1. Introduction
In this author’s previously published, referenced paper [1] 1, a derivation of the non-linear component
of the SU(3) field tensor for quantum chromodynamics was given which was elaborate and which required the somewhat artificial postulate of color confinement to complete the derivation. A much simpler and mathematically direct derivation which does not rely on color confinement and which mirrors SU(2)’s development exists and is given herein. The mathematical methodology used is taken from the subject original paper, which is covered in detail therein [1] .
2. The Derivation
The gauge field “cross product” for the non-linear term of the SU(3) field tensor has the form [1]
(1)
where i = 0 − 7 and the
are the structure constants of the Gell-Mann commutation relation
. A bijective relation between the Gell-Mann generators
and the octonion basis elements
was given with structure constants existing for the terms [1]
(2)
Using the formalism’s unique division-algebraic coupling equation [1]
(3)
we now consider the coupled operator
(where
defines the involution
of
) instead of the coupled operator
as was considered in the original paper. Setting
, we have for the applicable vector portion
of the coupled operator
(4)
in which we have used
. As we are using the 𝕆-based coupling equation, both terms of Equation (4) are 7-dimensional cross products.
The term
has components
. Since the 7-dim cross product only sums from i = 1 − 7, setting
only covers the structure constants
.
To cover the remaining
we look to the term
, which has components
. Recalling the total asymmetry of
, we
simply set
for
and
for
, with
for all other ik and the
being required due to the 2 in
.
The bijective mapping between eight Clifford fields
and the eight SU(3) gauge fields GjGk follows as in the original paper, with
(5)
otherwise,
thus generating the non-linear component
.
3. Results and Discussion
The derivation herein of the non-linear portion of SU(3)’s field tensor is more direct and mathematically straightforward than the original paper’s derivation. Further, it mirrors the SU(2) formalism’s use of
in generating the
portion of the SU(2) field tensor and does not require the somewhat artificial postulate of color confinement for the mathematical derivation. Lastly, given this derivation the previously established bijective relation between the octonion basis
and the Gell-Mann generators
[1] is now seen to be unnecessary and superfluous to the octonionic development of SU(3) gauge theory, since the vector section
of Equation (3) generates the entirety of SU(3)’s Lie algebra structure constants while residing solely within the
basis in doing so.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
NOTES
1See Ref. [1] , Sec. 3.i.