In General
* 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.
* 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].
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].
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].
@ Hamiltonian formulation: 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]; Schmidt gq/95 [4th-order
theories]; 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
Ostrograski formulation]; Muslih & El-Zalan IJTP(07);
Andrzejewski et al a0710;
Morozov a0712 [brief
review]; Dunin-Barkowski & Sleptsov a0801 [reparametrization-invariant
theories].
@ Noether theorem, symmetries: Miron IJTP(95); de León & Martín
de Diego JMP(95); Sardanashvily mp/03.
@ Geometrical: de León & Lacomba JPA(89)
[ito 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 a0708 [from
phase space path integrals]; Andrzejewski et al PRA(07).
@ In quantum field theory: Pais & Uhlenbeck PR(50).
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008