Physics – Mathematical Physics
Scientific paper
2003-02-21
J.Math.Phys. 45 (2004) 932-946
Physics
Mathematical Physics
Revised and expanded version, includes an application to symplectic transformations and groups, accepted for publication in J.
Scientific paper
We consider pseudo-unitary quantum systems and discuss various properties of pseudo-unitary 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 $i=\sqrt{-1}$ 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 $2\times 2$ pseudo-unitary matrices and discuss an example of a quantum system with a $2\times 2$ 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.
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