479 research outputs found
On the Fock representation of the renormalized powers of quantum white noise
No full textWe describe the "no-go" theorems recently obtained by Accardi-Boukas-Franz in [\cite{1}] for the Boson case, and by Accardi-Boukas in [\cite{2}] for the -deformed case, on the issue of the existence of a common Fock space representation of the renormalized powers of quantum white noise (RPWN)
The semi-martingale property of the square of white noise integrators
No full textThe abstract commutation relations of the algebra of the square of white
noise of Accardi, Lu, and Volovich are shown to be realized by operator
processes acting on the Fock space of Accardi and Skeide which is very
closely related to the Finite Difference Fock space of Boukas and Feinsilver.
The processes are shown to satisfy the necessary conditions for inclusion
in the framework of the representation free quantum stochastic calculus of Accardi,Fagnola,and Quaegebeur.
The connection between the Finite-Difference operators and the
creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied
Higher powers of -deformed white noise
No full textWe introduce the renormalized powers of q-deformed white noise, for
any q in the open interval (−1, 1), and we extend to them the no–go theorem recently
proved by Accardi–Boukas–Franz in the Boson case. The surprising fact is
that the lower bound (6.5), which defines the obstruction to the positivity of the
sesquilinear form, uniquely determined by the renormalized commutation relations,
is independent of q in the half-open interval (−1, 1], thus including the Boson case.
The exceptional value q = −1, corresponding to the Fermion case, is dealt with in
the last section of the paper where we prove that the argument used to prove the
no–go theorem for q 6= 0 does not extend to this case
Fock representation of the renormalized higher powers of White noise and the centreless Virasoro (or Witt)-Zamolodchikov-omega(infinity)*-Lie algebra
No full textThe identification of the *-Lie algebra of the renormalized higher powers of White noise (RHPWN) and the analytic continuation of the second quantized centreless Virasoro (or Witt)-Zamolodchikov-omega(infinity)*-Lie algebra of conformal field theory and high-energy physics, was recently established in [5] based on results obtained in [3] and [4]. In the present paper, we show how the RHPWN Fock kernels must be truncated in order to be positive semi-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 >= 2 host the continuous binomial and beta processes
CONTRACTIONS AND CENTRAL EXTENSIONS OF QUANTUM WHITE NOISE LIE ALGEBRAS
No full textWe show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − 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 RPQWNc with the scalar renormalization δnt = cn−1δt, 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.We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN, with the convolution type renormalization δnt − s = δsδt − 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 RPQWNc with the scalar renormalization δnt = cn−1δt, 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
On the central extensions of the Heisenberg algebra
No full textWe describe the nontrivial central extensions CE(Heis) of the Heisenberg algebra and their
representation as sub–algebras of the Schroedinger algebra. We also present the characteristic and moment
generating functions of the random variable corresponding to the self-adjoint sum of the generators of
CE(Heis)
Lie algebras associated with the renormalized higher powers of white noise
No full textWe recall the recently established (cf. [1] and [2]) connection between the renormalized higher powers of white noise (RHPWN) *-Lie algebra
and the Virasoro--Zamolodchikov--Lie algebra of conformal field theory (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 sub-*{Lie algebras of the (1{mode) full oscillator algebra. This program has two additional motivations:
(i) the full oscillator algebra is a fundamental object of mathematics and the structure of its subalgebras deserves deep investigation;
(ii) the no--go theorems show that the current algebras over some Lie sub--
algebras of Lie algebras may have a Fock representation individually without
this being true for the Lie algebra generated by them. The problem of classifying which sub--algebras of the full oscillator algebra have this property is open and a preliminary step towards its analysis is the classification of the ''natural" sub algebras of the full oscillator algebra.
We construct two hierarchies of such sub algebras, parametrized by the
natural integers. One of these hierarchies begins with the Virasoro algebra. Another possibility to bypass the no--go theorems is to consider different renormalizations of the higher powers of white noise commutation relations.
This approach is developed in Section 2, where we show with examples that
some of them lead to known (i.e., first or second order) commutation relations.
This fact is probably related with the gaussianization phenomenon discussed
in [7]
The Schrodinger Fock Kernel and the no-go Theorem for the First Order and Renormalized Square of White Noise Lie Alegbras
No full textUsing the non-positive definiteness of the Fock kernel associated
with the Schr ̈odinger algebra we prove the impossibility of a joint Fock representation
of the first order and Renormalized Square of White Noise Lie
algebras with the convolution type renormalization δ2(t−s) = δ(s) δ(t−s) for
the square of the Dirac delta function. We show how the Schr ̈odinger algebra
Fock kernel can be reduced to a positive definite kernel through a restriction of
the set of exponential vectors. We describe how the reduced Schr ̈odinger kernel
can be viewed as a tensor product of a Renormalized Square of White Noise
(sl(2)) and a First Order of White Noise (Heisenberg) Fock kernel. We also
compute the characteristic function of a stochastic process naturally associated
with the reduced Schr ̈odinger kernel
The emergence of the Virasoro and algebras through the renormalized higher powers of quantum white noise
No full textWe 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 of the extended conformal algebra with infinite symmetries
of Conformal Field Theory ( \cite{4a}-\cite{4d}, \cite{ketov}, \cite{7}, \cite{8}). In particular
we construct a white noise (boson) representation of the generators and commutation relations and of their second quantization
- …
