Observations sur la mécanique quantique finie. (Observations on finite quantum mechanics). (French) Zbl 0606.22017

We study the canonical transformations of the quantum mechanics on a finite phase space. For simplicity we assume that the configuration variable takes an odd prime number \(4K\pm 1\) of distinct values. We show that the canonical group is unitarily implemented. It admits a maximal abelian group of order 4K, commuting with the finite Fourier transform F, a finite analogue of the harmonic oscillator group. This provides a natural construction of \(F^{1/K}\) and of an orthogonal basis of eigenstates of F.


22E70 Applications of Lie groups to the sciences; explicit representations
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis