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 d
on
the space 
'(Rd)
of real-valued distributions f over
Rd,
satisfying:
(1) Euclidean invariance; (2) Osterwalder-Schrader
Positivity; (3) Regularity.
* Smoothness: The test
functions to which we apply the operator-valued fields 
(i.e.,
the configuration space) cannot be just smooth ones, because then the
2-point functions would go to a constant as 
x →
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
and
a class of unbounded self-adjoint (field) operators {(f),
f 
C0infty(Rd)},
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 mp/06.
@ 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-in; > s.a. quantum
field theory.
@ Other pictures: Rosenfelder et al ht/98-in
[world-line representation]; Jackson et al a0810 [sums
over multiparticle paths].
Specific Flat Space Theories > s.a. electroweak
theory; lattice
gauge theory; quantum field theory; parametrized
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
& Presnajder ht/05-in
[4,
non-perturbative]; > 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-in,
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]; > s.a. dirac
quantum field theory.
@ 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].
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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send feedback and suggestions to bombelli at olemiss.edu – modified 8
aug 2009