Path-Integral Approach to Quantum Field Theory| Path-Integral Approach to Quantum Field Theory |
In General, Flat Space
$ Def: A scalar quantum
field theory in d dimensions is a Borel measure dμ on
the space \({\cal D}'({\mathbb R}^d)\) of real-valued distributions f
over \(\mathbb R\)d, satisfying:
(1) Euclidean invariance; (2) Osterwalder-Schrader Positivity; (3) Regularity.
* Smoothness: The test
functions to which we apply the operator-valued fields φ (i.e.,
the configuration space) cannot be just smooth ones, because then the
2-point functions would go to a constant as Δx →
0, which they don't [except in (0+1)-dimensional quantum mechanics]; They are
instead very rough [& Streater]; > s.a. field theory.
* Results: It follows that
there exists a Hilbert space carrying a unitary representation of the
Poincaré group P, with a distinguished P-invariant
vector |0\(\rangle\) and a class of unbounded self-adjoint (field) operators
{φ(f), f ∈ C\(_0^\infty(\mathbb R^d\))},
which satisfy locality and transform correctly under P.
@ General references: Rzewuski 69;
Abers & Lee PRP(73);
Taylor 76; Nash 78;
Itzykson & Zuber 80;
Fradkin NPB(93);
Mosel 03;
LaChapelle IMTA-mp/06;
Vargas a1412 [measures on path spaces];
Blasone et al a1703 [and inequivalent representations].
@ Heuristic: Ramond 81;
Rivers 87;
Das 19.
@ Constructive: Glimm & Jaffe 87;
Rivasseau 91.
@ Field redefinitions: Apfeldorf et al MPLA(01)ht/00;
Latorre & Morris IJMPA(01)ht-proc;
> s.a. quantum field theory.
@ Other pictures:
Rosenfelder et al ht/98-proc [world-line representation];
Jackson et al a0810 [sums over multiparticle paths].
@ Related topics: Steinhaus JPCS(12) [discretization and reparametrization invariance];
Edwards et al a1812-proc [first quantization approach].
Specific Flat Space Theories > s.a. electroweak
theory; quantum field theory; parametrized
theories; types of quantum field theories.
@ Scalar fields: Klauder PRD(76) [augmented action, non-Gaussian measure];
Gosselin & Polonyi AP(98) [Klein-Gordon];
Hawking & Hertog PRD(02)ht/01 [4th-order, and ghosts];
Kaya PRD(04) [self-interacting];
Isham & Savvidou JMP(02) [foliation operator];
Bohacik & Prešnajder ht/05-conf
[φ4, non-perturbative];
Belokurov & Shavgulidze a1112 [continuous and discontinuous functions];
Kaya CQG(15)-a1212 [measure for in-in path integral];
> s.a. quantum klein-gordon fields.
@ Maxwell theory:
Bordag et al JPA(98) [in dielectrics];
Muslih NCB(00) [canonical form];
> s.a. gauge theories; QED.
@ Fermions / spinors:
Floreanini & Jackiw PRD(88);
Pugh PRD(88);
Nielsen & Rohrlich NPB(88);
Jacobson PLB(89);
Aliev et al NPB(94);
Bodmann et al qp/98-proc,
JMP(99)mp/98;
Polonyi PLB(99)ht/98,
ht/98 [Dirac equation];
Smirnov JPA(99);
Hiroshima & Lorinczi JFA-a0706 [spin-1/2 Pauli-Fierz model];
Briggs et al IJMPA(13)-a1109 [in cartesian and spherical coordinates];
> s.a. dirac quantum field theory; types of path integrals.
@ Supersymmetric theories: Rogers PLB(87);
O'Connor JPA(90),
JPA(90),
JPA(91);
Niemi & Pasanen PLB(91);
Ellicott et al AP(91) [gauge theory].
@ Related topics:
Bashkirov & Sardanashvily IJTP(04)ht [covariant Hamiltonian];
> s.a. lattice gauge theory.
Other Quantum Field Theories
@ For Riemannian geometries:
Carfora & Marzuoli PRL(89).
@ For topological field theories:
Cugliandolo et al PLB(90);
Kaul & Rajaraman PLB(90).
@ In curved spacetime: Jaffe & Ritter CMP(07)ht/06 [Euclidean];
Baldazzi et al a1901 [Lorentzian, without Wick rotation].
@ Related topics: Fleischhack & Lewandowski mp/01 [limits of validity].
> Gravity-related: covariant quantum gravity;
path-integral quantum gravity; quantum cosmology.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 22 dec 2019