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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