A common assumption that quantisation is simply a representation of the Lie algebra of classical observables by a Lie algebra of self-adjoint operators in Hilbert space is shown to be generally invalid.
Abstract We investigate the structure of the Schrödinger algebra. Two constructions are given that yield the physical realization via general methods starting from the abstract Lie algebra. Representations are found on a Fock space with basis given by a canonical Appell system. Generalized coherent states are used in the construction of the Hilbert space of functions on which certain commuting elements act as self‐adjoint operators. This yields a probabilistic interpretation of these operators as random variables. An interesting feature is how the semidirect product structure of the Lie algebra is reflected in the probability density function. A Leibniz function and orthogonal basis for the Hilbert space are found. Then certain evolution equations connected with canonical Appell systems on this algebra are shown.
Using the spectral theorem we compute the Quantum Fourier Transform or Vacuum Characteristic Function Φ, e itH Φ of an observable H defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where Φ is a unit vector in a Hilbert space H.We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting or disentanglement formulas for the operator exponential.
Using the spectral theorem we compute the Quantum Fourier Transform or Vacuum Characteristic Function ⟨Φ, e itH Φ⟩ of an observable H defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where Φ is a unit vector in a Hilbert space H.We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting or disentanglement formulas for the operator exponential.