How is there a conflict between unitarity and Lorentz invariance?

Question

I've read a paragraph in Schwartz-QFT where he argues that unitarity and Lorentz invariance are incompatible due to the norms being different:

Why does he assume that the boost in this basis is $(\cosh\beta,\sinh\beta)$ though? Can't you have a normal $(\cos\beta,\sin\beta)$ in this Hilbert space? I mean for some other definition of an inner product (i.e. not $\langle\psi|\psi\rangle)$ it may be $(\cosh\beta,\sinh\beta)$ again, but it doesn't have to be the same for $\langle\psi|\psi\rangle$?

Answer 1

You are correct that Schwartz shows here only that the standard 4-vector representation of the Lorentz group cannot be made unitary.

The general statement is that a non-trivial unitary representation of the Lorentz group cannot be on any finite-dimensional Hilbert space, essentially because the Lorentz group is not compact - see this answer by Valter Moretti for a proof of the precise mathematical claim.