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 CMP(09)-a0806 [relationship between approaches].
@ In curved spacetime: Hollands & Wald CMP(10)-a0803
@ Specific types of theories: Thiemann BCP(96)ht/95 [gauge theories]; Pickrell CMP(08)-mp/07 [P(2) quantum field theory and Segals' axioms].
@ Non-commutative generalizations: Mnatsakanova & Vernov mp/06-in; Grosse & Lechner JHEP(08).

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]; Kreimer a0906 [Hopf algebras and Hochschild cohomology, rev].
@ 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.
@ Gauge theories: Grundling & Hurst CMP(85); Morsella & Tomassini a0811 [local generators of global gauge symmetries].
@ Other theories: Rehren & Schroer NPB(89) [2D conformal field theory]; Lechner CMP(08)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]; Sanders a0809-PhD; Brunetti & Fredenhagen a0901-in [in terms of non-linear functionals].
@ Short distance structure: Buchholz ht/97-in; Schlingemann ht/99.
@ Perturbative: Dütsch & Fredenhagen CMP(01), ht/01-in; Brunetti et al ATMP-a0901 [and renormalization groups].
@ Related topics: Florig & Summers JMP(97) [statistical independence]; Schlingemann ht/99 [on the Euclidean sphere]; Clifton & Halvorson SHPMP(01)qp/00 [entanglement]; Buchholz & Summers CMP(04)mp/03 [vacuum states]; Ridgeway mp/05 [on infinite lattices].
> Related topics: see C*-algebra, Covariance; GNS construction; Hopf Algebra; theta sectors; types of cohomology.

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)mar [survey]; Borcherds & Barnard mp/02-ln; Abdesselam m.CO/02.
@ Texts, axiomatic: Bogoliubov, Logunov & Todorov 75; Strocchi 93.

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