Symmetry Transformations And Projective Representations


On this page, I'm going to try and sketch some arguments about symmetries in a Hilbert space. The objective is to introduce and motivate the subtleties of unitary and antiunitary representations in a Hilbert space, so as to bring continuous and discrete symmetries on an equal footing. This is of course not an original sketch, and you should by all means refer to books such as Weinberg and Sakurai.

Interestingly, conventional quantum mechanics courses harp at length about unitary operators, such as the time translation, space translation and rotation operators. However, operations such as parity and time reversal are often overlooked since standard textbooks do not adequately address the intricacies of these operations. This is partly my motivation for sketching this argument here, as it should hopefully provide a smooth transition for students entering quantum field theory (and particle physics for that matter) from conventional quantum mechanics (CQM).

Rays in Hilbert space

We define a system of rays in Hilbert space as the equivalence class $\{ \lambda |\psi\rangle\}$ of kets satisfying the following properties:

\begin{align} \langle \Phi|\Psi\rangle = \langle \Psi|\Phi\rangle^{*} \end{align}


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