Formal Series of Generalised Functions and Their Application to Deformation Quantisation

TL;DR

This paper explores the foundations of formal series calculus in deformation quantisation, introducing classes of linear functionals and examining the concept of nonnegativity, with implications for defining quantum states.

Contribution

It develops a rigorous framework for formal series calculus in deformation quantisation, including new classes of linear functionals and analysis of quantum state definitions.

Findings

Introduces classes of continuous linear functionals relevant to physics.

Proposes a notion of nonnegativity in formal series calculus.

Analyzes challenges in defining quantum states over formal series.

Abstract

Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity in formal series calculus is proposed. Problems with defining quantum states over the set of formal series are analysed.