By D. Brown
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Additional info for Space-Time Development of Heavy-Ion Collisions [thesis]
12 + M 2 )G≷(1, 1′) = d3 x2 QMF (x1 , x2 , t1 )G≷(x2 , t1 , 1′ ) t1 + t0 t′1 + t0 1 2 ≷ ∂ D (1, 1′) = 4π 1 µν d2 (Q> (1, 2) − Q< (1, 2)) G≷(2, 1′) (27a) d2 Q≷(1, 2) (G> (2, 1′ ) − G< (2, 1′ )) ≷ d3 x2 ΠMF (x1 , x2 , t1 )Dµν (x2 , t1 , 1′ ) t1 + t0 t′1 + t0 d2 Π>µ ν (1, 2) − Π<µ ν (1, 2) Dν≷′ ν (2, 1′ ) ′ ′ (27b) ′ d2 Π≷µ ν (1, 2) (Dν>′ ν (2, 1′) − Dν<′ ν (2, 1′)) ≷ (x2 , t1 , 1′ ) d3 x2 ΣMF (x1 , x2 , t1 )Sαβ ≷ (1, 1′) = (i ∂ 1 − me )Sαβ t1 + t0 t′1 + t0 ≷ ′ < d2 Σ> αβ ′ (1, 2) − Σαβ ′ (1, 2) Sβ ′ β (2, 1 ) (27c) > ′ < ′ d2 Σ≷ αβ ′ (1, 2) Sβ ′ β (2, 1 ) − Sβ ′ β (2, 1 ) Here the > and < self-energies have the same relation to the contour self-energy that the > and < Green’s functions have to the contour Green’s functions.
5) We identify f (x, p) with the number density of particle (or antiparticles) per unit volume in phase-space per unit invariant mass squared at time x0 : f (x, p) = dn(x, p) d3 p dp2 d3 x In particular, for p0 > 0, iG< (x, p) is associated with the particle densities and iG> (x, p) is associated with the hole density. For p0 < 0, iG< (x, p) is the antihole density and iG> (x, p) is the anti-particle density. The photon and electron densities are deﬁned in the same way, however because of their more complicated spin structure, their Wigner functions carry indices.
We present the corresponding diagrams in Figures 6(a-c). In Equations (23a)-(23c), the non-interacting contour Green’s functions have a 0 superscript. The self-energies describe all of the branchings and recombinations possible for 41 the photons, electrons and scalars. The self-energies are: ↔µ Q(1, 1′ ) =i(eZ ∂ ) C d2 d3 G(1, 3) Γνγφφ (2, 3, 1′)Dµν (1, 3) ′ ′ + i(2iαem Z 2 g µν ) C µν d2 d3 d4 G(1, 2) Γγγφφ (2, 3, 4, 1′)Dµµ′ (1, 3) Dνν ′ (1, 4) + QMF (1)δ 4 (1, 1′ ) (24a) Πµν (1, 1′ ) = − i(−ie(γµ )αβ ) ↔ ′ ′ C β d2 d3 Sαα′ (1, 2) Γαγe,ν (2, 3, 1′)Sβ ′ β (3, 1) d2 d3 G(1, 2) Γγφφ,ν (2, 3, 1′)G(3, 1) + i(eZ ∂ µ ) C ′ + i(2iαem Z 2 gµµ′ ) C d2 d3 d4 G(1, 2) G(3, 1) Γνγγφφ,ν (2, 3, 4, 1′)Dµ′ ν ′ (1, 4) + ΠMF (1)gµν δ 4 (1, 1′) (24b) ′ Σαβ (1, 1′ ) =i(−ie(γ µ )αα′ ) C d2 d3 Sα′ β ′ (1, 2) Γβγeβ,ν (2, 3, 1′)Dµν (1, 3) (24c) + ΣMF (1)δαβ δ 4 (1, 1′ ) In Figures 7(a-c), we show the diagrams corresponding to the non-mean-ﬁeld terms in Equations (24a)-(24c).