I was reading the book 'First Course in String Theory' by Zwiebach. In Chapter 11, the author discusses the quantisation of the relativistic particle in light-cone coordinates. The independent dynamical variables that are quantised are: x0−x^-_0 , xIx^I (the transverse coordinates), p+p^+ and pIp^I . One thing is immediately apparent: the transverse coordinates xIx^I and the light cone coordinate x−x^- are not treated on the same footing. While the transverse coordinates are quantised like usual QM, where xIx^I are promoted to time-independent Schrödinger picture operators with ∂xI∂τ=0\frac{\partial x^I}{\partial \tau} = 0 (in the Schrödinger picture); for the light-cone coordinate, the equation of motion is solved x−(τ)=x0−+p−m2τx^-(\tau) = x^-_0 + \frac{p^-}{m^2}\tau , and the initial position x0−x_0^- of the particle is quantised instead of the coordinate x−x^- being quantised. Moreover, in the Schrödinger picture, the author claims ∂x−∂τ=p−m2\frac{\partial x^-}{\partial \tau} = \frac{p^-}{m^2} . Why this difference between x−x^- and xIx^I , i.e., why is x−x^- explicitly dependent on τ\tau while xIx^I aren't? The quantisation of the initial position x0−x^-_0 after solving the equation of motion, instead of the coordinate x−x^- seems very odd, something which is not done in usual non-relativistic QM. Can someone please explain this (apparent) discrepancy?