When is a Field a Jacobi-Field? A Characterization of States on Tensor Algebras

Abstract

In order to analyse the positivity condition for states on the tensoralgebra $\underline{E}$ over certain (function-)spaces $E$ more efficiently a representation of the components of the states in terms of a set of “independent parameters” is suggested. For this purpose the concept of a Jacobifield is introduced. In the case of a finite dimensional space $E$ every state on $\underline{E}$ is “parametrized” this way. If however the basic space $E$ is infinite dimensional additional domain problems arise related to algebras of unbounded operators which are involved naturally. It is analysed to which extent this “parametrization in terms of Jacobi-fields” also works in the general case, and it is shown that for “many” basic spaces $E$ which occur in applications “most” of the states admit indeed such a “parametrization”. This then also means a corresponding decomposition for the associated algebra of unbounded operators into “independent components”. Several applications are indicated.