Parametrized
Theories |

**In General** > s.a. statistical
mechanical systems; symplectic
structures; time.

* __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].

@ __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}* m*_{i}
(d*x*_{i}/d*t*)^{2}]^{1/2}
d*τ* .

@ __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* = ∫ (*p*_{a}
d*x*^{a}/d*t* – *N*
*H*) d*τ* ,

with constraint *H* = *p*^{2}/2*m
*+ *V*(*x*) and *N* = Lagrange multiplier (= d*t*/d*τ*);
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* = ∫ (*p*_{a}
d*x*^{a}/d*t* – *N*
*H*) d*τ* , with
constraint *H* = (*p*^{2}+*m*^{2})/2*m*
,

which gives a hyperbolic superhamiltonian; We fix *x*(*τ*_{1})
= *x*_{1}, *x*(*τ*_{2})
= *x*_{2}, 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*), d*f*/d*τ* > 0, with *f*(*τ*_{1})
= *τ*_{1}, *f*(*τ*_{2})
= *τ*_{2} is

*L* = –*m* [*g*_{ab}
(d*x*^{a}/d*t*)(d*x*^{b}/d*t*)]^{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].

@ __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 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 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 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].

@ \(\langle\)*H*\(\rangle\) = 0 __approach__: Wang CQG(03)gq
[FLRW spacetime + scalar]; Nikolić gq/03.

@ __In quantum cosmology__: Blyth & Isham PRD(75)
[use
*H*^{1/2}].

@ __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]; Anderson CQG(07)gq/06.

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send feedback and suggestions to bombelli at olemiss.edu – modified 3 apr
2016