Why it is said $W$ should be linear partial time derivative so that wave function could be determined by initial wave function?
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$\begingroup$ It is said the wave equation must be linear in $W$ or $\partial/\partial t$. Not that $W$ should be "linear partial time derivative". $\endgroup$ – J-T Sep 14 '17 at 13:25
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If the equation is not linear (say quadratic) in $\partial_t$, then one needs to know more about the initial condition than just the initial wavefunction.
As an example, in Newtonian dynamics, one needs to know both the initial position and the initial velocity, whereas in non-relativistic quantum mechanics one only needs the initial wavefunction (and not the initial "rate of change" of the initial wavefunction).
This is this property of the quantum dynamical mechanics that Dirac is trying to preserve when he introduces his equation.
