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Communications on Stochastic Analysis

Normally Ordered Disentanglement of Multi-Dimensional Schrödinger Algebra Exponentials

Luigi Accardi, Andreas Boukas

Published Jan 1, 2019• 6 citations
Publisher

Abstract

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.

DOI
10.31390/cosa.12.3.05

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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.
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.
In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.
In the preceding paper [Commun. Theor. Phys. 51 (2009) 321] we have recommended a convenient method for disentangling exponential operators in the form of exp{B + C}, trying to find an operator A that satisfies [A, B] = C, and [A,[A,B]] = 0, then from the Baker–Hausdorff formula we have exp{B + C} = exp{B+[A, B]} = e A e B e −A . After arranging e A e B = e B e A e W , the disentangling exp{B + C} = e B e W is obtained. In this work we use this method to two-mode case, especially, derive the normal ordering form of exp[h(a † a + b † b) + ga † b † + kab] without appealing to Lie algebra method.