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(x₁ − x₂):= \(\langle\)0_in | T(φ*(x₁) φ(x₂)) | 0_out\(\rangle\) / \(\langle\)0_in | 0_out\(\rangle\) ,

and with the right boundary conditions satisfies (\(\square\)_x + m² + ξR) G_F(x, x') = −|g|^−1/2 δⁿ(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 + 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 γ^a ∂_a − m) S_F(x, x') = δⁿ(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 δⁿ(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.

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 24 apr 2021