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 π___{1}(\(\cal C\)):
Quantize on the universal covering space of \(\cal C\), and use
π_{1}(\(\cal C\)) 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. quantum gauge theory.

* __Idea__: In Yang-Mills
theory \(\cal C\)_{*}
always ends up being abelian (\(\mathbb Z\) if Σ = S^{3}), and
π_{1}(\(\cal C\)_{*})
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 a1806 [and vacuum energy density in chiral models];
Vonk et al JHEP-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 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,
π_{1}(\(\cal C\)_{*})
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*/*G*Λ)
*R*^{ij} ∧ *R*_{ij},
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];
> 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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