Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2007.01527 · Communications on Stochastic Analysis

Spectral Theorem Approach to the Characteristic Function of Quantum Observables

Andreas Boukas, Philip Feinsilver

Published Jun 1, 2019• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.31390/cosa.13.2.03

Related papers

Spectral Theorem Approach to the Characteristic Function of Quantum Observables

Andreas Boukas, Philip Feinsilver

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.

Civil War Book Review•Jun 1, 2019•2 citations

Spectral Theorem Approach to the Characteristic Function of Quantum Observables

Andreas Boukas

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.

arXiv:2204.10528•Journal of Stochastic Analysis•Mar 10, 2022•0 citations

Fourier transform of random variables associated with the multi-dimensional Heisenberg Lie algebra

Luigi Accardi, Andreas Boukas

We compute the Fourier transform (or vacuum characteristic function) of quantum random variables (observables), defined as self-adjoint finite sums of Fock space operators, satisfying the multi-dimensional Heisenberg Lie algebra commutation relations. The main tool is a splitting formula for the multi-dimensional Heisenberg group obtained by Feinsilver and Pap.

Proceedings of the American Mathematical Society•Apr 6, 2015•8 citations

Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials

Luigi Accardi, Andreas Boukas

We derive a normally ordered disentanglement or splitting formula for exponentials of the infinite-dimensional Schrödinger Lie algebra generators.As an application we compute the vacuum characteristic function of a quantum random variable defined as a self-adjoint finite sum of Fock space operators, satisfying the multi-dimensional Schrödinger Lie algebra commutation relations.

Communications on Stochastic Analysis•Jan 1, 2019•6 citations

HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO THE CHARACTERISTIC FUNCTION OF QUANTUM OBSERVABLES

Andreas Boukas

We show how Cauchy’s Integral Formula and the ideas of Dunford’s Holomorphic Functional Calculus (for unbounded operators) can be used to compute the Vacuum Characteristic Function (Quantum Fourier Transform) of quantum random variables defined as self-adjoint operators on L2(R,C). We consider in detail several quantum observables defined in terms of the position and momentum operators X, P, respectively, on L2(R,C).

arXiv:2407.03694•Journal of Stochastic Analysis•Jun 1, 2024•0 citations

Mathematical Formalism of Quantum Mechanics

Reinhold A. Bertlmann, Nicolai Friis

Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem

•Sep 7, 2023•0 citations
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.
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.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem