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 > s.a. spin foam.
* Scalar field: In the general case of different in and out states, the Feynman propagator is

i G_F(x₁–x₂):= 0_in|T(*(x₁)(x₂))|0_out / 0_in|0_out ,

and with the right boundary conditions satisfies (_x + m² + R) G_F(x, x') = –|g|^{–1/2 n}(x–x') (for , > see klein-gordon fields); In terms of other Green functions,

G_F = –i (t–t') G⁺ – i (t'–t) G^– = –G* – G⁽¹⁾ ;

For a thermal state (m = 0),

G_F^th(k) = exp()/[exp()–1] (k · k + i)^–1 + 1/[exp()–1] (k · k – i)^–1 ,

where = k⁰, and the second term is acausal, in the sense that it propagates backwards in time.
* Spinor field: It satisfies (i ^a_a – m) S_F(x, x') = ⁿ(x–x'), and is given by

S_F(x, x'):= –i 0| T((x)*(x')) |0 = (i^a_a + m) G_F(x, x') .

* Maxwell field: It is given by

D_Fab(x, x'):= –i 0| T(A_a(x)A_b(x')) |0   (gauge dependent) = –_ab D_F(x, x')   (in the Feynman gauge) ,

and satisfies [_{ac x} – (1–^–1) _ac] D_F^cb(x, x') = _a^bn(x–x').
@ Simple harmonic oscillator: Holstein AJP(98); Thornber & Taylor AJP(98); Barone et al AJP(03) [methods]; Moriconi AJP(04).

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