Canonical Operator Transformations in Representation of Secondary Quantization

Felix Alexandrovich Berezin

Steven Kenneth Kauffmann

The Feynman-Garrod path integral representation for time evolution is extended to arbitrary one-parameter continuous canonical transformations. One thereupon obtains a generalized Kerner-Sutcliffe formula for the unique quantum representation of the transformation generator, which can be an arbitrary classical dynamical variable. This closed-form quantization procedure is shown to be equivalent to a natural extension of the standard canonical quantization recipe -- an extension that resolves the operator-ordering ambiguity in favor of the Born-Jordan rule.

Donald Jack Kouri, Cameron L. Williams, Nikhil Pandya

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: ΔWΔPw ≥1/2 , h=1.

Stephen Bruce Sontz

Donald Jack Kouri, Cameron L. Williams, Pandyaq, Nikhil

We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. 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, p_W, 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. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.