An Interdisciplinary Journal

* 2004, Vol.7, No.2, pp.106-128*

A special approach to examine spinor structure of 3-space is proposed.
It is based on the use of the concept of a spatial spinor defined through taking the square
root of a real-valued 3-vector. Two sorts of spatial spinor
according to *P*-orientation of an initial 3-space are introduced:
properly vector or pseudo vector one.
These spinors, and , turned out to be different functions of
Cartesian coordinates. To have a spinor space model,
you ought to use a doubling vector space
{ (x_{1},x_{2},x_{3} ) (x_{1},x_{2},x_{3})'}.
The main idea is to develop some mathematical
technique to work with such extended models.
Spinor fields and , given as functions of Cartesian
coordinates x_{i}x_{i}', do not obey Cauchy-Riemann analyticity condition
with respect to complex variable (x_{1} + i x_{2}) (x_{1} + i x_{2})'.
Spinor functions are in one-to-one correspondence with coordinates
x_{i} x_{i}' everywhere excluding the
whole axis (0,0,x_{3}) (0,0,x_{3})'
where they have an exponential discontinuity. It is proposed to consider
properties of spinor fields (x_{i} x_{i}') and (x_{i}x_{i}')
in terms of continuity with respect to geometrical directions
in the vicinity of every point.
The mapping of spinor field into and inverse
have been constructed. Two sorts of spatial spinors
are examined with the use of curvilinear coordinates (y_{1},y_{2},y_{3}):
cylindrical parabolic, spherical and
parabolic ones. Transition from vector to spinor models is achieved by doubling
initial parameterizing domain G (y_{1},y_{2},y_{3}) (y_{1},y_{2},y_{3}) with new identification rules on the boundaries.Different spinor space models are built on explicitly different spinor
fields (y) and (y). Explicit form of the mapping spinor field (y)
of pseudo vector model into spinor (y) of properly vector one is given,
it contains explicitly complex conjugation.

*Key words: *
space, geometry, spinor, continuity, P-orientation,
spherical and parabolic coordinates

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Last updated: *June 29, 2004*