Position Space Interpretation for Generalized Parton Distributions

density interpretation of q(x,b ⊥ )
with usual (canonical) equal light-cone time x + anti-commutation
relations, e.g.
{
br (k + ,k ⊥ ), b † s(q + ,q ⊥ ) } = δ(k + − q + )δ(k ⊥ − q ⊥ )δ rs
and the normalization of the spinors is such that
ū (+) (p, r)γ + u (+) (p, s) = 2p + δ rs .
Note: ū (+) (p ′ , r)γ + u (+) (p, s) = 2p + δ rs for p + = p ′+ , one finds for
x > 0
q(x,b ⊥ ) = N ′ ∑ s
∫ d 2 ∫
k ⊥ d 2 k ′ 〈 ∣
⊥ p + ,0 ⊥ b †
2π 2π
s(xp + ,k ′ ⊥)b s (xp + ,k ⊥ ) ∣ ∣p + 〉
,0 ⊥
×e ib ⊥·(k ⊥ −k ′ ⊥ ) .
Position Space Interpretation for Generalized Parton Distributions – p.35/55