By Lorentz transformation I mean an element of the Poincaré group.
In Steven Weinberg's Quantum Theory of Fields vol 1., the Poincaré group is constructed in terms of its action on coordinates. I understand that in the context of acting on coordinates the set of Lorentz transformations form a group.
However, I do not understand why we can then just talk about the Poincaré group as an abstract group which has a physically significant action in other contexts. Emphasis on the qualifying clause of the aforementioned sentence. In particular, using Wigner's theorem to furnish a unitary representation of the Poincaré group over Hilbert space.
The only way that I can currently conceive of remedying this situation is that I misunderstand the definition of a Lorentz transformation. My current understanding is that a Lorentz transformation is defined to be an isometry of Minkowski space (or some equivalent definition). I proceed to give a different understanding of what a Lorentz transformation is w.r.t. to the understanding I currently hold.
Definitionally, a Lorentz transformation preserves predicted experimental results. This means two things.
- The speed of light in all inertial reference frames must be $c$ $\implies$ Lorentz transformations are isometries of Minkowski space.
- Born's rule, which (I think) essentially gives rise to all experimental predictions of Quantum Mechanics, must be preserved by Lorentz transformations $\implies$ Lorentz transformations are Wigner symmetries (physical state ray symmetries) $\implies$ by Wigner's theorem Lorentz transformations have a unitary representation on a Hilbert space of physical states.
Thus, a Lorentz transformation is both an isometry of Minkownski space AND a Wigner symmetry and potentially more. This understanding, if accurate, would solve my confusion because it unifies the two actions of a Lorentz transformation with a single definition of the Lorentz transform. But, in this case, it would be erroneous to merely call Lorentz transformations coordinate transformations.
In other words, it would be a necessary but not sufficient condition for a Lorentz transformation to be an isometry of Minkowski space. The truly necessary and sufficient condition would be that a Lorentz transformation is an isometry of Minkowski space and a Wigner symmetry (on projective Hilbert space).