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{\noindent\LARGE{\bf Norma Manko\v c Bor\v stnik}\\[1cm]
\noindent\bf{Spins and charges, unifying in Grassmann
space enables unification of all interactions}}
\\
The fact that ordinary space-time is not enough to describe
dynamics of particles is known for more than 70 years: in
addition to the vector space, spanned over the ordinary coordinate
space, the internal space of the spins and the charges is needed.
Today's theories try to unify the internal space of charges,
describing this space with the fundamental and the adjoint
representations of the appropriate groups, for fermionic and
bosonic fields, respectively.
The experimental fact that only the left handed fermions carry
the weak charge can only be understood if spins and charges are
unified; how otherwise could handedness, which concerns the spin
degrees of freedom, be connected with the weak charge?
I propose the approach in which all the internal degrees of
freedom of fermionic and bosonic fields are described in an
unique way: as dynamics in a space of noncommutative
coordinates.
In a space of $d $ Grassmann coordinates
two types of generators of Lorentz transformations can be
defined, one
of spinorial and the other of vectorial character. (While in
ordinary space only one kind of operators exist.)
Both kinds of operators appear as linear differential operators
in Grassmann space, defining the
fundamental and the adjoint
representations of the group $ SO(1,d$$-$$1) $, respectively.
(The canonical quantization of coordinates in Grassmann space
can be performed in complete agreement with
the canonical quantization of coordinate in ordinary space.)
The eigenvalues of commuting operators in Grassmann space
belonging to the subgroup
$(SO(1,4)) $ can be identified with spins of either fermionic or
bosonic fields, while the operators belonging to subgroups of $
SO(d$$-$$5) $ ${\supset SU(3)}$ $ { \times SU(2)}$ $ { \times U(1)} $,
determine the Yang-Mills charges.
It turns out that in the $SO(1,14)$ multiplets of spinorial
character, left handed weak
doublets appear together with right handed weak singlets, while
right handed weak doublets appear together with left handed weak
singlets. One also can find the left handed vectorial
representations which only interact with left handed spinorial
representations, while right handed vectorial representations
interact with only right handed spinorial representations.
The Lagrange function which describes a particle on a supergeodesics,
leads to the momentum of the particle in Grassmann space which is
proportional to the Grassmann coordinate. This brings the Clifford
algebra and the spinorial degrees of freedom into the theory.
The supervielbeins, transforming the geodesics from the freely
falling to the external coordinate system, carry the vectorial as
well as the spinorial degrees of freedom.
The Yang-Mills fields appear as the contribution of gravity
through spin connections. Some of spin connection terms
have even properties of Yukawa couplings.
The approach suggests that elementary
particles are either in the spinorial representations with
respect to the groups determining the spin and the charges,
or they are in the vectorial representations with respect to the
groups, which determine the spin and the charges. This disagrees
with supersymmetric models suggesting the existence of
fermions in the adjoint representations with respect to charges
and bosons in the fundamental representations with respect to
charges.
The approach offers the explanations for families of quarks and
leptons, suggesting four rather than three families.
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