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 JHEP(18)-a1801 [massless, 2D, short-distance singularity].

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

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