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'*.

* __Remark__: The various Green
functions can also 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\(\rangle\) , *G*_{β}(*x*,
*x'*):= tr(...) *ρ*_{β} ,

where *ρ*_{β} is
the canonical distribution (the v.e.v. corresponds to *β* = 0).

* __For a scalar field__: The vacuum ones can all be expressed by the integral

*G*(*x*,* x'*)
= –i (2π)^{–n} ∫ *G*(*k*)
exp{i *k* · (*x*–*x'*)}
d^{n}*k* , *G*(*k*)
= (*k* · *k* + *m*^{2})^{–1}

(the integrand has poles at *k*^{0}
= ± (**k**^{2}+*m*^{2})^{1/2})
with various choices of contours, i.e., of how to add i*ε*s
to the denominator, depending on boundary conditions.

* __For a massive Klain-Gordon field__: With a static source,

\(\int_{\mathbb R}\) Δ^{ret}(**x**, *t*)
d*t* = exp{–|**x**|*m*}/(4π |**x**|)
,

a Yukawa-type Green function for –∇^{2}+*m*^{2}:

(∇^{2} – *m*^{2})
\(\int_{\mathbb R}\) Δ^{ret}(**x**, *t*)
d*t* =
–*δ*^{3}(**x**) .

**Euclidean Green Function**

* __Idea__: It is defined
by substituting *τ* = –i*t* 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

*G*_{E}(i*τ*, *x*;
i*τ**'*, *x'*)
= i *G*_{F}(*t*, *x*; *t'*, *x'*) .

* __Properties__: It satisfies
(\(\square\)_{x}–*m*^{2}) *G*_{E}(*x*, *x'*)
= –δ^{n}(*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,

*G*_{ret}:= –*θ*(*t*–*t'*)
*G* ; *G*_{adv}:=
*θ*(*t'*–*t*) *G* .

* __Properties__: For the
scalar field case they satisfy (\(\square\)_{x} + *m*^{2}) *G*_{ret/adv}(*x*,* x'*)
= *δ*^{n}(*x*–*x'*).

* __One defines also their
average__: *G*_{avg}(*x*, *x'*):=
(*G*_{ret} + *G*_{adv})/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

@

@

**References** > s.a. covariant quantum gravity;
green functions for differential equations; scalar field theory.

@ __Simple__: Dyson PW(93)aug.

@ __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.

@ __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 a1801 [massless, 2D, short-distance singularity].

@ __Quantum gravity corrections__: Padmanabhan gq/97;
Rinaldi PRD(08)-a0803 [from modified dispersion relations].

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