Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics

We consider pseudo-unitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudo-unitary matrix is the exponential of i = √ −1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudo-unitary matrices. In particular, we present a thorough treatment of 2 × 2 pseudo-unitary matrices and discuss an example of a quantum system with a 2 × 2 pseudo-unitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudo-unitary group U(n, n), and elaborate on an approach to second quantization that makes use of the underlying pseudo-unitary dynamical groups.