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arXiv:2407.03694 · Journal of Stochastic Analysis

HOLOMORPHIC FUNCTIONAL CALCULUS APPROACH TO THE CHARACTERISTIC FUNCTION OF QUANTUM OBSERVABLES

Andreas Boukas

Published Jun 1, 2024• 0 citations
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Abstract

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).

DOI
10.31390/josa.5.2.01

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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.
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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.
We show that quantum measures and integrals appear naturally in any L2-Hilbert space H. We begin by defining a decoherence operator D(A, B) and it’s associated q-measure operator µ(A) = D(A, A) on H. We show that these operators have certain positivity, additivity and continuity properties. If ρ is a state on H, then Dρ(A, B) = tr [ρD(A, B)] and µρ(A) = Dρ(A, A) have the usual properties of a decoherence functional and q-measure, respectively. The quantization of a random variable f is defined to be a certain self-adjoint operator fb on H. Continuity and additivity properties of the map f 7→ fb are discussed. It is shown that if f is nonnegative, then fb is a positive operator. A quantum integral is de- fined by R f dµρ = tr(ρfb). A tail-sum formula is proved for the quantum integral. The paper closes with an example that illustrates some of the theory.
The semigroup of statistical operators, and the unitary group of time translation operators generated by the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphic half-plane semigroup. The statistical expectation values of products of time-dependent operators are then boundary values of holomorphic functions which allow a Bochner integral representation with a Cauchy kernel. The general time-temperature-dependent Green's functions permit a concise spectral representation. It is suggested that a thermodynamic perturbation theory should treat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphic half-plane semigroups generated by semibounded self-adjoint Hamiltonians.