Operator Lie Algebras of Rotations and Transformations in White Noise

2022/03/17 by Wolfgang Böck, Bock, Wolfgang, Janeth Canama +1
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Quantum Algebra (math.QA) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2203.09345

openalex publication_date 2022/03/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The infinitesimal generator of a one-parameter subgroup of the infinite dimensional rotation group associated with the complex Gelfand triple (E) ⊂ L²(E^*, μ) ⊂ (E)^* is of the form R_κ= ∫_{T× T} κ(s,t) (a_s^* a_t - a_t^* a_s) ds dt where κ∈ E ⊗ E^* is a skew-symmetric distribution. Hence R_κ is twice the conservation operator associated with a skew-symmetric operator S. The Lie algebra containing R_κ, identity operator, annihilation operator, creation operator, number operator, (generalized) Gross Laplacian is discussed. We show that this Lie algebra is associated with the orbit of the skew-symmetric operator S.

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