Green Functions in Quantum Field Theory  

In General > Usually, "Green function", with no further specification, means feynman propagator.
* Idea: The 2-point function giving the probability amplitude that, given that a particle is created at x, it will be observed at x'; In flat spacetime the choice of a particular Green's function depends on the choice of an integration contour in momentum space.
* Remark: The various Green functions can be expressed as expectation values of products of field operators in various states; The most common ones refer to the vacuum state (vacuum expectation values), but ensemble averages with thermal states at temperature β−1 can be used (thermal Green functions),

G(x, x'):= \(\langle\)0| ... |0\(\rangle\) ,   Gβ(x, x'):= tr(...) ρβ ,

where ρβ is the canonical distribution (the v.e.v. corresponds to β = 0).
* For a scalar field: The

G(x, x') = −i (2π)n G(k) exp{i k · (xx')} dnk ,   G(k) = (k · k + m2)−1

(the integrand has poles at k0 = ± (k2 + m2)1/2) with various choices of contours, i.e., of how to add iεs to the denominator, depending on boundary conditions.
* For a massive Klein-Gordon : With a static source,

\(\int_{\mathbb R}\) Δret(x, t) dt = exp{−|x|m}/(4π |x|) ,

a Yukawa-type Green function for −∇2 + m2:

(∇2m2) \(\int_{\mathbb R}\) Δret(x, t) dt = −δ3(x) .

Euclidean Green Function
* Idea: It is defined by substituting τ = −it in the Lorentzian Green function, i.e., rotating the contour (> see Wick Rotation); This can be done only for the Feynman propagator, and one finds

GE(iτ, x; iτ', x') = i GF(t, x; t', x') .

* Properties: It satisfies (\(\square\)xm2) GE(x, x') = −δn(xx') (notice that \(square\) is elliptic).
@ References: Candelas & Raine PRD(77) [Feynman propagator in curved spacetime]; Wald CMP(79).

Advanced and Retarded Green Function
$ Def: In terms of the Pauli-Jordan function,

Gret:= −θ(tt') G ;   Gadv:= θ(t't) G .

* Properties: For the scalar field case they satisfy (\(\square\)x + m2) Gret/adv(x, x') = δn(xx').
* One defines also their average: Gavg(x, x'):= (Gret + Gadv)/2.

Other Types > s.a. feynman propagator; Hadamard's Elementary Function; Pauli-Jordan, Wightman Function [including measurement].
* n-point Green functions: They are generated by the vacuum-to-vacuum transition amplitude Z[J]; > s.a. vacuum.
@ Wheeler Green function: Bollini & Rocca IJTP(98)ht; Bollini & Rocca a1012 [and relation to Feynman propagators]; Koksma & Westra a1012 [and causality].
@ Schwinger's function: Tsamis & Woodard CQG(01)hp/00.

References > s.a. covariant quantum gravity; green functions for differential equations; scalar field theory.
@ General references: Dyson PW(93)aug; Fabbri & Bueno a2011 [the most general propagator].
@ Non-perturbative methods: Rochev JPA(97)ht/96; Brouder in(09)-a0710 [equations for Green functions in general states].
@ Lattice theories: Glasser & Boersma JPA(00) [cubic]; Maassarani JPA(00)hl; Martinsson & Rodin PRS(02); Sakaji et al JMP(02).
@ Mechanical systems: Alhaidari mp/02 [Dirac-oscillator problem]; > s.a. quantum oscillator.
@ Yang-Mills gauge theories: Huber PhD(10)-a1005 [infrared behavior]; Frasca PoS-a1011 [from quartic scalar field theory]; Cornwall et al 11 [pinch technique].
@ Related topics: Kröger PLA(96) [fermions, fractal geometry]; Doniach & Sondheimer 98 [solid state applications]; Fried 02 [and ordered exponentials]; Grozin IJMPA(04) [methods, up to 3 loops]; Sardanashvily ht/06 [Euclidean field theory]; Ottewill & Wardell PRD(11)-a0906 [transport equation approach]; Bender a1003 [series expansions in powers of the spacetime dimension]; > s.a. Phonons; renormalization.

In Curved Spacetime > s.a. electromagnetic field; topology change.
* Remark: The definition of the different Green functions requires a careful discussion because momentum space is not available as in flat spacetime.
@ Higher-order Green functions: in Mankin et al PRD(01)gq/00.
@ Various fields: Krtouš gq/95 [scalar]; Antonsen & Bormann ht/96 [scalar, Dirac, Yang-Mills fields in various backgrounds]; Gabriel & Spindel JMP(97)ht/99 [massive spin-2, dS spacetime]; Kratzert AdP(00)mp [Dirac, globally hyperbolic spacetime]; Loran JHEP(18)-a1801 [massless, 2D, short-distance singularity]; Niardi IJGMP(21)-a2101 [Yang-Mills fields].
@ Quantum gravity corrections: Padmanabhan gq/97; Rinaldi PRD(08)-a0803 [from modified dispersion relations].

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