


Victor L.
Mironov 
Octons 

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Algebra of
octons
The eightcomponent octon Ğ is defined in the following form Ğ=d + ai + bj + ck + DE + AI + BJ + CK where values i, j, k are polar unit vectors; I, J, K are axial unit vectors and E is the pseudoscalar unit.
The octon components d, a, b, c, D, A, B, C are the numbers (complex in
general). Thus the octon is the sum of scalar, vector, pseudoscalar and
pseudovector and the i, j, k, I, J, K values are polar and
axial bases of octon respectively. The algebra of octons was discussed in
detail in [1]. The basic commutation and multiplication rules are represented
in the table. The value ξ is the imaginary unit: ξ^{2} = 1. Table. The rules of multiplication and commutation
for the octon unit vectors. [1] V.L.Mironov, S.V.Mironov
 Octonic electrodynamics // ArXiv: 0802.2435
(2008). http://arxiv.org/abs/0802.2435 The applications of octons in electrodynamics and in relativistic
quantum mechanics see in the following publications:
V.L.Mironov, S.V.Mironov  Octonic
representation of electromagnetic field equations // Journal of Mathematical Physics, 50, 012901 110 (2009). <PDF> In this paper we represent eightcomponent values "octons",
generating associative noncommutative algebra. It
is shown that the electromagnetic field in a vacuum can be described by a
generalized octonic equation, which leads both to the wave equations for
potentials and fields and to the system of Maxwell equations. The octonic
algebra allows one to perform compact combined calculations simultaneously
with scalars, vectors, pseudoscalars, and pseudovectors. Examples of such calculations are
demonstrated by deriving the relations for energy, momentum, and Lorentz
invariants of the electromagnetic field. http://jmp.aip.org/resource/1/jmapaq/v50/i1/p012901_s1 V.L.Mironov, S.V.Mironov  Octonic secondorder
equations of relativistic quantum mechanics // Journal of Mathematical Physics, 50, 012302 113 (2009). <PDF> We demonstrate a generalization of relativistic
quantum mechanics using eightcomponent value "octons" that
generate an associative noncommutative spatial
algebra. It is shown that the octonic secondorder equation for the
eightcomponent octonic wave function, obtained from the Einstein relation
for energy and momentum, describes particles with spin 1/2. It is established
that the octonic wave function of a particle in the state with defined spin
projection has a specific spatial structure that takes the form of an octonic
oscillator with two spatial polarizations: longitudinal linear and transverse
circular. http://jmp.aip.org/resource/1/jmapaq/v50/i1/p012302_s1 V.L.Mironov, S.V.Mironov  Octonic
firstorder equations of relativistic quantum mechanics // International Journal of Modern Physics A, 24(22), 41574167 (2009). <PDF> In this paper we demonstrate a generalization of
relativistic quantum mechanics using eight component octonic wave function
and octonic spatial operators. It is shown that the secondorder equation for
octonic wave function describing particles with spin 1/2 can be reformulated
in the form of a system of firstorder equations for quantum fields, which is
analogous to the system of Maxwell equations for the electromagnetic field.
It is established that for the special types of wave functions the secondorder
equation can be reduced to the single firstorder equation analogous to the
Dirac equation. At the same time it is shown that this firstorder equation
describes particles, which do not have quantum fields. http://www.worldscinet.com/ijmpa/24/2422/S0217751X09045480.html 


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