By starting from a quaternionic separable Hilbert space as a base model,
the paper uses the capabilities and the restrictions of this model in order to
investigate the origins of the electric charge and the electric fields. Also,
other discrete properties such as color charge and spin are considered. The
paper exploits all known aspects of the quaternionic number system and it uses
quaternionic differential calculus rather than Maxwell based differential
calculus. The paper presents fields as mostly continuous quaternionic
functions. The electric field is compared with another basic field that acts as
a background embedding continuum. The behavior of photons is used in order to investigate
the long range behavior of these fields. The paper produces an algorithm that
calculates the electric charge of elementary particles from the symmetry
properties of their local parameter spaces. The paper also shows that the usual
interpretation of a photon as an electric wave is not correct.
Quantum
logic was introduced by Garret Birkhoff and John von Neumann in their paper:
Birkhoff, G. and von Neumann, J. (1936) The Logic of Quantum Mechanics. Annals of Mathematics, 37, 823-843.
This paper also indicates the relation between this orthomodular lattice and
separable Hilbert spaces.
In
the sixties Israel Gelfand and GeorgyiShilov introduced a way to model continuums
via an extension of the separable Hilbert space into a so called Gelfand
triple. The Gelfand triple often gets the name rigged Hilbert space. It is a
non-separable Hilbert space. http://www.encyclopediaofmath.org/index.php?title=Rigged_Hilbert_space
Paul
Dirac introduced the braket notation, which popularized the usage of Hilbert
spaces. Dirac also introduced its delta function, which is a generalized
function. Spaces of generalized functions offered continuums before the Gelfand
triple arrived.
See: Dirac, P.A.M. (1958) The Principles of Quantum Mechanics. 4th Edition,
Oxford University Press, Oxford, ISBN 978 0 19 852011 5.
Quaternionic
function theory and quaternionic Hilbert spaces are treated in: van Leunen, J.A.J.
(2015) Quaternions and Hilbert Spaces. http://vixra.org/abs/1411.0178 .
In
the second half of the twentieth century Constantin Piron and Maria Pia Solèr
proved that the number systems that a separable Hilbert space can use must be
division rings. See: Baez, J. (2011) Division Algebras and Quantum Theory. http://arxiv.org/abs/1101.5690 and Holland, S.S. (1995)
Orthomodularity in Infinite Dimensions: A Theorem of M. Solèr. Bulletin of the American Mathematical
Society, 32, 205-234. http://www.ams.org/journals/bull/1995-32-02/S0273-0979-1995-00593-8/
