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/dtN 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/dtN 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 ττ' = 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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