In General > s.a. classical mechanics [higher-order equations
of motion].
* Idea: A generalized form of mechanics, which uses higher-order
Lagrangians (in the
-order
case, the theory is non-local in time).
* Motivation: One
motivation was to see if non-local theories were free of divergence problems
in quantum field theory.
* Remark: One usually wants
first-order Lagrangians in order to get second-order equations of motion,
define conserved quantities in the usual
way, and
perform Legendre transformations; Even the general relativity Lagrangian
is of this kind,
though it may not be obvious from the Einstein-Hilbert expression, because
the second derivative terms are a pure divergence.
Non-Local Theories > s.a. non-commutative
physics; partial differential equations.
* Issue: Non-local actions generally
possess acausal equations of motion and non-real in-out operator amplitudes.
@ Of finite extent: Woodard PRA(00); Llosa PRA(03)ht/02; > s.a. hamiltonian
systems.
@ Related topics: Soussa ht/03-in
[causality]; Calcagni et al PRD(07)-a0705 [FRW
+ scalar cosmology]; > s.a. FRW spacetimes.
Other Specific Types of Theories > s.a. higher-order
gravity;
oscillators; spinning
particles.
@ Field theories: in Stelle GRG(78)
[gravity]; Grigore FdP(99)ht/96 [trivial
second-order Lagrangians]; de Urries & Julve JPA(98)
[scalar]; de Urries et al JPA(01)
[bosonic, as constrained second-order]; Villaseñor JPA(02)ht [fermionic];
Bazeia et
al JPA(03)
[2+1, dualities]; Nguyen a0807 [principle
of least action]; Li et al IJTP(08)
[gauge theories]; Campos et al a0906-in [unambiguous intrinsic formalism].
Hamiltonian Formulation
@ General references: Ostrogradskii MASP(1850)
[momenta]; Coelho de Souza & Rodrigues JPA(69);
Jaén et al PRD(87), JMP(89);
Llosa & Vives JMP(94);
Belvedere et al ZPC(95)
[for field theory, canonical transformations]; Rashid & Khalil IJMPA(96);
Woodard PRA(00)ht/00 [finite
non-locality]; Bering ht/00;
Cheng et al NPB(02)
[and non-commutative field theory]; Leclerc gq/06 [modified
Ostrogradski formulation]; Muslih & El-Zalan IJTP(07);
Andrzejewski et al a0710;
Morozov TMP(08)-a0712 [brief
review]; El-Zalan et al IJTP(08);
Vitagliano a0905.
@ Types of theories: Schmidt gq/95 [4th-order
theories]; Dunin-Barkowski & Steptsov TMP(09)-a0801 [reparametrization-invariant].
References > s.a. hamiltonian and lagrangian
formulation.
@ General: Whittaker 37; de León & Rodrigues 85; Negri & da
Silva PRD(86);
Jaén et al PRD(86);
Gràcia et al JMP(91);
Hojman et al JMP(92)
[Lagrangian from differential equations of any order].
@ Noether theorem, symmetries: Miron IJTP(95); de León & Martín
de Diego JMP(95); Sardanashvily mp/03.
@ Geometrical: de León & Lacomba JPA(89)
[in terms of sympletic higher-order tangent bundles].
@ Meaning / degrees of freedom: Chervyakov & Nesterenko PRD(93);
de Urries & Julve
gq/95.
@ Related topics: Nakamura & Hamamoto PTP(96)ht/95 [path
integrals]; Benito et al IJGMP(06)
[geometric integrators]; Nesterenko PRD(07)
[instability].
Quantization [> s.a. formulations
of quantum mechanics and quantum systems.]
@ General references: Hayes JMP(69);
Tesser JMP(72);
Acatrinei JPA(07)-a0708 [from
phase space path integrals]; Andrzejewski et al PRA(07).
@ In quantum field theory: Pais & Uhlenbeck PR(50);
> s.a. Pais-Uhlenbeck Model, path integrals, quantum
oscillators.
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send feedback and suggestions to bombelli at olemiss.edu – modified 7
aug 2008