Itô calculus and quantum white noise calculus (original) (raw)
LIE ALGEBRAS ASSOCIATED WITH THE RENORMALIZED HIGHER POWERS OF WHITE NOISE
2007
We recall the recently established (cf. (1) and (2)) connection be- tween the renormalized higher powers of white noise (RHPWN) ⁄-Lie algebra and the Virasoro {Zamolodchikov{ w1 ⁄-Lie algebra of conformal fleld the- ory (cf. (10)). Motivated by this connection, with the goal of investigating a possible connection with classical independent increments processes, we begin a systematic study of the
In the first part of the paper we discuss possible definitions of Fock representation of the * -Lie algebra of the Renormalized Higher Powers of White Noise (RHP W N. We propose one definition that avoids the no-go theorems and we show that the vacuum distribution of the analogue of the field operator for the n-th renormalized power of WN defines a continuous binomial process. In the second part of the paper we present without proof our recent results on the central extensions of RHP W N, its subalgebras and the w∞ Lie algebra of conformal field theory. In the third part of the paper we describe our results on the non-trivial central extensions of the Heisenberg algebra. This is a 4-dimensional Lie algebra, hence belonging to a list which is well known and has been studied from time to time by several groups. However the canonical nature of this algebra, i.e. the fact that it is the unique (up to a complex scaling) non-trivial central extension of the Heisenberg algebra seems to be new. We also find the possible vacuum distributions corresponding to a family of injective * -homomorphisms of different non-trivial central extensions of the Heisenberg algebra into the Schrödinger algebra.
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 powers of quantum white noise
Giving meaning to the powers of the creation and annihilation densities (quantum white noise) is an old and important problem in quantum field theory. In this paper we present an account of some new ideas that have recently emerged in the attempt to solve this problem. We emphasize the connection between the Lie algebra of the renormalized higher powers of quantum white noise (RHPWN), which can be interpreted as a suitably deformed (due to renormalization) current algebra over the 1-mode full oscillator algebra, and the current algebra over the centerless Virasoro (or Witt)-Zamolodchikov-w∞ Lie algebras of conformal field theory. Through a suitable definition of the action on the vacuum vector we describe how to obtain a Fock representation of all these algebras. We prove that the restriction of the vacuum to the abelian subalgebra generated by the field operators gives an infinitely divisible process whose marginal distribution is the beta (or continuous binomial).
We prove the triviality of the second cohomology group of the Virasoro-Zamolodchikov and Renormalized Higher Powers of White Noise * -Lie algebras. It follows that these algebras admit only trivial central extensions. We also prove that the Heisenberg-Weyl * -Lie algebra admits nontrivial central extensions which are parametrized in a 1-to-1 way by C \{0}. Explicit unitary * -representations of these extensions and their implications for our renormalization program are discussed in Ref. 8.
CONTRACTIONS AND CENTRAL EXTENSIONS OF QUANTUM WHITE NOISE LIE ALGEBRAS
We show that the Renormalized Powers of Quantum White Noise Lie algebra RP QW N * , with the convolution type renormalization δ n (t − s) = δ(s) δ(t − s) of the n 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RP QW Nc with the scalar renormalization δ n (t) = c n−1 δ(t), c > 0. Using this renormalization, we also obtain a Lie algebra W∞(c) which contains the w∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W∞ algebra of Pope, Romans and Shen, we show that W∞(c) can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W∞, the central extension coincides with that of the Virasoro algebra.
Higher powers of quantum white noise derivatives
Communications on Stochastic Analysis, 2014
By a Wick differential equation, we characterize the operator W l,m (f) studied in [1, 3, 9] where l, m ∈ N ∪ {0} and f ∈ S(R). As an application we give in our setting a new renormalization in order to get the higher powers of white noise. Then, we investigate the commutation relations obtained from the quantum white noise (QWN) derivatives in order to introduce two operators acting on white noise operators, from which we get the higher powers of quantum white noise derivatives and a *-Lie algebra generalizing the renormalized higher power white noise Lie algebra.
2010
In previous papers whe have shown that the one mode Heisenberg algebra Heis(1) admits a unique nontrivial central extensions CeHeis(1) which can be realized as a sub-Lie-algebra of the Schrödinger algebra, in fact the Galilei Lie algebra. This gives a natural family of unitary representations of CeHeis(1) and allows an explicit determination of the associated group by exponentiation. In contrast with Heis(1), the group law for CeHeis(1) is given by nonlinear (quadratic) functions of the coordinates. The vacuum characteristic and moment generating functions of the classical random variables canonically associated to CeHeis(1) are computed. The second quantization of CeHeis(1) is also considered.