Andreas Boukas

RENORMALIZED POWERS OF QUANTUM WHITE NOISE

We prove some no-go theorems on the existence of a Fock representation of the *-Lie algebra generated by , where , bs are the Hida white noise densities. In particular we prove the nonexistence of such a representation for any *-Lie algebra containing . This drastic difference with the quadratic case proves the necessity of investigating different renormalization rules for the case of higher powers of white noise.

An example of a quantum exponential process

In [3],R. L. Hudson andK. R. Parthasarathy showed that the Fock space based on the Heisenberg—Weyl algebra hosts Brownian motion and Poisson processes. In this paper we construct a quantum exponential process acting on the Fock space based on the finite-difference algebra ofP. J. Feinsilver ([2]).

RENORMALIZED HIGHER POWERS OF WHITE NOISE (RHPWN) AND CONFORMAL FIELD THEORY

The Virasoro–Zamolodchikov *-Lie algebra w∞ 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∞ algebra. Arguments discussed at the end of this note suggest the conjecture that this inclusion is in fact an identification.

Renormalized higher powers of white noise and the virasoro-zamolodchikov-w∞ algebra

We have recently proved that the generators of the second quantized centedess Virasoro (or Witt)-Zamolodchikov- w ∞ algebra can be expressed in terms of the Renormalized Higher Powers of White Noise (RHPWN) 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 centerless 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 physics literature and some heuristic results were obtained in the direction of the extension of this inclusion to the 1-mode centerless Virasoro (or Witt)-Zamolodchikov- w ∞ algebra. However, the possibility of an identification of the second quantizations of these two algebras was not even conjectured in the physics literature.

The Unitarity Conditions for the Square of White Noise

The renormalized stochastic differentials of the square of white noise are defined in Boson–Fock space representation. The Ito multiplication table of these differentials and the module form of the unitarity conditions are obtained.

Unitarity conditions for stochastic differential equations driven by nonlinear quantum noise

We prove the stochastic independence of the basic integrators of the renormalized square of white noise (SWN). We use this result to deduce the unitarity conditions for stochastic differential equations driven by the SWN.

On the unitarity of stochastic evolutions driven by the square of white noise

Using the closed Itos table for the renormalized square of white noise, recently obtained by Accardi, Hida, and Kuo in Ref. 4, we consider the problem of providing necessary and sufficient conditions for the unitarity of the solutions of a certain type of quantum stochastic differential equations.

Quantum Formulation of Classical Two Person Zero-Sum Games

The concept of a classical player, corresponding to a simple random variable on a finite cardinality probability space, is shown to extend to that of a quantum player, corresponding to a self-adjoint operator on a quantum probability Hilbert space. Quantum versions of Von Neumanns minimax theorem are proved.

Control of Quantum Langevin Equations

The problem of controlling quantum stochastic evolutions arises naturally in several different fields such as quantum chemistry, quantum information theory, quantum engineering, etc. In this paper, we apply the recently discovered closed form of the unitarity conditions for stochastic evolutions driven by the square of white noise [9] to solve this problem in the case of quadratic cost functionals (cf. (5.5) below). The optimal control is explicitly given in terms of the solution of an operator Riccati equation. Under general conditions on the system Hamiltonian part of the stochastic evolution and on the system observable to be controlled, this equation admits solutions with the required properties and they can be explicitly described.

Quantum Probability, Renormalization and Infinite-Dimensional -Lie Algebras ⋆

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.