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Deformation Expression for Elements of Algebras (VII) --Vacuum/Pseudo-vacuum Representations-- | | Hideki Omori
; Yoshiaki Maeda
; Naoya Miyazaki
; Akira Yoshioka
; | | Date: |
12 Oct 2012 | | 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. | | Source: | arXiv, 1210.3426 | | Services: | Forum | Review | PDF | Favorites | |
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