The Frame-Independent Spatial Coordinate $\tilde{z}$: Implications for Light-Front Wave Functions, Deep Inelastic Scattering, Light-Front Holography, and Lattice QCD Calculations

A general procedure for obtaining frame-independent, three-dimensional light-front coordinate-space wave functions is introduced. The third spatial coordinate, $\tilde{z}$ , is the conjugate to the light-front momentum coordinate $x={k^+\over P^+}$ which appears in parton distributions. These light-front wave functions are used to derive a general expression for the quark distribution function of hadrons as an integral over the spatial separation $s= \tilde{z} -\tilde{z'}$, the frame-independent longitudinal distance (the Ioffe time) between virtual-photon absorption and emission in the forward virtual photon-hadron Compton scattering amplitude. The integrand, $g(s,x)$ of the quark distribution contains a factor of $\cos{s x}$ which remains significant for very large values of $s$ at small $x$, thus demonstrating that the spatial extent of a proton in the longitudinal direction can be very large -- a key feature of the Ioffe time. Specific examples using models derived from light-front holographic QCD exhibit a large extent in $\tilde{z}$.