arXiv subjects
Publications and source records attributed to Andreas Boukas.
Using the spectral resolution of the multiplication operator on the Schwartz class of $L^2(\mathbb{R},\mathbb{C})$, we compute the characteristic function of the cube of a Gaussian random variable.
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 $L^2(\mathbb{R},\mathbb{C})$. We consider in detail several quantum observables defined in terms of the position and momentum operators $X$, $P$, respectively, on $L^2(\mathbb{R},\mathbb{C})$.
We compute the resolvent of the anti-commutator operator $XP+PX$ and of the quantum harmonic oscillator Hamiltonian operator $\frac{1}{2}(X^2+P^2)$. Using Stone's formula for finding the spectral resolution of an, either bounded or unbounded, self-adjoint operator on a Hilbert space, we also compute their Vacuum Characteristic Function (Quantum Fourier Transform). We also show how Stone's formula is applied to the computation of the Vacuum Characteristic Function of finite dimensional quantum observables. The method is proposed as an analytical alternative to the algebraic (or Heisenberg) approach relying on the associated Lie algebra commutation relations.
We prove that the commutation relations among the generators of the quadratic Heisenberg algebra of dimension $n\in\mathbb{N}$, look like a kind of \textit{non-commutative extension} of $\hbox{sl}(2, \mathbb{C})$ (more precisely of its unique $1$--dimensional central extension), denoted $\hbox{heis}_{2;\mathbb{C}}(n)$ and called the complex $n$--dimensional quadratic Boson algebra. This \textit{non-commutativity} has a different nature from the one considered in quantum groups. %In particular we prove that, for %most values of $n$, this Lie algebra cannot be isomorphic to %$\hbox{sl}(N, \mathbb{C})$ for almost any value of $N$. We prove the exponentiability of these algebras (for any $n$) in the Fock representation. We obtain the group multiplication law, in coordinates of the first and second kind, for the quadratic Boson group and we show that, in the case of the adjoint representation, these multiplication laws can be expressed in terms of a generalization of the Jordan multiplication. We investigate the connections between these two types of coordinates (disentangling formulas). From this we deduce a new proof of the expression of the vacuum characteristic function of homogeneous quadratic boson fields.
Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle Φ, e^{itH}Φ\rangle$ 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 $\mathcal{H}$. 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.
The concept of a classical player, corresponding to a classical random variable, is extended to include quantum random variables in the form of self adjoint operators on infinite dimensional Hilbert space. A quantum version of Von Neumann's Minimax theorem for infinite dimensional (or continuous) games is proved.
Within the framework of the Accardi-Fagnola-Quaegebeur (AFQ) representation free calculus of \cite{b}, we consider the problem of controlling the size of a quantum stochastic flow generated by a unitary stochastic evolution affected by quantum noise. In the case when the evolution is driven by first order white noise (which includes quantum Brownian motion) the control is shown to be given in terms of the solution of an algebraic Riccati equation. This is done by first solving the problem of controlling (by minimizing an associated quadratic performance criterion) a stochastic process whose evolution is described by a stochastic differential equation of the type considerd in \cite{b}. The solution is given as a feedback control law in terms of the solution of a stochastic Riccati equation.
After providing a general formulation of Fermion flows within the context of Hudson-Parthasarathy quantum stochastic calculus, we consider the problem of determining the noise coefficients of the Hamiltonian associated with a Fermion flow so as to minimize a naturally associated quadratic performance functional. This extends to Fermion flows results of the authors previously obtained for Boson flows .
The basic aspects of the Hudson-Parthasarathy quantum stochastic calculus and of the Accardi-Fagnola-Quaegebeur representation free stochastic calculus are presented. The basic features of the stochastic calculus for the square of white noise recently developed by Accardi-Boukas are described. The feedback control problem for stochastic processes driven by quantum noise is solved
The present paper reviews some intriguing connections which link together a new renormalization technique, the theory of *-representations of infinite dimensional *-Lie algebras, quantum probability, white noise and stochastic calculus and the theory of classical and quantum infinitely divisible processes.
We consider the problem of determining the minimal time for which an energy supply source should operate in order to supply a system with a desired amount of energy in finite time.
We study the non-trivial central extensions $CEHeis$ of the Heisenberg algebra $Heis$ recently constructed in {AccBouCE}. We prove that a real form of $CEHeis$ is one the fifteen classified real four--dimensional solvable Lie algebras. We also show that $CEHeis$ can be realized (i) as a sub--Lie--algebra of the Schroedinger algebra and (ii) in terms of two independent copies of the canonical commutation relations (CCR). This gives a natural family of unitary representations of $CEHeis$ and allows an explicit determination of the associated group by exponentiation. In contrast with $Heis$, the group law for $CEHeis$ is given by nonlinear (quadratic) functions of the coordinates.
The identification of the $*$--Lie algebra of the renormalized higher powers of white noise (RHPWN) and the analytic continuation of the second quantized Virasoro--Zamolodchikov--$w_{\infty} *$--Lie algebra of conformal field theory and high-energy physics, was recently established in \cite{id} based on results obtained in [1] and [2]. In the present paper we show how the RHPWN Fock kernels must be truncated in order to be positive definite and we obtain a Fock representation of the two algebras. We show that the truncated renormalized higher powers of white noise (TRHPWN) Fock spaces of order $\geq 2$ host the continuous binomial and beta processes.
Motivated by the work of Segal and Segal on the Black-Scholes pricing formula in the quantum context, we study a quantum extension of the Black-Scholes equation within the context of Hudson-Parthasarathy quantum stochastic calculus. Our model includes stock markets described by quantum Brownian motion and Poisson process.
Recently (cf. \cite{ABIDAQP06} and \cite{ABIJMCS06}) L. Accardi and A. Boukas proved that the generators of the second quantized Virasoro--Zamolodchikov--$w_{\infty}$ algebra can be expressed in terms of the Renormalized Higher Powers of White Noise and conjectured that this inclusion might in fact be an identity, in the sense that the converse is also true. In this paper we prove that this conjecture is true. We also explain the difference between this result and the Boson representation of the Virasoro algebra, which realizes, in the 1--mode case (in particular without renormalization), an inclusion of this algebra into the full oscillator algebra. This inclusion was known in the physical literature and some heuristic results were obtained in the direction of the extension of this inclusion to the 1--mode Virasoro--Zamolodchikov--$w_{\infty}$ algebra. However the possibility of an identification of the second quantizations of these two algebras was not even conjectured in the physics literature.
We introduce a new renormalization for the powers of the Dirac delta function. We show that this new renormalization leads to a second quantized version of the Virasoro sector $w_{\infty}$ of the extended conformal algebra with infinite symmetries $W_{\infty}$ of Conformal Field Theory. In particular we construct a white noise (boson) representation of the $w_{\infty}$ generators and commutation relations and of their second quantization.
The Virasoro--Zamolodchikov Lie algebra $w_{\infty}$ has been widely studied in string theory and in conformal field theory, motivated by the attempts of developing a satisfactory theory of quantization of gravity. The renormalized higher powers of quantum white noise (RHPWN) *-Lie algebra has been recently investigated in quantum probability, motivated by the attempts to develop a nonlinear generalization of stochastic and white noise analysis. We prove that, after introducing a new renormalization technique, the RHPWN Lie algebra includes a second quantization of the $w_{\infty}$ algebra. Arguments discussed at the end of this note suggest the conjecture that this inclusion is in fact an identification