References - Bogoliubov Laboratory of Theoretical Physics - JINR

in the four-dimensional and general covariant form [1]. The operator rot is defined for the
vector fields as follows:
(rotM) i = e ijkl tj∂kMl = 1
2 eijkl tj(∂kMl − ∂lMk),
where eijkl is the orientation of physical space (”element of volume”). It is easy to show
that
(M, rotN)+div[MN] = (rotM, N),
where
[MN] i = e ijkl tjMkNl
is a vector product of two vector fields M and N, div M = ∇iM i . Thus, the operator rot
is self-adjoint. We also mention that (gradϕ)i = △i ϕ, △i = ∇i − ti∇t and rot grad =
0, div rot = 0.
Equations of spindynamics include four scalar and four vector equations:
(∇t + 1ϕ)κ
=divK−mμ 2
(∇t + 1ϕ)λ
=divL−mν 2
(∇t + 1ϕ)μ
=divM + mκ
2
ϕ)ν =divN + mλ
(∇t + 1
2
(∇t + 1
ϕ)K = −rot L +gradκ + m M
2
(∇t + 1ϕ)L
=rotK +gradλ + m N
2
(∇t + 1ϕ)M
=rotN +gradμ−mK 2
ϕ)N = −rot M +gradν−mL, (∇t + 1
2
where ∇t = t i ∇i, ϕ = ∇it i . From the first principles it follows that Equations (2) and
(3) describe all phenomena connected with spin symmetry and spin.
3. Strong Interactions as subject of Spindynamics. We can consider (following
the successive approximations method) that physical space is 4-dimensional Euclidean
space E 4 with the Euclidean distance function. Evolution and causality in the physical
space E 4 is defined by the scalar temporal field f(q1, q2, q3, q4) which is a solution to the
equation (1) . In this case, Equation (1) has the general solution f(q1, q2, q3, q4) =a·q+a,
where a =(a1, a2, a3, a4) is a unit constant vector,
and the singular solution
a · a =1,
f(q1, q2, q3, q4) = � (q · q) =
�
q 2 1 + q 2 2 + q 2 3 + q 2 4.
The space cross-section of E 4 is defined by the field of time. For a given real number t,
the space cross-section is defined by the equation f(q 1 ,q 2 ,q 3 ,q 4 )=t. Thus, in the first
case the space sections of E 4 are the Euclidean space E 3 and in the second one the space
sections are the 3d spheres √ q · q = t. For the stream of time we have
t i = ti = ∂f
∂qi
(2)
(3)
= qi
f , ∂it i = 3
. (4)
f
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