In Schwartz's book, 'Quantum Field Theory and the Standard Model' P.696, he starts to derive an expression for a parton distribution function in terms of matrix elements evaluated on the lightcone. Most of the derivation is clear to me, except a couple of things at the start and midway. The derivation begins by saying that the probability for finding a quark within a proton with a given momentum fraction (=PDF) is given by $$f(\zeta) = \sum_X \int \text{d}\Pi_X | \langle X|\psi|P \rangle |^2 \delta( \zeta n \cdot P - n \cdot p),$$ with ##P = p + p_X##. Why is this the correct mathematical representation for a PDF? I understand the delta function constrains the parton to take a fraction ##\zeta## of the proton's momentum in the ##n## direction and that probabilities are squared matrix elements but what does the sum over X mean here (we have no scattering taking place) and what is the meaning of ##\langle X | \psi | P \rangle##?(adsbygoogle = window.adsbygoogle || []).push({});

The other thing was he states that since the proton moves mostly in the ##\bar n = (1,0,0,-1) ## direction, ##\not \bar n \psi \approx 0##. Why does this approximation hold?

Thanks!

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# I Towards a matrix element definition of PDF

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