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 JMP(10)ht/06 [space and time]; Sardanashvily a0707 [jet formalism]; Schreiber CMP(09)-a0806 [relationship between approaches]; Unterberger a1102 [user's guide]; Summers a1203-en [overview].
@ In curved spacetime: Hollands & Wald CMP(10)-a0803; Khavkine & Moretti ch(15)-a1412 [and quasifree Hadamard states]; Doplicher et al CMP-a1906 [on quantum spacetime].
@ Interacting theories: Johnson a1502; Alazzawi a1503-PhD.
@ Other types of theories: Thiemann BCP(96)ht/95 [gauge theories]; Pickrell CMP(08)-mp/07 [P(φ2) quantum field theory and Segals' axioms].
@ Non-commutative generalizations: Mnatsakanova & Vernov mp/06-conf; Grosse & Lechner JHEP(08); Ohl & Schenkel GRG(10)-a0912 [on non-commutative curved spacetime].

Algebraic Approach > s.a. quantum field theory / C*-algebra; dirac quantization; formulations of quantum theory; 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; One then chooses a state (satisfying the Hadamard condition) and uses the GNS construction to recover the full theory.
* Motivation: It can address the issue of locality, which in quantum field theory in curved spacetime appears both in the localization of observables in bounded regions (this can go wrong very quickly even in QED with long-range interactions) and in causality (which is expected to fail in quantum gravity, with observables localized in all of spacetime).
@ 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-proc [Hopf algebras and Hochschild cohomology, rev]; Park a1101; Wallace SHPMP(11) [critique]; Fraser SHPMP(11) [defense]; Ojima et al a1501 [local states]; Okamura & Ozawa JMP(16)-a1501 [measurement theory]; Lechner a1503 [rev, constructions and theories]; Fredenhagen & Rejzner a1503 [perturbative construction of models]; Benini & Schenkel a1903-proc [higher structures].
@ Intros: Emch 72; Horuzhy 90; Baez et al 92; Haag 92; Schroer AP(97)ht/96 [motivation], ht/97; Buchholz mp/00-conf; Brunetti & Fredenhagen mp/04-en; Halvorson & Mueger mp/06-ch; Moretti 13; Fewster & Rejzner a1904-ln [pedagogical].
@ Gauge theories: Grundling & Hurst CMP(85); Morsella & Tomassini RVMP(10)-a0811 [local generators of global gauge symmetries]; Buchholz et al LMP(16)-a1506 [universal C*-algebra of the electromagnetic field].
@ Other theories: Rehren & Schroer NPB(89) [2D conformal field theory]; Lechner CMP(08)mp/06 [factorizable S-matrix]; Buchholz & Fredenhagen a1902 [interacting]; Benini et al a2010 [smooth, 1D]; Brunetti et al a2103 [interacting, inclusion of Fermi fields]; > s.a. quantum gravity.
@ 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)gq/99 [without metric]; Brunetti et al CMP(03)mp/01 [general covariance]; Ruzzi CMP(05)mp/04 [Haag duality]; Sanders PhD(08)-a0809; Brunetti & Fredenhagen a0901-proc [in terms of non-linear functionals]; Dappiaggi et al CMP(11)-a1001 [causal structures and covariance]; Ciolli et al ATMP(12)-a1109 [net of causal loops]; Khavkine & Moretti ch(15)-a1412 [and quasifree Hadamard states]; Bahns et al a1501-in [and quantum spacetime, rev]; Fewster & Verch a1504 [rev]; Hack a1506 [cosmological applications]; Benini et al AHP(18)-a1712 [on spacetimes with timelike boundary]; Benini et al CMP(19)-a1903 [and factorization algebras]; > s.a. quantum fields in curved backgrounds [anti-de Sitter].
@ Sectors: Buchholz & Summers CMP(04)mp/03 [vacuum states]; Buchholz a1301-proc [physical state space, charge classes, and infrared problems]; Sardanashvily et al a1508 [inequivalent vacuum states]; > s.a. superselection sectors; theta sectors.
@ Short distance structure: Buchholz ht/97-proc; Schlingemann ht/99.
@ Perturbative: Dütsch & Fredenhagen CMP(01), ht/01-proc; Brunetti et al ATMP-a0901 [and renormalization groups]; Fredenhagen & Rejzner ch(15)-a1208-ln [intro]; Fredenhagen & Lindner CMP(14)-a1306 [general construction of KMS states]; Fredenhagen & Rejzner CMP(13)-a1110 [Batalin-Vilkovisky formalism]; Rejzner a1603-proc [renormalization and periods].
@ Related topics: Florig & Summers JMP(97) [statistical independence]; Schlingemann ht/99 [on the Euclidean sphere]; Clifton & Halvorson SHPMP(01)qp/00 [entanglement]; Ridgeway mp/05 [on infinite lattices]; Brunetti & Moretti a1009 [Klein-Gordon theory in causal diamonds]; Hofer-Szabó & Vecsernyés FP(12) [non-validity of Reichenbach's Weak Common Cause Principle]; Benini et al a2003 [2-categorical]; Stergiou a2004 [causal processes]; > s.a. Covariance; GNS construction; Hopf Algebra.
> Related topics: see boundaries in field theory; casimir effect; entanglement in quantum field theory; locality; 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 dμ on a set of paths (fields) f : \(\mathbb R\)n → \(\mathbb R\) satisfies the O-S positivity if each L2 function Fin the algebra

A+:= {F: φ → \(\mathbb R\) | there exist N, FN: \(\mathbb R\)Nn → \(\mathbb R\) such that F(ω) = FN(ω(x1), ..., ω(xn)) xi0 > 0}

satisfies \(\int\)dμ(ω) (θF)* F ≥ 0, where θF(ω(x1), ..., ω(xn)) changes the signs of all the xi0s.
@ 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. Algebraic Quantum Theory; 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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