Parametrized
Theories| Parametrized
Theories |
In General > s.a. symplectic
structures; time.
* Idea: Theories with time as
a dynamical variable; t is a function
of 
and the theory is invariant
under → '().
* Hamiltonian formulation:
We need to introduce Lagrange multipliers to get a Hamiltonian formulation;
These theories have as a common feature that H is a constraint,
and vanishes (on shell); The reduced phase space gives the space of histories.
@ General references: Hájícek pr(87), PRD(88), JMP(89);
Hájícek NPPS(97)gq/96 [time
evolution and observables]; Fulop
et al IJTP(99)
[reparametrization as gauge]; Muslih IJMMS(02)mp/01 [Lagrangian
and Hamiltonian]; Albrecht & Iglesias PRD(08)-a0708 [clock
ambiguity]; Castrillón et al JPA(08)
[parametrization method with a background metric].
@ Relational theories: Anderson CQG(08)-a0706 [foundations];
Gryb CQG(09)-a0810.
Barbour-Bertotti Models > s.a. mach's
principle.
* Idea: A relational
class of models, with action of the type
S = 
[–V(x) 
i mi (dxi/dt)2]1/2 d .
@ General references: Barbour Nat(74)may, NCB(75);
Barbour & Bertotti NCB(77), PRS(82);
in Smolin pr(88); Gergely CQG(00)gq, & McKain
CQG(00)gq;
Pooley & Brown
BJPS(02)
[relationalist implications].
@ Quantization: Barbour & Smolin pr(88); Rovelli pr(88), in(91)
[group quantization]; in Smolin
pr(88); Gryb a0804 [and time].
Non-Relativistic Point Particle > s.a. particle
models.
* Action: It is of the type
S = (pa dxa/dt – N H)
d ,
with constraint H = p2/2m + V(x)
and N =
Lagrange multiplier (= dt/d);
Gives a parabolic superhamiltonian.
* Applications: Often used as a test model for proposed solution to
the
problem of time in quantum gravity.
@ General references: Hartle & Kuchar JMP(84)
[path integrals]; Anderson CQG(06)gq/05,
CQG(06)gq/05 [relational
particle models].
@ Time: Elze & Schipper PRD(02)gq [stochastic];
Gambini et al in(03)gq [from
discrete formulation].
Free Relativistic Point Particle > s.a. particle
models.
* Action: One possible form is of the type
S = (pa dxa/dt – N H)
d , with
constraint H = (p2+m2)/2m ,
which gives a hyperbolic superhamiltonian; We fix x(1)
= x1, x(2)
= x2, and vary the trajectory in
between; The meaning of the Lagrange multiplier is related to a 1D metric on
the world-line.
* Alternative: A Lagrangian
invariant under reparametrizations → '
= f(),
df/d > 0,
with f(1)
= 1, f(2)
= 2 is
L = –m [gab (dxa/dt)(dxb/dt)]1/2 .
@ Invariant time formulation: Fanchi FP(93).
@ Quantum theory: Hartle & Kuchar PRD(86)
[path integrals]; Fanchi 93.
Field Theories > s.a. diffeomorphisms [parametrized Maxwell
theory]; string
theory.
* Minkowski space field theory:
The theory can be parametrized by treating the 'embedding variables' which
describe the foliation as dynamical variables to be varied in the action, in
addition to the scalar field; Yields
a parabolic superhamiltonian; > s.a. constrained
systems [quantization].
@ Scalar field: Isham & Kuchar AP(85);
Kuchar PRD(89)
[on a cylinder];
Varadarajan PRD(04)
[path-integral quantization], PRD(07)
[Dirac quantization, lqg techniques].
@ Gravity: Kuchar in(81); [> s.a. canonical
general relativity].
Quantization > s.a. quantization
of constrained systems.
@ General references: Hájícek JMP(95)gq/94,
et al JMP(95)gq/94 [group
quantization];
Ruffini PhD(95)gq/05;
Ashworth
PRA(98)qp/96 [coherent
state]; Muslih NCB(00); De Cicco & Simeone IJMPA(99)gq/01 [path
integral];
Savvidou & Anastopoulos
CQG(00)
[histories]; Montesinos GRG(01)
[evolving constants]; > s.a. histories
quantum mechanics.
@ 
H
=
0 approach: Wang CQG(03)gq [FRW
spacetime + scalar];
Nikolic
gq/03.
@ In quantum cosmology: Blyth & Isham PRD(75)
[use H1/2].
@ Quantum observables: Marolf CQG(95)gq/94.
@ Parametrized
non-relativistic quantum mechanics: Hartle & Kuchar in(84); Hartle
CQG(96)gq/95 [time];
Hartle & Marolf PRD(97)gq [decoherent
histories];
Lawrie & Epp PRD(96)gq/95 [interpretation];
Colosi a0711 [Pegg-Barnett
phase operator formalism].
@ And quantum gravity: Gaioli & García-Álvarez GRG(94)gq/98 [and
quantum gravity];
Gambini & Porto PRD(01)gq [generally
covariant theories];
Anderson CQG(07)gq/06.
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send feedback and suggestions to bombelli at olemiss.edu – modified 8
aug
2009