Deformation Expression for Elements of Algebras (VII) --Vacuum/Pseudo-vacuum Representations--

TL;DR

This paper explores advanced algebraic structures like vacuums and pseudo-vacuums within deformation quantization, proposing new mathematical frameworks that could extend the foundations of mathematical physics beyond classical calculus.

Contribution

It introduces the concepts of vacuums and pseudo-vacuums in extended algebraic calculus, suggesting they open new frontiers in mathematical physics and phase space quantization.

Findings

Identification of polar elements and pseudo-vacuums as new algebraic entities.

Proposition that vacuum functions form Frobenius-like algebras.

Highlighting the importance of expression parameter restrictions for classical interpretations.

Abstract

Thinking back the long history of physics, we see that the calculation used by physicists was nothing but the ordinary calculus. Another word, physicists have never wrote theories beyond the basic axioms of the calculus. This is not to declare of the victory of calculus or algebraic topology. On the contrary, we are thinking that every theory of mathematical physics must suggest new frontier of ordinary calculus, which are never viewed by classical geometers. Weyl algebras or Heisenberg algebras are naturally involved in slightly extended systems of the algebra of ordinary calculus, and are supported by the classical notion of phase spaces on which the general mechanics are based. The theory of deformation quantizations gives a notion of quantization of "phase space". To explain its essence in brief we proposed in the previous note the notion of $\mu$-regulated algebra. In this series, we have introduced elements, called "vacuums" to consider the state vectors and the configuration spaces within the world of extended algebra of calculus with various expressions. We have found several strange elements, called polar elements, and an extended notions of vacuums, which were called pseudo-vacuums in our paper. These are not established notions in mathematical physics, but we are thinking that these must propose new frontier for mathematical physics. We are thinking that vacuums and pseudo-vacuums are not unique, but the function algebra of the configuration spaces must be an algebra similar to the Frobenius algebra defined by vacuums. The point in this note is that to obtain classical pictures one has often to restrict the expression parameters, and there are two essentially different expression parameters.