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 GF(p)); For m = 0, it is also denoted DF.

Specific Types of Theories
* Scalar field: In the general case of different in and out states, the Feynman propagator is

i GF(x1x2):= \(\langle\)0in | T(φ*(x1) φ(x2)) | 0out\(\rangle\) / \(\langle\)0in|0out\(\rangle\) ,

and with the right boundary conditions satisfies (\(\square\)x + m2 + ξR) GF(x, x') = –|g|–1/2 δn(xx') (for ξ, > see klein-gordon fields); In terms of other Green functions,

GF = –i θ(tt') G+ – i θ(t't) G = –G* – \(1\over2\)G(1) ;

For a thermal state (m = 0),

GFth(k) = exp(βω)/[exp(βω)–1] (k · k + iε)–1 + 1/[exp(βω)–1] (k · k – iε)–1 ,

where ω = k0, and the second term is acausal, in the sense that it propagates backwards in time.
* Spinor field: It satisfies (i γaam) SF(x, x') = δn(xx'), and is given by

SF(x, x'):= –i \(\langle\)0| T(ψ(x)ψ*(x')) |0\(\rangle\) = (i γaa + m) GF(x, x') .

* Maxwell field: It is given by

DFab(x, x'):= –i \(\langle\)0| T(Aa(x)Ab(x')) |0\(\rangle\)   (gauge dependent) = –ηab DF(x, x')   (in the Feynman gauge) ,

and satisfies [ηac \(\square\)x – (1–ζ–1) ∂ac] DFcb(x, x') = δab δn(xx').
@ 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 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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