Quantum Field Theory Formalism and Techniques  

General Features > s.a. perturbative approach / interpretations of quantum mechanics.
* Linearity: We can have kinematical linearity (the space of fields is linear), and dynamical non-linearity (field equations), e.g. in λφ4 scalar field theories; For non-Abelian theories or gravity, on the other hand, there are already kinematical non-linearities; Traditionally, non-linear fields have been treated only perturbatively, although non-perturbative techniques are being developed, especially for gravity; > s.a. axiomatic approach.
* Quantum field tomography: The reconstruction of unknown quantum field states based on data on correlation functions.
@ General references: Cheng et al CP(10) [quantum mechanics as limiting case, spacetime resolution]; Dvali a1101 [classicalization vs weakly-coupled UV completion]; Padmanabhan EPJC(18)-a1712 [relationship with quantum mechanics].
@ Probabilistic techniques: Damgaard et al ed-90; Garbaczewski et al PRE(95)qp; Man'ko et al PLB(98)ht [probability representation]; Dickinson et al JPCS(17)-a1702 [working directly with probabilities].
@ Euclidean field theory: Guerra mp/05; > s.a. Wick Rotation.
@ Covariant Schrödinger formalism: Freese et al NPB(85); Kyprianidis PRP(87).
@ Light front formulation: Allen PhD(99)ht; Ullrich JMP(04); Polyzou a2102 [relation with instant formulation].
@ Worldline formalism: Bonezzi et al JPA(12)-a1204, Franchino-Viñas a1510-PhD [for non-commutative theories]; Bastianelli & Bonezzi a1504-proc [graviton self-energy]; Bonora et al a1802 [for a massive fermion model]; > s.a. approaches to quantum gravity [correlated worldlines].
@ Quantum field tomography: Steffens et al NJP(14)-a1406, nComm(15)-a1406; Berra-Montiel & Cartas a2006 [and deformation quantization].
@ Other formulations: Aldaya et al JPA(88) [group manifold approach]; Shajesh & Milton ht/05 [Fradkin's representation]; Nikolić EPL(09)-a0705 [in terms of integral curves of particle currents]; Villalba-Chávez et al JPG(10)-a0807 [Hamiltonian vs Lagrangian formalisms]; Anselmi EPJC(13)-a1303 [general field-covariant approach, and renormalization as a changes of variables]; Dickinson et al JPCS(15)-a1503 [with negative-frequency modes].
> Related concepts: see angular momentum; anomalies; arrow of time; causality in quantum field theory; quantum chaos; topology.

Non-Perturbative Approach > s.a. locality in quantum field theory; QED; renormalization; symmetry breaking.
@ General references: Rajaraman PRP(75); Gervais & Neveu PRP(76); Brézin & Gervais PRP(79); Fröhlich 92; Ferrara ht/96-conf; Borne et al 01 [and structure of matter]; Frishman & Sonnenschein 10, summary a1004; Dzhunushaliev a1003 [quantum corrections]; Bakulev & Shirkov a1102-conf; Dunne & Ünsal JHEP(12) [resurgence theory, the trans-series framework, and Borel-Écalle resummation]; Strocchi 13; Fried 14 [functional approach]; Trachenko & Brazhkin AP(14) [insights from liquid theory]; Shojaei-Fard a1811 [phenomenology, mathematical perspective].
@ Non-local aspects: Visser PLA(03)ht [covariant wavelets]; Paugam JGP(11)-a1010 [observables]; > s.a. instanton; monopole; soliton; Sphaleron.
@ In curved spacetime: Aastrup & Grimstrup a1712 [and non-commutative geometry].
@ Related topics: Bender et al JMP(90) [δ expansion]; Turner PhD(96)ht/01; Ksenzov PLB(97) [vacuum]; Dzhunushaliev & Singleton IJTP(99)ht/98; Salwen & Lee PRD(00)ht/99 [2-dimensional φ4]; Kizilersu et al PLB(01)ht/00; Hogervorst et al PRD(15)-a1409 [Truncated Conformal Space Approach]; Clavier PhD-a1511 [and Hopf algebra of renormalization]; Bellon FrPh(16)-a1701 [Borel transforms and alien calculus]; > s.a. other approaches [loop quantization].

Changing Variables / Field Redefinitions > s.a. Coleman-Mandula Theorem; CPT; path-integral quantization.
* Idea: Leads to the same physics (equivalence theorem, Chisholm theorem) if the origin in field space is not changed, otherwise masses can change; An appropriate Lee-Yang term must be introduced in the lagrangian.
* Chisholm theorem: Given the S-matrix elements for a field φ, the interpolating field is not unique; A point transformation \(\phi \mapsto \phi\,F(\phi)\), with F(0) = 1, does not change the physics.
@ General references: Lee & Yang PR(62); Salam & Strathdee PRD(70); Honerkamp & Meetz PRD(71); Gerstein et al PRD(71).
@ Chisholm theorem: Chisholm NP(61); Kamefuchi et al NP(61); Coleman et al PR(69); Lam PRD(73); Kallosh & Tyutin SJNP(73); Bergere & Lam PRD(76); Bando et al PRP(88); Tyutin PAN(02)ht/00; in Donoghue et al 14.

Other Techniques and Concepts > s.a. approaches [enhanced quantization, general-boundary formulation]; canonical and stochastic quantum mechanics.
@ General references: Gitman & Tyutin CQG(90) [from first quantization]; deLyra et al PRD(91) [lattice, differentiability]; Lam JMP(98)ht, ht/98-conf [integrals of time-ordered products]; Neumaier gq/03; Jaffe & Jäkel CMP(06) [exchange identity for non-linear bosonic fields]; Sibold & Solard PRD(09) [conjugate variables]; Hiroshima et al a1203-ln [enhanced binding]; Dybalski & Gérard CMP(14)-a1308 [criterion for asymptotic completeness]; Vatsya a1404 [geometrical]; Dunne & Unsal a1501-conf [resurgent trans-series and Picard-Lefschetz theory]; Várilly & Gracia-Bondía NPB(16)-a1605 [refined notion of divergent amplitudes].
@ Frameworks: Piazza & Costa PoS-a0711 [regions as subsystems]; Stoyanovsky Dokl-a0810 [definition of dynamical evolution].
@ Techniques: Oehme ht/00-en [reduction of parameters]; Drummond JPA(17)-a1611 [coherent functional expansions]; Sakhnovich a1710 [approach to divergences]; Polyzou et al FBS(18)-a1712-conf [multiscale methods]; Gough & Kupsch 18 [combinatorial approach]; Blümlein a2103 [intro to analytic calculation methods]; Jackson et al a1805 [algebraic and combinatorial techniques].
@ Mathematical concepts: Carey PLB(87) [cocycles]; Brown & Schnetz a1304 [modular forms]; Lanéry a1604 [with projective limits of state spaces].
> Techniques: see fractional calculus; cellular automata; clifford algebra; Coarse-Graining; cohomology theories; Colombeau Algebra; computational physics; distributions [products]; effective quantum field theory; Elliptic Genera; field theory [current algebra]; green functions; Hopf Algebra; K-Theory; Motives; path integrals; quantum field theory [including beable-based pilot-wave]; regularization; states [semiclassical quantization]; Wavelets.
> Related concepts: see boundaries; bundle [gerbe]; complex structure; Determinant; Dirac Sea; lattice field theory; logic; Machine Learning; measurement; N-Point Functions; non-commutative field theory; quantum information; representations [and pictures]; resonance; Schwinger-Dyson Equation; Schemes; states [including non-equilibrium]; symplectic structures; types of fields [including polymer representation]; types of theories; Unitarity.

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