According to the chapter II.1 in "Quantum Field Theory in a Nutshell" by A. Zee, Dirac was trying to write down the relativistic wave equation linear in spacetime derivative.
The author stated that
The equation is supposed to have the form "some linear combination of $\partial_\mu$ acting on some field $\psi$ is equal to some constant times the field." Denote the linear combination by $c^\mu \partial_\mu$. If the $c^\mu$'s are four ordinary number, then the four-vector $c^\mu$ defines some direction and the equation cannot be Lorentz invariant
What I do not understand is the last part of this statement. If $a_\mu$ and $b_\mu$ are four-vectors, it is obvious that $a_\mu b^\mu$ must be Lorentz invariant. Why can it not be true in this case since both $c_\mu$ and $\partial_\mu$ are also four-vectors?
