How can Schrodinger equation be derived based on the argument of Lorentz transformation?

Question

I came across this note which talk about obtaining the Schrodinger equation based on the argument of Lorentz transformation. However, I am not able to follow how it exactly works. Any help would be appreciated.

Answer 1

The text you quote is just asking you to complete the picture here: If $H$ is the 0-th component of a four-vector where the momenta $P_i$ are the spatial part, and you know the $P_i$ are represented by the spatial derivatives $-\mathrm{i}\hbar \partial_i$, then what 0-th component makes the vector of spatial derivatives $\partial_i$ into a 4-vector and is hence a possible representation of $H$? Of course it's the time-derivative $\partial_0$ that completes the 4-vector of derivatives.

I find it a bit silly to paraphrase this as "an argument based on Lorentz transformations", but you can arrive at this by explicitly thinking about how the $\partial_i$ transform under Lorentz transformations: They mix with $\partial_0$ under boosts in exactly the same way as the $P_i$ mix with $H$.

However, note that the argument presented is questionable in its other steps: $\partial_t$ in quantum mechanics is not an operator of the same kind as $\partial_i$ - while the spatial derivatives act on the wavefunction $\psi(x)$ of a single state, the time derivative acts only on a trajectory of states $\psi(t)$ (regardless of representation). See this question and its answers for more discussion on this, and see e.g. this answer by Valter Moretti for a discussion of why relativistic "time" (and indeed, position) operators are problematic, i.e. the assumption of a "$(x,t)$-representation" in the quote is likewise questionable.