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 ∈ C0(\(\mathbb R\)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 06.
@ 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].

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].
@ Related topics: Fleischhack & Lewandowski mp/01 [limits of validity].
> Gravity-related: covariant quantum gravity; path-integral quantum gravity; quantum cosmology.

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