Wigner's symmetry representation theorem and representations of the homogeneous Lorentz group

Question

In Weinberg's QFT textbook (volume 1), he proved the symmetry representation theorem in Appendix A, chapter 2, which states

Any symmetry transformation can be represented on the Hilbert space of physical states by an operator that is linear and unitary or antilinear and antiunitary.

But in section 5.4 (on page 218), he showed that the representation of the homogeneous Lorentz group is not unitary.

Do these two statements contradict with each other? Is the catch here the fact that the representation of the Lorentz group don't really act on Hilbert space?

Answer 1

The point is that the notion of symmetry is not the same in the 2 situations:

1. For Wigner's theorem, a symmetry transformation refers to a property of a transformation on a ray space.

2. But a symmetry could also refer to invariance of an object (typically an action functional) under a group action. In particular, if a group $G$ acts on a vector space $V$, the corresponding group representation need not be unitary.

TL;DR: The no-go theorem (that Weinberg implicitly alludes to) does not affect Wigner's theorem in any way.