# Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise

Banach Center Publications (2010)

- Volume: 89, Issue: 1, page 13-43
- ISSN: 0137-6934

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topLuigi Accardi, and Andreas Boukas. "Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise." Banach Center Publications 89.1 (2010): 13-43. <http://eudml.org/doc/281709>.

@article{LuigiAccardi2010,

abstract = {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 (RHPWN). 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 RHPWN, 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 by several research 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.},

author = {Luigi Accardi, Andreas Boukas},

journal = {Banach Center Publications},

keywords = {renormalized powers of white noise; second quantization; Heisenberg algebra; w-algebra; Virasoro algebra; Zamolodchikov algebra; Fock space; moment systems; continuous binomial distribution; central extension; group law},

language = {eng},

number = {1},

pages = {13-43},

title = {Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise},

url = {http://eudml.org/doc/281709},

volume = {89},

year = {2010},

}

TY - JOUR

AU - Luigi Accardi

AU - Andreas Boukas

TI - Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise

JO - Banach Center Publications

PY - 2010

VL - 89

IS - 1

SP - 13

EP - 43

AB - 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 (RHPWN). 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 RHPWN, 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 by several research 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.

LA - eng

KW - renormalized powers of white noise; second quantization; Heisenberg algebra; w-algebra; Virasoro algebra; Zamolodchikov algebra; Fock space; moment systems; continuous binomial distribution; central extension; group law

UR - http://eudml.org/doc/281709

ER -

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