Parametrized Theories| Parametrized Theories |
In General > s.a. statistical
mechanical systems; symplectic
structures; time in physical theories.
* Idea: Theories with time
as a dynamical variable, in which t is a function of a parameter
τ and the theory is invariant under a reparametrization
τ \(\mapsto\) τ'(τ).
* 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: Hájíček pr(87),
PRD(88),
JMP(89);
Hájíček 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];
Kothawala a2004
[general covariance and Euler-Lagrange equations from suitable Lie derivatives].
@ Relational theories: Anderson CQG(08)-a0706 [foundations];
Gryb CQG(09)-a0810;
> s.a. Relationalism.
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 dτ .
@ 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 PRD(10)-a0804 [and time].
Non-Relativistic Point Particle > s.a. particle models.
* Action: It is of the type
S = ∫ (pa dxa/dt − N H) dτ ,
with constraint H = p2/2m
+ V(x) and N = Lagrange multiplier (= dt/dτ);
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 & Kuchař 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) dτ , with constraint H = (p2+m2)/2m ,
which gives a hyperbolic superhamiltonian; We fix
x(τ1)
= x1,
x(τ2)
= 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 τ → τ'
= f(τ), df/dτ > 0,
with f(τ1)
= τ1,
f(τ2)
= τ2 is
L = −m [gab (dxa/dt)(dxb/dt)]1/2 .
@ Invariant time formulation: Fanchi FP(93).
@ Quantum theory: Hartle & Kuchař PRD(86) [path integrals];
Fanchi 93.
Field Theories > s.a. string
theory; types of quantum field theories.
* 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].
@ General references: Margalef-Bentabol a1807-PhD.
@ Scalar field:
Isham & Kuchař AP(85);
Kuchař PRD(89) [on a cylinder];
Varadarajan PRD(04) [path-integral quantization],
PRD(07) [Dirac quantization, lqg techniques];
Thiemann a1010 [and lessons for lqg];
Barbero et al CQG(16)-a1507
[with spatial boundaries, Hamiltonian description].
@ Gravity: Bertotti & Easthope IJTP(78) [prerelativistic Machian theory];
Kuchař in(81);
> s.a. canonical general relativity.
@ Electromagnetic field: Barbero et al CQG(16)-a1511;
> s.a. diffeomorphisms [parametrized Maxwell theory].
Quantization > s.a. quantization of constrained systems;
types of quantum field theories.
@ General references: Hájíček 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);
Savvidou & Anastopoulos CQG(00) [histories];
Montesinos GRG(01) [evolving constants];
Cattaneo & Schiavina LMP(17)-a1607 [1D models, and time];
> s.a. histories quantum mechanics.
@ Path integral: De Cicco & Simeone IJMPA(99)gq/01;
Steinhaus JPCS(12)-a1110 [discretized propagator and discretization independence].
@ H\(\rangle\) = 0 approach: Wang CQG(03)gq [FLRW spacetime + scalar];
Nikolić 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 & Kuchař 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, relational time];
Anderson CQG(07)gq/06.
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