Quantum Field Theory Formalism and Techniques

Perturbative Approach > s.a. Feynman Diagram; renormalization; series.
* Idea: Methods that allow us to calculate amplitudes for processes as power series in the strength of the interaction; They are approximate methods that work well except for non-linear fields in strong-field situations.
* Types: One normally uses covariant perturbation theory, but light front and others are also possible; Causal perturbation theory is an approach in which a specific causality condition is imposed at every order of perturbation theory and divergent integrals are avoided from the outset.
* Loop expansions: Tree diagrams are normally associated with classical physics, while loop effects are considered quantum mechanical in nature; This is not always the case.
* Remark: Renormalizability does not imply superrenormalizability.
@ General references: Fischer IJMPA(97) [rev]; Kastening & Kleinert PLA(00)qp/99 [calculation of Feynman integrals]; Sterman IJMPA(01) [intro]; Schubert PRP(01) [string-inspired]; Dunne ht/02-in [and non-perturbative]; Holstein & Donoghue PRL(04) [tree diagrams vs loop effects]; Szabo ht/05-in [intro]; Hollands a0802 [consistency conditions framework]; Stora IJGMP(08), a0901 [renormalized]; Kreimer a0909-in [algebraic structure].
@ Divergences, infinities: Jackiw in(00); Hurst RPMP(06) [history]; Sharatchandra a0707 [proposal for divergence-free approach]; Weinberg a0903.
@ And algebraic quantum field theory: Dütsch & Fredenhagen CMP(01)ht/00; Bergbauer & Kreimer a0704-in.
@ Perturbatively non-renormalizable theories: Paban et al ZPC(87); Gegelia & Japaridze IJTP(00)ht/98 [new method].
@ Causal perturbation theory: Aste & Trautmann CJP(03)ht [UV finite results]; Grangé & Werner qp/06; Aste a0810-in; Aste et al a0906 [examples].
@ Schemes: Bender et al PRD(88), & Jones JMP(88), follow-up Brown PRD(88); Schoonderwoerd & Bakker PRD(98), PRD(98) [covariant and light front]; Meurice PRL(02) [improved method]; Holstein & Donoghue PRL(04)ht [loop vs expansions]; Weinstein NPPS(06)ht/05 [adaptive]; Frasca NPPS(09)-a0807 [strong-coupling expansion]; Hollands & Olbermann a0906 [in terms of vertex algebras].
> Related topics: see deformation quantization; fock space; instanton [including WKB]; S-Matrix; scattering; Time Slice Axiom.

Non-Perturbative Features > s.a. locality and localization; renormalization; symmetry breaking.
@ General references: Rajaraman PRP(75); Gervais & Neveu PRP(76); Brézin & Gervais PRP(79); Fröhlich 91; Ferrara ht/96-in; Borne et al 01 [and structure of matter]; Frishman & Sonnenschein 10.
@ Non-local configurations: Visser PLA(03)ht [covariant wavelets]; > s.a. instanton; monopole; soliton; Sphaleron.
@ Approaches: Bender et al JMP(90) [ expansion]; Turner PhD(96)ht/01; Ksenzov PLB(97) [vacuum]; Dzhunushaliev & Singleton IJTP(99)ht/98; Salwen & Lee PRD(00)ht/99 [1+1 ⁴]; Kizilersu et al PLB(01)ht/00; > s.a. other approaches [loop quantization].

Other Techniques and Concepts > s.a. approaches [formulations]; canonical quantum mechanics; path integrals; stochastic quantum mechanics.
@ Probabilistic: Damgaard et al ed-90; Garbaczewski et al PRE(95)qp; Man'ko et al PLB(98)ht [probability representation].
@ Euclidean field theory: Guerra mp/05; > s.a. Wick Rotation
@ Covariant Schrödinger formalism: Freese et al NPB(85); Kyprianidis PRP(87).
@ Hamiltonian light front: Allen ht/99-PhD; Ullrich JMP(04).
@ Related topics: Clarke IJTP(79), JPA(90) [group bundle theories]; Carey PLB(87) [cocycles]; Aldaya et al JPA(88) [group manifolds]; Gitman & Tyutin CQG(90) [from first quantization]; deLyra et al PRD(91) [lattice, differentiability]; Lam JMP(98)ht, ht/98-in [integrals of time-ordered products]; Oehme ht/00-in [reduction of parameters]; Neumaier gq/03; Shajesh & Milton ht/05 [Fradkin's representation]; Jaffe & Jäkel CMP(06) [exchange identity for non-linear bosonic fields]; Nikolic EPL(09)-a0705 [in terms of integral curves of particle currents]; Piazza & Costa a0711 [regions as subsystems]; Villalba-Chavez et al a0807 [Hamiltonian vs Lagrangian formalisms]; Stoyanovsky a0810 [definition of dynamical evolution].
> Techniques: see Coarse-Graining; cohomology theories; Colombeau Algebra; effective quantum field theory; field theory [current algebra]; green functions; Hopf Algebra; Motives; quantum feld theory [including beable-based pilot-wave]; regularization; renormalization; Wavelets.
> Related concepts: see boundaries; bundle [gerbe]; complex structure; Determinant; Dirac Sea; distributions [products]; Elliptic Genera; K-Theory; lattice field theory; N-Point Functions; representations [and pictures]; resonance; Schwinger-Dyson Equation; states; symplectic structures.

Features > s.a. angular momentum; anomalies; arrow of time; causality; interpretations of quantum mechanics; quantum chaos; topology.
* Linearity: We can have kinematical linearity (the space of fields is linear), and dynamical non-linearity (field equations), e.g. in ⁴ 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.
@ Particle production: Srinivasan & Padmanabhan PRD(99)gq/98, gq/99; Prodan JPA(99)mp; Nikolic ht/01; Haro IJTP(03) [charged Klein-Gordon in electric field]; > s.a. quantum field theory effects in curved spacetime.
@ Radiative corrections: Jackiw IJMPB(00)ht/99.
@ Infrared limit: Mansfield ht/96-in [large distance expansion]; Kjaergaard & Mansfield JHEP(00)ht/99.

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