Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Abstract

Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W , and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x -representation and that all are Hermitian in the W -representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W -harmonic oscillators yielding standard orthonormal bases (as functions of W ) and associated coherent states and Wigner distributions. The MUB lead to W -generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,± i ) , as well as “W-linear chirps”. As expected, W , Pw satisfy the uncertainty product relation: Δ W Δ Pw ≥1/2 , h =1.