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arXiv:2204.10528 · Journal of Stochastic Analysis

Spectral Theorem Approach to the Characteristic Function of Quantum Observables

Andreas Boukas

Published Mar 10, 2022• 0 citations
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Abstract

Using Stone's formula for finding the spectral resolution of an, either bounded or unbounded, self-adjoint operator on a Hilbert space, we compute the vacuum characteristic function (quantum Fourier transform) of the anti-commutator XP + P X and of the Hamiltonian 1 2 (X 2 + P 2 ) quantum random variables .We also show how Stone's formula is applied to the computation of the vacuum characteristic function of finite dimensional quantum random variables.The method is proposed as an analytical alternative to the algebraic (or Heisenberg) approach relying on the associated Lie algebra commutation relations.

DOI
10.31390/josa.3.1.06

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