawking CMP(82); Page GRG(82); Banks et al NPB(84); Hawking NPB(84); Gross NPB(84); Neacsu IJTP(84); Banks NPB(85); Mavromatos NIMB(04)hp/03, SPP-hp/03; Mavromatos hp/04-conf, LNP(05)hp/04 [tests with neutrinos]; Mavromatos LNP(05)gq/04-ln [review, emphasis on spacetime foam]; Klinkhamer & Rupp PRD(04); Bernabeu et al PRD(06)ht [entangled neutral mesons]; Mavromatos PoS-a0707 [and decoherence]; Rovelli & Wilson-Ewing PRD(12)-a1205 [time reversal and parity in covariant lqg].
@ In string theory: Colladay & Kostelecký PRD(97)hp; Mavromatos JPCS(09)-a0904, FP(10)-a0906.
@ Other scenarios: Alexanian & Balachandran PLB(02)ht/01 [geons].

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