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*_{F}(*p*)); For *m* = 0,
it is also denoted *D*_{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} + *ξ**R*)
*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].

@ __In discrete spacetimes__: Johnston PRL(09)-a0909 [on a causal set];
> s.a. spin-foam models.

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