Dirac Quantization of First-Class Constrained Systems| Dirac Quantization of First-Class Constrained Systems |
In General > s.a. quantum particle models.
* Idea: Impose the constraints
as operators on functions Ψ: Γ → \(\mathbb C\) (this requires
a choice of operator ordering and regularization); Their kernel defines physical
states, and only on these we define an inner product such that the observables
are selfadjoint.
* Example: Wave functions
are densities of weight 1/2 on phase space, which have to satisfy
(va
Pa) ψ
= −i\(\hbar\) \(\cal L\)v
ψ(q) = 0, and the Hamiltonian is different,
H = −\(\hbar\)2 gab ∇a∇b + potential = −\(\hbar\)2 gab Da Db − \(\hbar\)2 λ−1 (Dbλ) Db + potential .
* Remark: An anomaly
in the commutators would mean, e.g., that the wave function on a
given surface depends on gauge equivalent paths used to get there!
* Criticism: Might lead
to non-normalizable states if the gauge orbits are non-compact.
@ General references: Dirac CJM(50);
Bergmann HPA(56);
Dirac PRS(58),
PR(59),
64;
Matschull qp/96 [review];
Deriglazov PLB(05) [without primary constraints];
Lantsman a1110
[Dirac variables and gauge-invariant and Poincaré-covariant states];
Kiriushcheva et al a1112 [field-parametrization dependence].
@ Geometric version: Tulczyjev SM(74);
Lichnerowicz CRAS(75);
Barvinsky gq/96;
Gozzi NPPS(97)dg;
Lian et al a1703 [geometric potential].
@ Objections, problems:
Shanmugadhasan JMP(73); Kundt.
@ Applications, examples: DeWitt PR(67);
Kuchař PRD(89);
Hájíček CQG(90) [quadratic constraint];
Hájíček & Kuchař PRD(90),
JMP(90)
[quadratic + linear constraints, operator ordering and transversal affine connection];
Montesinos et al PRD(99)gq [2 non-commuting Hamiltonian constraints];
Mišković & Zanelli JMP(03) [irregular systems];
Rosenbaum et al JPA(07)ht/06 [spacetime non-commutative theories];
Barbero et al CQG(19)-a1904 [with boundaries];
> s.a. modified electromagnetism; parametrized theories.
Refined Algebraic Quantization
* Idea: A variation of the
Dirac prescription, motivated by the fact that in many cases, physical states
of interest are not in \(\cal H\)kin;
Given a kinematical \(\cal H\)kin,
instead of imposing constraints on the kinematical states to get physical
ones, rig \(\cal H\)kin to obtain Ω
⊂ \(\cal H\)kin⊂ Ω*, and
choose a suitable \(\cal H\)phy ⊂ Ω*,
with inner product (η(ψ1),
η(ψ2)):=
η(ψ2)[ψ1],
where η is the rigging map; One also defines an action of physical observables;
The rigging map can be defined when the constraints form a Lie algebra (not an algebra
with non-trivial structure functions).
@ General references: Marolf gq/95;
Embacher gq/97-MG8,
HJ(98)gq/97;
Giulini & Marolf CQG(99)gq/98,
CQG(99)gq;
Giulini NPPS(00)gq [rev];
Louko & Martínez-Pascual JMP(11)-a1107 [constraint rescaling];
Martínez-Pascual JMP(13)-a1305 [with non-constant gauge-invariant structure functions].
@ Specific theories: Ashtekar & Tate JMP(94)gq [examples];
Louko & Rovelli JMP(00)gq/99 [SL(2, \(\mathbb R\)) gauge theory];
Shvedov ht/01 [systems with structure functions];
Louko & Molgado CQG(05)gq [Ashtekar-Horowitz-Boulware model];
Rumpf gq/97
[relativistic particle in curved spacetime];
Gambini & Olmedo CQG(14)-a1304 [totally constrained model, and other quantization methods].
Group Averaging
* Try: Do the case C
= px + α
py on T2,
with α irrational.
* Idea: An implementation of the
refined algebraic quantization method, which one can use (modulo expression
being well-defined) when the constraint algebra closes, but has been formally
used even in a more general case.
* Prescription:
Define the rigging map η: Ω → Ω*
by (|ψ\(\rangle\) ∈ Ω) \(\mapsto\)
(ψ|:= V−1
∫ dμG(g)
\(\langle\)ψ| U(g), with U a representation of G.
@ References:
Giulini NPPS(00)gq [rev];
Marolf gq/00-MG9;
Marolf PRD(09)-a0902 [perturbation theory for rigging map].
@ Non-compact groups: Gomberoff ht/00-MG9;
Louko JPCS(06)gq/05.
@ Examples: Gomberoff & Marolf IJMPD(99)gq [SO(n,1)];
Louko & Molgado JMP(04)gq/03
[(p, q)-oscillator representation of SL(2, \(\mathbb R\))],
IJMPD(05)gq/04 [subgroup of SL(2, \(\mathbb R\))];
Kamiński et al CQG(09)-a0907 [lqc-relatedexamples];
> s.a. quantum klein-gordon theory.
Other Approaches and Comparisons
> s.a. canonical quantum mechanics [group quantization].
* Master constraint approach: Replace
the individual constraints by a weighted sum of absolute squares of the constraints.
@ Master constraint approach: Dittrich & Thiemann CQG(06)gq/04 [framework],
CQG(06)gq/04 [finite-dimensional],
CQG(06)gq/04 [SL(2, \(\mathbb R\)) models],
CQG(06)gq/04 [free field theories],
CQG(06)gq/04 [interacting field theories];
Han & Thiemann JMP(10)-a0911 [and refined algebraic quantization];
> s.a. loop quantum gravity.
@ And reduced phase space:
Buchbinder & Lyakhovich TMP(89) [including inner product];
Mladenov IJTP(89);
Romano & Tate CQG(89);
Loll PRD(90);
Schleich CQG(90);
Kunstatter CQG(92);
Ordóñez & Pons PRD(92),
JMP(95)ht/93;
Plyushchay & Razumov IJMPA(96)ht/93,
ht/94-conf;
Epp PRD(94);
Vathsan JMP(96)ht/95 [for simple gauge theory];
Shimizu PTP(97)gq/96;
> s.a. dirac quantum field theory.
@ And path integral: Faddeev TMP(69);
Maskawa & Nakajima PTP(76);
Hájíček JMP(86)
[path-integral version of the projector];
Blau AP(91);
Cabo et al PLB(91);
Halliwell & Hartle PRD(91);
Govaerts JPA(97)ht/96 [phase-space coherent states];
Barvinsky NPB(98)ht/97,
PLB(98)ht/97 [solution];
Han & Thiemann CQG(10)-a0911 [and master constraint and reduced phase space].
@ And Faddeev-Jackiw approach:
García & Pons IJMPA-ht/96,
IJMPA(98)ht;
Liao & Huang AP(07);
Manavella IJMPA(14)
[composite particles, with Grassmann dynamical variables];
> s.a. Stückelberg Model.
@ And other approaches: in Faddeev & Slavnov 80 [BRST];
Barvinsky & Krykhtin CQG(93) [BFV, 1-loop];
Ogawa et al PTP(96)ht/97 [Schwinger];
Louis-Martinez PLA(00) [and Moyal];
Shvedov AP(02)ht/01 [BRST-BFV];
Lantsman FizB(09)ht/06 [and Faddeev-Popov].
@ Related topics: Tuynman JMP(90) [modified];
Barvinsky CQG(93) [operator ordering];
Tate PhD(92)gq/93 [algebraic approach];
Seiler & Tucker JPA(95) [from the pde point of view];
Rovelli gq/97 [space of solutions];
Kempf & Klauder JPA(01)qp/00 [0 ∈ continuous spectrum and projection];
Brody et al JPA(09)-a0903 [variant, metric approach].
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