Theta Sectors / Vacua  

In General > s.a. representations in quantum mechanics; vacuum [in Yang-Mills theory].
* Idea: A θ-sector refers to a choice of vacuum, lebelled by a parameter θ, for quantum mechanics or quantum field theories with degenerate vacua, in which a superselection rule prevents a physical state from going from one vacuum to another.
* Origin: It arises as a quantization ambiguity when the configuration space is non-simply connected; Unitarily inequivalent quantizations are characterized by unitarily irreducible representations of the fundamental group.
* Quantum mechanics with non-trivial π1(\(\cal C\)): Quantize on the universal covering space of \(\cal C\), and use π1(\(\cal C\)) as a symmetry, with representations labeled by l; The Hilbert space is then \(\cal H\) = ⊕l \(\cal H\)l.
* For a gauge theory: The configuration space is the space \(\cal C\) = \(\cal A\) / \(\cal G\) of connections modulo gauge transformations, which usually is not a manifold because the group action has fixed points; One can then restrict the gauge group to a \(\cal G\)* ⊂ \(\cal G\) with no fixed points, and use \(\cal C\)* = \(\cal A\) / \(\cal G\)*.

@ General: Schulman JMP(71); Dowker JPA(72); Horvathy PLA(80); Klimek-Chudy & Kondracki JGP(84); Imbo & Sudarshan PRL(88); Sudarshan et al PLB(88); Blau MPLA(89).
@ Quantum mechanics in multiply connected spaces: Doebner et al JMP(89) [harmonic oscillator in pointed plane]; Ho & Morgan JPA(96)ht.
@ And constrained quantization: Landsman & Wren NPB(97)ht; Wren NPB(98).
@ And algebraic quantization: Landsman LMP(90); Aldaya et al CMP(96)ht/95.
@ And star quantization: Alcalde JMP(90).
@ And path integrals: Laidlaw & Morette DeWitt PRD(71); Tanimura & Tsutsui AP(97)ht/96.
@ Related topics: Giulini HPA(95)qp [finite fundamental group]; Azcoiti et al PRL(02) [numerical simulations]; Huerta & Zanelli PRD(12)-a1202 [optical properties].

In Yang-Mills Theories > s.a. gauge [large gauge transformations]; quantum gauge theory.
* Idea: In Yang-Mills theory \(\cal C\)* always ends up being abelian (\(\mathbb Z\) if Σ = S3), and π1(\(\cal C\)*) has only 1D IURRs; Instantons induce tunneling between configurations related by large gauge transformations, giving rise to the θ-sectors.
@ General references: Callan et al PLB(76); Jackiw & Rebbi PRL(76); Dowker pr(80); Isham & Kunstatter PLB(81), JMP(82); Jackiw in(84); Zhang ZPC(89); Krive & Rozhavskii TMP(91); Arai JMP(95); Imbo & Teotonio-Sobrinho NPB(97) [2D]; Mazur & Staruszkiewicz ht/98 [electrodynamics]; Vicari & Panagopoulos PRP(09) [SU(N) gauge theory]; Morchio & Strocchi AP(09) [QCD, and chiral symmetry breaking]; Canfora et al PRD(11)-a1105 [theta term in a bounded region]; Luciano & Meggiolaro PRD(18)-a1806 [and vacuum energy density in chiral models]; Vonk et al JHEP(19)-a1905 [QCD].
@ On a circle / cylinder: Witten NCA(79) [2D QCD]; Rajeev PLB(88); Hetrick & Hosotani PLB(89); Langmann & Semenoff PLB(92); Gupta et al JMP(94); Chandar & Ercolessi NPB(94); Landsman & Wren NPB(97)ht; Horie IJMPA(99)ht; > s.a. Gribov Problem.
@ On other spacetimes: Etesi IJTP(07)ht/00 [asymptotically flat, stationary, classification].
@ Gauge-independence: Adam MPLA(99)ht/98.
@ Argument for θ = 0: Khoze PLB(94).
@ In fuzzy / non-commutative physics: Balachandran & Vaidya IJMPA(01)ht/99.
@ Phenomenology: Buckley et al PRL(00)hp/99 [heavy ion collisions, proposal]; Aguado et al MPLA(03) [and CP violation]; Hsu a1012 [for QED]; Meggiolaro PRD(19)-a1903 [vacuum energy density in chiral effective Lagrangian models].

In Quantum Gravity > s.a. [canonical quantum gravity]; Mapping Class Group; quantum-gravity phenomenology; quantum geometry.
* Idea: The configuration space \(\cal C\)* = Riem(Σ) / Diff*(Σ) is generically non-simply connected, because of the existence of diffeomorphisms not connected to the identity (large diffeomorphisms); However, unlike in Yang-Mills gauge theory, π1(\(\cal C\)*) can be non-abelian and different representations of the fundamental group not only lead to θ-sectors, but can also lead to sectors with spin-1/2; In particular, the Kodama state will have sectors with spin 1/2 for generic topologies; Gravitational θ-sectors can arise from a term in the Lagrangian of the form (γ/GΛ) RijRij, where γ is the Immirzi parameter.
@ General references: Isham PLB(81), in(82); Friedman & Witt PLB(83); Witt JMP(86); Friedman & Witt in(88); Hartle & Witt PRD(88); Hájíček CQG(92); Sorkin & Surya IJMPA(98)gq/96 [and representations of the mapping class group]; Chatzistavrakidis et al a2007 [in gravitomagnetism]; > s.a. diffeomorphisms.
@ In 2+1 quantum gravity: Giulini & Louko CQG(95)gq [à la Witten]; Peldán PRD(96)gq.
@ Physical effects: Giulini & Louko PRD(92) [in quantum cosmology]; Fischler & Kundu IJMPD(16)-a1612 [black-hole stretched horizon].
@ From internal gauge: Ashtekar, Balachandran & Jo IJMPA(89); Balachandran, Jo & Srivastava IJMPA(89).

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