Theta Sectors / Vacua| 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 \(\cal C\)*
= \(\cal A\) / \(\cal G\)*.
References
@ 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)
Rij ∧ Rij,
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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