Analytic central extensions of infinite dimensional white noise *–Lie algebras (original) (raw)
Related papers
Journal of Physics A: Mathematical and Theoretical, 2008
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 ∞ * -Lie algebra of conformal field theory and high-energy physics, was recently established in [3] 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 ≥ 2 host the continuous binomial and beta processes.
The 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-w ∞ * -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.
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.
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.
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
2006
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_∞ of the extended conformal algebra with infinite symmetries W_∞ of Conformal Field Theory. In particular we construct a white noise (boson) representation of the w_∞ generators and commutation relations and of their second quantization.