Feynman Propagator |

**In General**

* __Idea__: A Green function for a
quantum system, obtained as a combination of the advanced and retarded Green
functions, such that the vacuum one propagates positive frequencies into the
future, negative ones into the past (see the form of \(G_{\rm F}(p)\)); For
*m* = 0, it is also denoted \(D_{\rm F}\).

**Specific Types of Theories**

* __Scalar field__: In the
general case of different in and out states, the Feynman propagator is

i *G*_{F}(*x*_{1}
− *x*_{2}):=
\(\langle\)0_{in}
| *T*(*φ**(*x*_{1})
*φ*(*x*_{2}))
| 0_{out}\(\rangle\)
/ \(\langle\)0_{in}
| 0_{out}\(\rangle\) ,

and with the right boundary conditions satisfies
(\(\square\)_{x}
+ *m*^{2} + *ξR*)
*G*_{F}(*x*, *x'*)
= −|*g*|^{−1/2}
δ^{n}(*x*−*x'*)
(for *ξ*, > see klein-gordon fields);
In terms of other Green functions,

*G*_{F}
= −i *θ*(*t*−*t'*)
*G*^{+} − i
*θ*(*t'*−*t*)
*G*^{−} = −*G**
− \(1\over2\)*G*^{(1)} ;

For a thermal state (*m* = 0),

*G*_{F}^{th}(*k*)
= exp(*βω*)/[exp(*βω*)−1] (*k* ·
*k* + i*ε*)^{−1}
+ 1/[exp(*βω*)−1] (*k* · *k* −
i*ε*)^{−1} ,

where *ω* = *k*^{0},
and the second term is acausal, in the sense that it propagates backwards in time.

* __Spinor field__: It satisfies
(i *γ*^{a}
∂_{a} − *m*)
*S*_{F}(*x*, *x'*)
= δ^{n}(*x* −
*x'*), and is given by

*S*_{F}(*x*, *x'*):= −i
\(\langle\)0| *T*(*ψ*(*x*)*ψ**(*x'*))
|0\(\rangle\) = (i *γ*^{a}
∂_{a} + *m*)
*G*_{F}(*x*, *x'*) .

* __Maxwell field__: It is given by

*D*_{Fab}(*x*, *x'*):=
−i \(\langle\)0|
*T*(*A*_{a}(*x*)
*A*_{b}(*x'*))
|0\(\rangle\) (gauge dependent)
= −*η*_{ab}
*D*_{F}(*x*,
*x'*) (in the Feynman gauge) ,

and satisfies [*η*_{ac}
\(\square\)_{x} −
(1−*ζ*^{−1})
∂_{a}∂_{c}]
*D*_{F}^{cb}(*x*, *x'*)
= δ_{a}^{b}
δ^{n}(*x*−*x'*).

@ __Simple harmonic oscillator__:
Holstein AJP(98)jul;
Thornber & Taylor AJP(98)nov;
Barone et al AJP(03)may [methods];
Moriconi AJP(04)sep.

@ __Scalar fields__: Dereziński & Siemssen RVMP(18)-a1608 [Klein-Gordon, coupled to Maxwell field, in static spacetime];
Padmanabhan a2104
[world-line path integral, expressed as an ordinary integral].

@ __And quantum gravity__: Johnston PRL(09)-a0909 [on a causal set];
Zhang & Yuan a1911 [Planck-scale corrections];
Curiel et al Symm(20)-a1910 [corrections from sum over all dimensions];
> s.a. particle phenomenology in quantum gravity;
spin-foam models.

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