3,785 research outputs found
Clifford geometric parameterization of inequivalent vacua
We propose a geometric method to parameterize inequivalent vacua by dynamical
data. Introducing quantum Clifford algebras with arbitrary bilinear forms we
distinguish isomorphic algebras --as Clifford algebras-- by different
filtrations resp. induced gradings. The idea of a vacuum is introduced as the
unique algebraic projection on the base field embedded in the Clifford algebra,
which is however equivalent to the term vacuum in axiomatic quantum field
theory and the GNS construction in C^*-algebras. This approach is shown to be
equivalent to the usual picture which fixes one product but employs a variety
of GNS states. The most striking novelty of the geometric approach is the fact
that dynamical data fix uniquely the vacuum and that positivity is not
required. The usual concept of a statistical quantum state can be generalized
to geometric meaningful but non-statistical, non-definite, situations.
Furthermore, an algebraization of states takes place. An application to physics
is provided by an U(2)-symmetry producing a gap-equation which governs a phase
transition. The parameterization of all vacua is explicitly calculated from
propagator matrix elements. A discussion of the relation to BCS theory and
Bogoliubov-Valatin transformations is given.Comment: Major update, new chapters, 30 pages one Fig. (prev. 15p, no Fig.
Clifford Algebraic Remark on the Mandelbrot Set of Two--Component Number Systems
We investigate with the help of Clifford algebraic methods the Mandelbrot set
over arbitrary two-component number systems. The complex numbers are regarded
as operator spinors in D\times spin(2) resp. spin(2). The thereby induced
(pseudo) normforms and traces are not the usual ones. A multi quadratic set is
obtained in the hyperbolic case contrary to [1]. In the hyperbolic case a
breakdown of this simple dynamics takes place.Comment: LaTeX, 27 pages, 6 fig. with psfig include
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