Green Functions in Quantum Field Theory| 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 · (x−x')} 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:
(∇2 − m2) \(\int_{\mathbb R}\) 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
(\(\square\)x−m2)
GE(x, x')
= −δn(x−x')
(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(t−t') G ; Gadv:= θ(t'−t) G .
* Properties: For the scalar
field case they satisfy (\(\square\)x
+ m2)
Gret/adv(x, x')
= δn(x−x').
* 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].
main page
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