Theta
Sectors / Vacua |

**In General** > s.a. representations
in quantum mechanics.

* __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].

@ __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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