Quantization of First-Class Constrained Systems| Quantization of First-Class Constrained Systems |
In General
> s.a. quantization of second-class systems; dirac procedure.
* Methods: There are various
methods; The Dirac and reduced-phase-space formalisms are not equivalent,
but this is not obvious from some of the simplest examples (like QED without
sources); When they differ, the Dirac procedure seems to be the correct one
if the constrained degrees of freedom are in principle excitable; They are
equivalent for cotangent bundle phase spaces with canonical symplectic structure
[@ Puta LMP(84)];
When the constraints are power of a linear function (irregular, type II), the
Hamiltonian and Lagrangian descriptions may be dynamically inequivalent.
Reduced Phase Space
* Idea: Use the space
of orbits of the constraint vector fields on the constraint surface
Γ' as phase space.
* Example: Consider the gauge vector
field v on Γ', with gab
vavb
= λ2 (or λ?); Then, go to the space of
orbits of v, on which there is a metric hab;
Wave functions are densities of weight 1/2 on this reduced phase space, and the
Hamiltonian is H = −\(\hbar\)2
hab
Pa
Pb + potential; When defining the inner
product, the measure should be λ1/2
dvh, not just
dvh.
@ References: Blyth & Isham PRD(75) [applications];
Pons et al JPA(99)mp/98 [theory for gauge theory];
Chingangbam & Sharan qp/99 [examples];
Muslih NCB(02)mp/01;
Thiemann CQG(06)gq/04 [and partial observables];
Anastopoulos gq/04 [geometric procedure].
Batalin-Vilkovisky, Batalin-Fradkin-Vilkovisky, BRST Methods
> s.a. BRST quantization; lagrangian dynamics;
renormalization; symplectic structures.
* Idea: The BV
method is a powerful Lagrangian method, generalizing the BRST approach, to analyze
functional integrals with (infinite-dimensional) gauge symmetries, invented to fix gauges
associated with symmetries that do not close off-shell; The BFV method is Hamiltonian.
* Fradkin-Vilkovisky theorem:
The Batalin-Fradkin-Vilkovisky path integral is complete independent of the gauge
fixing 'fermion', even within a non-perturbative context.
@ General references: Hasiewicz et al JMP(91) [and Gupta-Bleuler];
Govaerts & Troost CQG(91) [BFV and Faddeev];
Khudaverdian & Nersessian MPLA(93) [geometrical];
Batalin & Tyutin IJMPA(96)ht/95 [perturbative equivalence];
Govaerts & Scholtz JPA(04)ht [Fradkin-Vilkovisky theorem];
Bashkirov et al ht/05 [field theories, necessary and sufficient conditions];
Bashkirov & Sardanashvily ht/06 [and Ward identities];
Fredenhagen & Rejzner CMP(12)-a1101 [on generic globally hyperbolic spacetimes];
Bonechi et al a1907 [equivariant extension];
Rejzner a2004-proc [intro and motivation].
@ BV-BFV formalism, intros: in Cattaneo & Schiavina LMP(16)-a1607;
Mnev a1707-in [and topological quantum field theory];
Cattaneo & Moshayedi a1905-ln.
@ Batalin-Vilkovisky method: Albert et al JMP(10)-a0812 [in finite dimensions];
Anselmi PRD(14)-a1311 [and background-field method];
Getzler JHEP(16)-a1511 [for the spinning particle];
Clavier & Nguyen a1609-proc [as integration for polyvectors];
Iseppi RVMP(19)-a1610 [application to a matrix model];
Ikeda & Strobl a2007 [from BFV, and spacetime covariance].
@ And geometric quantization:
Duval et al CMP(90);
Figueroa-O'Farrill & Kimura CMP(91).
@ Specific types of theories: McMullan JMP(87) [BFV and Yang-Mills theories];
Browning & McMullan JMP(87) [BFV for other theories];
Hüffel APS(02)ht-in [2-point non-commutative Yang-Mills model];
Dayi IJMPA(04)ht/03 [generalized fields];
Bashkirov ht/04 [BV, quadratic \(\cal L\)];
Fredenhagen &Rejzner CMP(13)-a1110 [perturbative algebraic quantum field theory];
Rejzner PhD-a1111 [locally covariant field theory];
> s.a. 3D quantum gravity; BF theories;
modified QED [scalar]; perturbative
quantum gravity; quantization of gauge theories.
Path-Integral Quantization > s.a. Faddeev-Popov;
Ghost Fields; path integrals.
* First-class: Choose gauge-fixing
conditions χi(q,
p) = f i,
for fixed f i,
with {χi,
χj}
= 0 and det|{C, χ}| ≠ 0; Then
Z = ∫ \(\cal D\)p \(\cal D\)q δ(χi − f i) δ(Cj) det|{C, χ}| exp{ i ∫ dt \((\dot pq-H)\)} .
@ General references:
Garczyński PLB(87);
Abrikosov PLA(93);
Ferraro et al PLB(94);
Muslih & Güler NCB(97);
Klauder & Shabanov qp/98;
Muslih mp/00;
Rabei NCB(00);
Klauder LNP(01)ht/00 [rev];
Ohnuki JPA(04) [particle on the D-sphere];
LaChapelle a1212.
@ Coherent states: Klauder AP(97)qp/96,
qp/96,
qp/96;
Klauder & Shabanov PLB(97)ht/96 [including Yang-Mills];
Junker & Klauder EPJC(98)qp/97,
in(99)ht/98 [with fermions].
Other Methods > s.a. deformation
quantization [Fedosov, Moyal]; Faddeev-Jackiw Method.
* Expectation values: Define as physical
states those for which \(\langle\)ψ |C | ψ\(\rangle\)
= 0; One drawback is that it is not a linear condition on the states, so it is not
preserved by linear combinations and the solutions don't obey the superposition
principle–they don't form a subspace of \(\cal H\).
* Triplectic: The
Sp(2)-covariant version of the field-antifield quantization in the
Lagrangian formalism.
@ Expectation values: Marinov FPL(89);
Kheyfets & Miller PRD(95)gq/94;
Kheyfets et al IJMPA(96) [for gravity].
@ Triplectic: Batalin & Marnelius NPB(96)ht/95;
Geyer et al MPLA(99)ht/98;
Grigoriev PLB(99) [Lie group structure];
Geyer & Lavrov IJMPA(04) [general coordinates];
[> s.a. BRST formalism].
@ Hamilton-Jacobi approach: Baleanu & Güler NCB(99),
NCB(00);
Güler NCB(05).
@ Rieffel induction: Landsman dg/96;
Wren JGP(98).
@ Related topics: Dayi PLB(89) [gauge fixing];
Klauder qp/98-fs [infinite-dimensional];
Savvidou & Anastopoulos CQG(00)gq/99 [histories quantization];
Rabei et al PRA(02) [WKB approximation, semiclassical];
Little & Klauder PRD(05)gq [second-class on quantization, model];
Bojowald et al RVMP(09)-a0804 [effective constraints];
Wachsmuth & Teufel PRA(10)-a1005 [configuration-space constraints in terms of confining potential].
References > s.a. coherent states;
geometric quantization; quantum states [semiclassical].
* Remark: Grundling
has proposed a method for obtaining an algebra on reduced phase space,
which works even for classically ergodic systems, where other methods
like group averaging fail (from Ray).
@ Texts, revs: Gitman & Tyutin 90;
Klauder LNP(01)ht/00;
Grundling RPMP(06) [Grundling, Hurst approach];
Rothe & Rothe 10;
Prokhorov & Shabanov 11.
@ General:
in Ashtekar & Horowitz PRD(82);
Ashtekar & Stillerman JMP(86);
Kuchař PRD(87) [factor ordering],
in(88) [covariant];
Dresse et al PLB(90);
Hájíček in(94);
Klauder qp/96-proc;
Kaplan et al PRA(97)qp/98;
Klauder NPB(99)ht;
Corichi CQG(08)-a0801 [geometrical];
Brody et al JPA(08),
JPA(09);
Gustavsson JPCS(09)-a0903 [symplectic vs metric formulation];
Fairbairn & Meusburger a1204-in;
Bojowald & Tsobanjan a1906 [symplectic reduction].
@ Semiclassical aspects: Kirwin MZ-a0810 [reduction and quantization, semiclassical];
Bojowald & Tsobanjan RVMP(09),
PRD(09)-a0906 [effective constraint methods];
Wachsmuth & Teufel MAMS-a0907 [effective Hamiltonian];
Tsobanjan AIP(09)-a0911 [leading-order corrections to dynamics].
@ Related topics:
Goldberg et al JMP(91);
Rovelli PRL(98) [gauge transformations in quantum mechanics];
Lavrov et al MPLA(99) [osp(1,2) supersymmetry];
Facchi et al JOB(04)qp/03 [and Zeno dynamics];
Konopka & Markopoulou gq/06 [states, from noiseless subsystems];
> s.a. regularization;
superselection rules; theta sectors.
Specific Types of Systems
> s.a. quantum gauge theories; canonical quantum gravity;
Proca Theory; supergravity.
* Parametrized field
theories: Torre and Varadarajan showed that for generic foliations
emanating from a flat initial slice in D > 2 spacetimes,
scalar field evolution along arbitrary foliations is not unitarily
implemented on the Fock space, which implies an obstruction to Dirac
quantization; The no-go result can be overcome however using lqg techniques.
@ General references: Banerjee & Chakraborty AP(96) [Chern-Simons];
Arik & Ünel ht/96 [quadratic C];
Grundling & Hurst JMP(98) [not preserved by dynamics];
Montesinos et al PRD(99)gq [general relativity toy model];
Mišković & Zanelli JMP(03),
Klauder & Little CQG(06)gq [irregular];
Serhan et al IJTP(09) [holonomic];
Belhadi et al AP(14)-a1406 [classically soluble constrained systems];
Bojowald & Brahma JPA(16)-a1407 [fluctuations and structure functions].
@ Special configuration spaces:
Kleinert & Shabanov PLA(97) [on Sd];
Maraner ht/98 [on a line];
Ikemori et al MPLA(99)
[on S2];
Scardicchio PLA(02) [on S1];
> s.a. quantum systems.
@ Reparametrization-invariant: Klauder JPA(01)qp/00 [ultralocal fields];
Varadarajan PRD(07)gq/06 [parametrized field theory];
> s.a. parametrized theories.
@ Totally constrained: Kodama PTP(95)gq,
PTP(95)gq [classical and quantum theory];
Doldán et al IJTP(96)ht/94;
Olmedo IJMPD(16)-a1604 [Schrödinger vs Heisenberg pictures].
@ Generally covariant: Montesinos GRG(01)gq/00 [relational evolution];
Sforza PhD(00)gq;
> s.a. types of quantum field theories.
@ Time-dependent constraints: Muslih CzJP(02)mp/01 [canonical path integral].
@ Discrete spacetime lattice: Di Bartolo et al CQG(02)gq.
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