Algebraic
and Axiomatic Quantum Field
Theory| Algebraic
and Axiomatic Quantum Field
Theory |
Axiomatic / Constructive Approach in General > s.a. Bochner
Theorem; Wightman Axioms.
* Approaches: There
are essentially two approaches to the axiomatization of quantum field theory,
the algebraic one, going back to Haag and Kastler (uses local nets
of operator algebras which assign an algebra of "observables" to
each patch),
and the functorial one, going back to Atiyah and Segal (more recently refined
to "extended" functorial
quantum field theory by Freed, Hopkins, Lurie and others, based on ideas
by Baez and Dolan; uses n-functors which assign to each patch a "propagator
of states").
* Applications: Has
given interesting results in 2D and 3D theories, in particular the rigorous
construction of the Gross-Neveu model in 3D, which is non-renormalizable.
@ General references: Streater RPP(75);
Albeverio et al RPMP(97)mp/05 [Lorentzian,
modified Wightman axioms]; Schroer & Wiesbrock RVMP(00)ht/98,
Schroer ht/99-in,
Kähler & Wiesbrock JMP(01)
[modular theory]; Buchholz LNP(00)
[status]; Rivasseau JMP(00);
Jaffe in(00); Puccini & Vucetich FP(04)
[spin-statistics, commutation relations, CPT], NCB(05);
Morgan qp/05 [weakened
linearity]; Schwarz ht/06 [space
and time]; Sardanashvily a0707 [jet
formalism]; Schreiber a0806 [relationship
between approaches].
@ In curved spacetime: Hollands & Wald a0803
@ Specific types of theories:
Thiemann BCP(96)ht/95 [gauge
theories]; Pickrell mp/07 [P(
2)
quantum field theory and Segals' axioms].
@ Generalized theories: Mnatsakanova & Vernov mp/06-in
[non-commutative
quantum field theory].
Algebraic Approach > s.a. [quantum
field theory]; boundaries
in field theory; dirac
quantization; observable algebras.
* Idea: The basic structure
is a "local net", an association to every
spacetime region of a C*-algebra of local observables;
States are generalized to positive linear functionals on the algebra of observables;
Causality is encoded
in the fact
that observables related to spacelike related regions commute;
Dynamics
gives the other commutation relations.
@ General references: Haag & Kastler JMP(64);
Segal in(67); Kastler ed-76; Isham in(77); Roberts CMP(87);
Buchholz JMP(90);
Thomas & Wichmann JMP(98)
[nets and spacetime geometry].
@ Intros: Emch 72; Horuzhy 90; Baez et al 92; Haag 92; Schroer AP(97)ht/96 [motivation], ht/97;
Buchholz mp/00-in;
Brunetti & Fredenhagen mp/04-in;
Halvorson & Mueger mp/06-in.
@ Theories: Grundling & Hurst CMP(85)
[gauge theories]; Rehren & Schroer NPB(89)
[2D conformal field theory]; Lechner mp/06 [factorizable
S-matrix].
@ In curved spacetime: Fellegara FdP(89);
Wollenberg JGP(89);
Yurtsever CQG(94)gq/93 [non-globally
hyperbolic]; Brunetti & Fredenhagen CMP(00)mp/99;
Rainer CQG(00)
[without metric]; Brunetti et al CMP(03)mp/01 [general
covariance]; Ruzzi CMP(05)mp/04 [Haag
duality].
@ Short distance structure: Buchholz ht/97-in;
Schlingemann ht/99.
@ Related topics: Florig & Summers JMP(97)
[statistical independence]; Schlingemann ht/99 [on
the Euclidean sphere]; Clifton & Halvorson SHPMP(01)qp/00 [entanglement];
Dütsch & Fredenhagen CMP(01), ht/01-in
[perturbative]; Buchholz & Summers CMP(04)mp/03 [vacuum
states]; Ridgeway mp/05 [on
infinite
lattices].
> Related topics: see C*-algebra, Covariance; GNS
construction; theta
sectors.
Osterwalder-Schrader Construction
* Idea: Allows to construct
a Hilbert space quantum field theory from a measure on the space of (Euclidean)
histories, and thus justifies the use of the Wick
rotation
to go back and forth between Euclidean and Lorentzian quantum field theory.
$ Positivity: A measure
d
on
a set of paths (fields) f : Rn →
R satisfies the O-S positivity if each L2 function F in
the algebra
A+:= {F: → R |
exists N, FN: RNn →
R such that F(
)
= FN((x1),
...,
(xn)) xi0 > 0}
satisfies 
d()
(
F)* F 
0,
where F((x1),
..., (xn))
changes the signs of all the xi0's.
@ Axioms: Osterwalder & Schrader CMP(73),
CMP(75) [with Wightman functions]; Glimm & Jaffe
87 [with measures; better; difficult].
@ Theorem: Schlingemann RVMP(99)ht/98 [C*-algebraic
version].
@ For diffeomorphism-invariant theories: Ashtekar et al CQG(00)qp/99.
References and Other Approaches > s.a. formalism
and techniques [including perturbative]; path
integrals.
@ General: Haag & Schroer JMP(62)
[survey of postulates].
@ Mathematical: Federbush BAMS(87) [survey]; Borcherds & Barnard
mp/02-ln;
Abdesselam m.CO/02.
@ Texts, axiomatic: Bogoliubov, Logunov & Todorov 75; Strocchi 93.
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008