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 
:
→ 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)
= –i
v (q)
= 0, and the
Hamiltonian is different,
H = –2
gab 
ab +
potential = –2
gab Da Db – 2 
–1(Db)
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.
Comparisons, Relationships > s.a. canonical
quantum mechanics [group
quantization].
@ 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-in;
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);
Hájícek JMP(86)
[path integral version of 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 a0911 [and master constraint and reduced phase space].
@ And Faddeev-Jackiw: García & Pons IJMPA(98)ht;
Liao & Huang AP(07).
@ 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 ht/06 [and
Faddeev-Popov].
References
@ General: Dirac CJM(50); Bergmann HPA(56);
Dirac PRS(58), PR(59),
64; Matschull qp/96 [review];
Deriglazov PLB(05)
[without primary constraints].
@ Geometric version: Tulczyjev SM(74); Lichnerowicz CRAS(75); Barvinsky
gq/96; Gozzi
NPPS(97)dg.
@ Objections, problems: Shanmugadhasan JMP(73); Kundt.
@ Applications, examples: DeWitt PR(67);
Kuchar PRD(89);
Hájícek CQG(90)
[quadratic constraint]; Hájícek & Kuchar PRD(90), JMP(90)
[quadratic + linear constraints,
operator ordering and transversal affine connection]; Montesinos et al PRD(99)gq [2
non-commuting Hamiltonian constraints];
Miskovic & Zanelli JMP(03)
[irregular systems]; Rosenbaum et al JPA(07)ht/06 [spacetime
non-commutative
theories]; > s.a. parametrized theories.
@ Master constraint approach: Dittrich & Thiemann CQG(06)gq/04 [framework],
CQG(06)gq/04 [finite-dimensional],
CQG(06)gq/04 [SL(2, R)
models], CQG(06)gq/04 [free
field theories], CQG(06)gq/04 [interacting
field theories]; Han & Thiemann a0911 [and refined algebraic quantization]; > s.a. loop
quantum
gravity.
@ Related topics: Tuynman JMP(90)
[modified]; Barvinsky CQG(93)
[operator ordering];
Tate gq/93-PhD
[algebraic]; 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 a0903 [variant, metric approach].
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 
kin;
Given
a kinematical kin,
instead of imposing constraints on the kinematical states to get physical ones,
rig kin to
obtain
kin *,
and choose
a suitable phy *,
with inner product
(
(1),
(2)):=
(2)[1],
where 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, HJ(98)gq/97;
Giulini & Marolf CQG(99)gq/98,
CQG(99)gq;
Giulini NPPS(00)gq [rev].
@ Specific theories:
Ashtekar & Tate JMP(94)gq [examples];
Louko & Rovelli JMP(00)gq/99 [SL(2, 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].
Group Averaging
* Try: Do the case C = px +
py on T2, with 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 :
→ *
by (|
)
(|:= V–1 
d
G(g)
| U(g),
with
U a representation of G.
@ References: Giulini NPPS(00)gq [rev];
Marolf gq/00-MG9;
Marolf 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, R)], IJMPD(05)gq/04 [subgroup
of SL(2, R)]; Kaminski et al a0907 [lqc-related examples]; > s.a. quantum
klein-gordon
theory.
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nov 2009