Applications of the local algebras of vector fields to the modelling of physical phenomena Bayak I. V. Abstract In this paper we discuss the local algebras of linear vector fields that can be used in the mathematical modeling of physical space by building the dynamical flows of vector fields on 8-dimensional cylindrical or toroidal manifolds. It is shown that the topological features of the vector fields obey the Dirac equation when moving freely within the surface of a pseudo-sphere in the 8-dimensional pseudo-Euclidean space. MSC: 57R25 Keywords: vector fields, algebra of linear vector fields, dynamic flow, topological features
1
Introduction
The paper contains a collection of algebraic, geometric and dynamical facts concerning linear vector fields on simple locally affine manifolds (Rn , cylinders, tori) where main ingredient is the algebra defined by these vector fields with the product X ? Y = ∇X Y , where ∇ is the flat, torsionless connection of the locally affine structure of the manifold. Definition 1. Let X be a vector field on a snooth manifold M , where (U, u) be a local chart of M and {ui } be the local coordinates system associated with it. A vector field X with local representation X = Xi
∂ ∂ui
is called linear relative to {ui } if its local components have a R-linear dependence X i = Cji uj , 1
where Cji ∈ R. Starting with the simplest two-dimensional case, we recall that if smooth vector fields A(x, y) = ax (x, y)∂x + ay (x, y)∂y ,
B(x, y) = bx (x, y)∂x + by (x, y)∂y , (1.1)
together with a function f (x, y) are defined on the Cartesian plane (x, y) then the gradient of the function f (x, y) is a differential 1-form (a covector field): ∂f (x, y) ∂f (x, y) ∇f (x, y) = dx + dy, (1.2) ∂x ∂y a derivative of f (x, y) in the direction of the vector field A(x, y) is a function of the convolution of the gradient ∇f (x, y) and the vector field A(x, y): ∇A f (x, y) = hA(x, y), ∇f (x, y)i = ax
∂f (x, y) ∂f (x, y) + ay . ∂x ∂y
(1.3)
In turn, a derivative of the vector field B(x, y) along a vector field A(x, y) is a vector field: ∇A B = ∇A bx ∂x + ∇A by ∂y . (1.4) Respectively, a derivative of the field A(x, y) in the direction of B(x, y) is a vector field: ∇B A = ∇B ax ∂x + ∇B ay ∂y . (1.5) Finally, the Lie bracket of two vector fields is a vector field: [A, B] = ∇A B − ∇B A = (∇A bx − ∇B ax )∂x + (∇A by − ∇B ay )∂y .
(1.6)
Let all the vector fields considered hereafter be linear [7], i.e. their coordinate functions are linear functions of the coordinates of the Cartesian plane. Then the linear vector fields of the Cartesian plane with respect to the derivative of a vector field in the direction of the vector field form the algebra A∇ , which is isomorphic to the matrix algebra M2 (R) and is invariant under nondegenerate linear transformations of coordinates Cartesian plane. However, the linear tangent vector fields (i.e. linear vector fields without a radial component, which are collinear unit vector field x∂x + y∂y ) with respect to the Lie bracket form an algebra AT , which is isomorphic to the matrix algebra Lie sl2 (R) and is invariant with respect to arbitrary non-degenerate linear transformations of the coordinates of the Cartesian plane. As an illustration of these algebras we construct the algebra of linear vector fields with the structure of the algebra of complex numbers. Take the vector fields: E = x∂x + y∂y , I = y∂x − x∂y , (1.7) 2
for which the following relations hold: E 2 = E ? E = ∇E E = x∂x + y∂y = E, E ? I = I ? E = ∇E I = ∇I E = y∂x − x∂y = I, I 2 = I ? I = ∇I I = −x∂x − y∂y = −E.
(1.8) (1.9) (1.10)
Hence the vector field I is the generator of the algebra of linear vector fields (subalgebras A∇ ), which is isomorphic to the algebra of complex numbers: h(I, ?)iR ∼ = C, and the Lie algebra of linear vector fields (subalgebras AT ), which is isomorphic to the corresponding unitary algebra h(I, [, ])iR ∼ = u(1). The notion of an algebra of linear vector fields on the Cartesian plane A∇ can be easily generalized to the case of the Cartesian space Rn . Indeed, if we take an arbitrary linear vector field: A = a1 ∂x1 + · · · + an ∂xn = ai ∂xi ,
(1.11)
where ai = ai1 x1 + · · · + ain xn , and an arbitrary linear vector field B = b1 ∂x1 + · · · + bn ∂xn = bi ∂xi ,
(1.12)
where bi = bi1 x1 + · · · + bin xn , and define the operation of differentiation of a vector field A in the direction of the vector field B: n X
∇B A =
i=1
i
∇B a ∂xi =
n X i=1
bj
∂ai ∂xi ∂xj
n X ∂(ai1 x1 + · · · + ain xn ) = (b11 x1 + · · · + b1n xn ) ∂xi + · · · ∂x1 i=1
+ (bn1 x1 + · · · + bnn xn ) =
∂(ai1 x1 + · · · + ain xn ) ∂xi ∂xn
n X (ai1 b11 + · · · + ain bn1 )x1 + · · · + (ai1 b1n + · · · + ain bnn )xn ∂xi i=1
=
n X
cij xj ∂xi = ci ∂xi = C, (1.13)
i=1
then, since C = (cij )n = (aij )n (bij )n = AB ∼ = ∇B A = B ? A, we find an equivalence between the algebra of linear vector fields in the Cartesian space Rn and the matrix algebra Mn (R), which is the algebra of endomorphisms of a linear space Rn . Note also that we consider below only subalgebras of linear vector fields. 3
2
Local algebra of vector fields
Let an element of the local algebra of linear vector fields A∇ (Tx0 M n ) be defined on the tangent bundle of the manifold M n through the following formal expression: n X 0 c(x ) = (2.1) cij (x0 )∆x0j ∂x0i , i=1
where (cij )n = C ∈ Mn (R), and Mn (R) — is the matrix algebra of endomorphisms of Rn , ∂x0i ∈ Tx0 M n , and ∆x0j are the coordinates of an arbitrary point in the tangent space Tx0 M n . Then, an element of the local algebra of linear vector fields on the tangent bundle of the space Rn is induced by the non-degenerate map F : Rn 7→ M n : (x → x0 ) from A∇ (Tx0 M n ) to A∇ (Tx0 Rn ). Indeed, since ∂xi ∈ Tx Rn , ∆x0j = Jji (x)∆xi , ∂x0i = Jij−1 (x)∂xj , where Jij (x) is the Jacobian of the map F , then n X c (x) = (J −1 CJ)ij (x)∆xj ∂xi , 0
(2.2)
i=1
with c0 (x) ∈ A∇ (Tx0 Rn ). Conversely, if we already have a local algebra A∇ (Tx0 Rn ) such that n X c0 (x) = c0ij (x)∆xj ∂xi , (2.3) i=1 0
−1
then, by the virtue of C = JC J , this algebra is transferred to A∇ (Tx0 M n ) in the form of the following expression: 0
c(x ) =
n X
(JC 0 J −1 )ij (x0 )∆x0j ∂x0i .
(2.4)
i=1
Besides, in the case of the algebra A being a subalgebra of the matrix algebra of endomorphisms of Rn and for all x0 be fulfilled C(x0 ) ∈ A, some additional conditions should be imposed on the elements of the local algebra A∇ (Tx0 M n ). Indeed, if C(x0 ), C 0 (x) ∈ A for all x0 , x, then J ∈ GA, where GA is the group of invertible elements of the subalgebra A, since only J acts on the algebra A, according to the formula of inner automorphisms of the group GA: C 0 = J −1 CJ. In this case we deal with a global algebra A∇ (M n ) on the locally affine manifold M n . For example, when the condition J ∈ GA is satisfied, and if we take a subalgebra of linear vector fields A∇ (R2 ) with the structure of the algebra of
4
complex numbers, then we obtain the Cauchy-Riemann conditions. In fact, if C 0 (x) ∈ C, then the following condition ! 0 0 ∂x1 ∂x1 ∂x02 ∂x1
J(x) =
∂x1 ∂x2 ∂x02 ∂x2
∈ C\{0},
(2.5)
has to be satisfied, which means that ∂x01 ∂x02 = , ∂x1 ∂x2
∂x01 ∂x0 = − 2. ∂x2 ∂x1
(2.6)
Similarly, if we take a subalgebra of A∇ (R8 ) with the structure of the matrix algebra M4 (C), then by virtue of ! ∂x0 ∂x0 2i−1
2i−1
∂x2j−1 ∂x02i ∂x2j−1
J(x) =
∂x2j ∂x02i ∂x2j
∈ M4 (C)\{0},
(2.7)
8
where i, j = 1, . . . , 4, we obtain the Cauchy-Riemann conditions ∂x02i−1 ∂x02i = , ∂x2j−1 ∂x2j
∂x02i−1 ∂x02i =− . ∂x2j ∂x2j−1
(2.8)
Suppose now we have: x1 = ez cos ϕ, x2 = ez sin ϕ, x1 −x2 0 C (x) = ∈ C. x2 x1
(2.9) (2.10)
Then the map F : R2 7→ R × S 1 : (x1 , x2 ) → (z, ϕ) has the Jacobian cos ϕ sin ϕ −z J(z, ϕ) = e ∈ C\{0}, (2.11) − sin ϕ cos ϕ and, therefore, 0
C(z, ϕ) = JC (x)J
−1
z
=e
cos ϕ sin ϕ − sin ϕ cos ϕ
∈ C.
(2.12)
Thus, a complex plane regarded as a subalgebra of A∇ (R2 ) with the structure of C and a complex cylinder regarded as a subalgebra of A∇ (R×S 1 ) with the structure of C are equivalent to each other, with the correspondence between them being equivalent to a complex-analytical map: z −ϕ cos ϕ sin ϕ z iϕ z z + iϕ → e e : →e , (2.13) ϕ z − sin ϕ cos ϕ 5
where z, ϕ ∈ R, in which the complexification of the covering of the cylinder is mapped onto the complexification of the cylinder itself. Now let a subalgebra of linear vector fields A∇ (M 08 ) be induced by the map F : M 8 7→ M 08 : (zi , ϕi )4 → (zi0 , ϕ0i )4 ) with the Jacobian J(z 0 , ϕ0 ). Then the Cauchy-Riemann equations would be: ∂zi0 ∂ϕ0i = , ∂zj ∂ϕj
∂zi0 ∂ϕ0 = − i, ∂ϕj ∂zj
(2.14)
where i, j = 1, 2, 3, 4. We can also note that, if C 0 (x) belongs to the algebra of dual numbers: x1 x2 0 C (x) = ∈ D, (2.15) x2 x1 then, in order to satisfy the belonging condition C(x0 ) ∈ D, it is required that the following hyperbolic Cauchy-Riemann equations hold: ∂x01 ∂x02 = , ∂x1 ∂x2
∂x01 ∂x02 = . ∂x2 ∂x1
(2.16)
Let us now consider an application of the local algebra of linear vector fields to the case when this algebra is used to represent the Riemann zeta function. First of all, we should note that the flow of the linear vector field on the Cartesian plane which is represented by the matrix a −b C= , (2.17) b a has a spiral shape described by the following parametric equation: x1 (t) = eat+z0 cos(bt + ϕ0 ),
x2 (t) = eat+z0 sin(bt + ϕ0 )
(2.18)
with x1 (0) = ez0 cos ϕ0 and x2 (0) = ez0 sin ϕ0 . On a cylinder, they have a helical shape described as z(t) = at + z0 ,
ϕ(t) = |bt + ϕ0 |2π
(2.19)
with z(t) ˙ = a, ϕ(t) ˙ = b. However, the mapping of a plane onto a torus 2 1 1 F : R 7→ S × S induces a complex analytic map: z + iϕ → |ez |2 eiϕ = |ρ|2 eiϕ ,
(2.20)
where z, ϕ ∈ R, ρ ∈ R+ , |ρ|2 ∈ [0, 2[, |ρ|2 : ρ0 ≡ ρ (mod 2), or a complex analytical map: z + iϕ → |ez |±1 eiϕ = ρ˜ eiϕ , (2.21) 6
where ρ˜ = |ez |±1
( |ez |2 if |ez |2 ∈ [0, 1] = |ez |2 − 2 if |ez |2 ∈]1, 2[.
(2.22)
The above sawtooth-shaped function can be represented by a Fourier series: |x|±1 =
∞ 2 X (−1)k+1 sin(kx). π 1 k
(2.23)
In this case, a helical trajectory on the cylinder: z(t) = at,
ϕ(t) = |bt|2π ,
(2.24)
is transformed into a spiral on the torus: ρ˜(t) = |eat |±1 ,
ϕ(t) = |bt|2π ,
(2.25)
whereas the sum ∞ ∞ ∞ X X 1 tn −atn 2 X (−1)k+1 ibtn S(a, b) = |e |±1 e = sin(kn1−a )nib , (2.26) n πn k=1 k 1 n=1
with tn = ln(n) represents the Riemann zeta function ζ(s). Indeed, since ∞ X 1 , S(s > 1) = ns 1
(2.27)
where s = a − ib, then we only have to prove the convergence of the sum S(s < 1). This sum is convergent because of the fact that |S(s < 1)| is smaller than the sum of a harmonic series because 1 (1−s)t 1 n |e |±1 < (2.28) n n is satisfied almost always. Another example of using the local algebra of linear vector fields is the mathematical derivation of the equation of the vibrations of a cylindrical string. Proposition 1. Let curvilinear coordinates on an infinite cylinder satisfy the Cauchy-Riemann equations: ∂z 0 ∂ϕ0 = , ∂z ∂ϕ
∂z 0 ∂ϕ0 =− , ∂ϕ ∂z 7
(2.29)
and let a function ψ(x, t) = z 0 (x, t) + ϕ0 (x, t) be defined with x = e−ε z − eε ϕ,
t = e−ε z + eε ϕ
(2.30)
in such a way that z 0 (x, t) ∼ = 1 and ϕ0 (x, t) ∼ = i. That is, the function ψ(x, t) and its partial derivatives are equivalent to an element 1 + i of the algebra of complex numbers. Then the function ψ(x, t) = z 0 (x, t) + ϕ0 (x, t) satisfies the equation ∂ 2ψ ∂ 2ψ eiα 2 + ei(α+π) 2 = 0, (2.31) ∂t ∂x where α = sinh 2ε. Proof. Indeed, as the following identities hold: ∂z 0 ∂z 0 ∂z 0 = e−ε + e−ε , ∂z ∂x ∂t ∂z 0 ∂z 0 ∂z 0 = eε − eε , ∂ϕ ∂t ∂x ∂ϕ0 ∂ϕ0 ∂ϕ0 = e−ε + e−ε , ∂z ∂x ∂t ∂ϕ0 ∂ϕ0 ∂ϕ0 = eε − eε , ∂ϕ ∂t ∂x
(2.32) (2.33) (2.34) (2.35)
then the following conditions: ∂ϕ0 ∂z 0 ∂ϕ0 ∂z 0 − e−ε = eε + e−ε , ∂x ∂x ∂t ∂t ∂z 0 ∂ϕ0 ∂z 0 ∂ϕ0 −e−ε + eε = e−ε + eε ∂t ∂t ∂x ∂x eε
(2.36)
are satisfied. Then, after differentiating the first equation with respect to t, and the second – with respect to x, we obtain a system of differential equations: ∂ 2z0 ∂ 2 ϕ0 ∂ 2z0 ∂ 2 ϕ0 − e−ε = eε 2 + e−ε 2 , ∂x∂t ∂x∂t ∂t ∂t 2 0 2 0 2 0 2 0 ∂ z ∂ ϕ ∂ z ∂ ϕ −e−ε + eε = e−ε 2 + eε 2 , ∂t∂x ∂t∂x ∂x ∂x eε
(2.37)
which is equivalent to the system: c¯
∂ 2ψ ∂ 2ψ =c 2, ∂x∂t ∂t
∂ 2ψ ∂ 2ψ −¯b = b 2, ∂t∂x ∂x 8
(2.38)
where c = cosh ε + i sinh ε, b = sinh ε + i cosh ε. By multiplying the first equation of the system (2.38) by c, the second equation by — b, and then by adding the resulting equations to each other we obtain the equation of oscillations of a cylindrical string (2.31) if we take into account the fact that c2 = ei sinh 2ε , b2 = ei(sinh 2ε+π) . Corollary 1. In the case of sinh 2ε = 2πk, where k ∈ Z, the equation of an oscillating cylindrical string (2.31) has the form: ∂ 2ψ ∂ 2ψ − = 0. ∂t2 ∂x2
3
(2.39)
The geometry of the algebra of vector fields
First of all, let us turn to the question of the representation of multidimensional spaces that preserve quadratic forms. If only one quadratic form is preserved in a n-dimensional linear space, this form takes the canonical form: x21 + · · · + x2k − x2k+1 − · · · − x2m ,
(3.1)
where n − m = d, m − k = l. Then, following the approach of [6], we can represent this space as a semi-Euclidean space with defect d and index l, which is denoted as l+{d} Rn . In the zero-defect case we are dealing with the Euclidean spaces of index l. However, if we impose the condition of preserving several quadratic forms in the n-dimensional linear space, the representation becomes ambiguous. Firstly, we have to distinguish the condition of simultaneous preservation of quadratic forms and the condition of the preservation of these forms separately from each other. For example, if, in a 2-dimensional linear space, either the form x21 + x22 or the form x21 − x22 is preserved, then we consider our space as a system of two incompatible Euclidean spaces {1 R2 , R2 }. However, if a system of equations x21 + x22 = const,
x23 + x24 = const
(3.2)
should be satisfied in the 4-dimensional linear space then we consider this space as a joint system of two semi-Euclidean spaces {{2} R4 ,{2} R4 }. Secondly, it should be realised that a joint system of equations corresponds to the whole class of equivalence. For example, the system (3.2) is equivalent to the system x21 + x22 − x23 − x24 = const,
x21 + x22 + x23 + x24 = const,
9
(3.3)
which corresponds to the representation of a joint system of Euclidean spaces {2 R4 , R4 } and, hence, the representations {2{2} R4 } ∼ = {2 R4 , R4 } are equivalent. Similarly, due to the equivalence of the system x21 + x22 = const, x25 + x26 = const,
x23 + x24 = const, x27 + x28 = const,
(3.4)
and the system x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 x21 + x22 − x23 − x24 + x25 + x26 − x27 − x28 x21 + x22 + x23 + x24 − x25 − x26 − x27 − x28 x21 + x22 − x23 − x24 − x25 − x26 + x27 + x28
= const, = const, = const, = const,
(3.5)
the representations {4{6} R8 } ∼ = {R8 , 34 R8 } are equivalent too. We let us consider an algebra A∇ (R2 ) of the linear vector fields with the structure of the algebra of complex numbers C. The geometric structure of this algebra is defined by its generating vector fields, namely, by the radial and tangent vector fields E = x∂x + y∂y ,
I = y∂x − x∂y .
(3.6)
Since the vector field I is orthogonal to E in the metric of the Euclidean plane (x, y) and, therefore, it is tangent to the circles x2 + y 2 = const, then in this case A∇ (R2 ) — is the algebra of linear vector fields of the Euclidean plane R2 . Similarly, the geometric structure of algebra A∇ (1 R2 ) of the linear vector fields with the structure of the algebra of dual numbers D is determined by its generating vector fields, namely, by the radial and tangent vector fields: E = x∂x + y∂y ,
J = y∂x + x∂y ,
(3.7)
where the vector field J is orthogonal to E in the metric of the pseudoEuclidean plane (x, y), and, hence, it is tangent to an equilateral hyperbola x2 − y 2 = const. Thus, in this case A∇ (1 R2 ) — is the algebra of linear vector fields of the pseudo-Euclidean plane 1 R2 . In turn, the geometric structure of the algebra A∇ (R2 ) of the linear vector fields with the structure of the matrix algebra M2 (R) is determined by its generating vector fields, namely, by the radial and tangent vector fields E = x∂x + y∂y , I = y∂x − x∂y , J = y∂x + x∂y , IJ = x∂x − y∂y , 10
(3.8)
where the vector field IJ is orthogonal to E in the metric of the pseudoEuclidean plane (x + y, x − y), and, hence, it is tangent to an equilateral hyperbola xy = const. Thus, in this case A∇ (R2 ) — is the algebra of linear vector fields of a system of planes {21 R2 , R2 } ∼ = R2 . If referring to the Lie algebras AT , we it should note that since a set of elements of this algebra is obtained by excluding the radial component of E from the algebra A∇ , then the geometry of the Lie algebra AT is completely determined by the geometry of the corresponding tangent vector fields. In particular, if we consider the Lie algebra of the tangent vector fields of the algebra A∇ (R4 ) with the structure of the algebra of quaternions H, then we would find that these vector fields are tangent to hyperspheres of the 4-dimensional Euclidean space R4 . Indeed, in this case AT (R4 ) = hiσ1 , iσ2 , iσ3 iR = su(2),
(3.9)
where σj are the Pauli matrices, and iσj is the matrix representation of vector fields x4 ∂x1 − x3 ∂x2 + x2 ∂x3 − x1 ∂x4 , x3 ∂x1 + x4 ∂x2 − x1 ∂x3 − x2 ∂x4 , x2 ∂x1 − x1 ∂x2 − x4 ∂x3 + x3 ∂x4 ,
iσ1 : iσ2 : iσ3 :
(3.10)
each of which is orthogonal to the radial vector field σ0 :
x1 ∂x1 + x2 ∂x2 + x3 ∂x3 + x4 ∂x4
(3.11)
in the metric of the Euclidean space R4 . So, in this case A∇ (R4 ) is the algebra of linear vector fields in the Euclidean space R4 . Similarly, if
AT ({3{2} R6 }) = [iσj ]k R = su(3), (3.12) where [iσj ]k is the matrix representation of vector fields with zero components ∂x2k−1 and ∂x2k , namely [iσ1 ]1 [iσ1 ]2 [iσ1 ]3 [iσ2 ]1 [iσ2 ]2 [iσ2 ]3 [iσ3 ]1 [iσ3 ]2 [iσ3 ]3
: : : : : : : : :
x6 ∂x3 − x5 ∂x4 + x4 ∂x5 − x5 ∂x6 , x6 ∂x1 − x5 ∂x2 + x2 ∂x5 − x1 ∂x6 , x4 ∂x1 − x3 ∂x2 + x2 ∂x3 − x1 ∂x4 , x5 ∂x3 + x6 ∂x4 − x3 ∂x5 − x4 ∂x6 , x5 ∂x1 + x6 ∂x2 − x1 ∂x5 − x2 ∂x6 , x3 ∂x1 + x4 ∂x2 − x1 ∂x3 − x2 ∂x4 , x4 ∂x3 − x3 ∂x4 − x6 ∂x5 + x5 ∂x6 , x2 ∂x1 − x1 ∂x2 − x6 ∂x5 + x5 ∂x6 , x2 ∂x1 − x1 ∂x2 − x4 ∂x3 + x3 ∂x4 , 11
(3.13)
wherein [iσ3 ]1 + [iσ3 ]3 = [iσ3 ]2 ,
(3.14)
it is plain to verify that each of these vector fields is tangent to a hypersphere of the space {2} R6 , i.e. of a 6-dimensional semi-Euclidean space of defect 2. Thus, in this case A∇ ({3{2} R6 }) is an algebra of linear vector fields of a system of semi-Euclidean spaces {3{2} R6 }. In turn, if AT ({2{2} R4 }) = h(σj )3 , (iσj )3 iR = sl2 (C),
(3.15)
then the vector fields represented by the matrix (iσj )3 , are tangent to hypersphere in the Euclidean space R4 , and the vector fields represented by the matrix (σj )3 , are tangent to hyperspheres of the space 2 R4 , i.e., the Euclidean spaces of index 2 which are, in other words, 4-dimensional pseudo-Euclidean spaces with the metric signature (2, 2). In this case A∇ ({2{2} R4 }) is the algebra of linear vector fields of a system of the Euclidean spaces {2 R4 , R4 } or of an equivalent system of the semi-Euclidean space {2{2} R4 }. Finally, let
AT ({4{6} R8 }) = γ0 , iγ0 , γ[1,2,3] , iγ[1,2,3] , γ0 γ[1,2,3] , iγ0 γ[1,2,3] R = sl4 (C), (3.16) where γ[1,2,3] is a set of seven γ-Dirac matrices (γ1 , γ2 , γ3 , γ1 γ2 , γ1 γ3 , γ2 γ3 , γ1 γ2 γ3 ). Then one can notice that a half of this algebra, namely the vector fields represented by the matrices γ[1,2,3] , iγ0 , iγ0 γ[1,2,3] , is tangent to hyperspheres in the Euclidean space R8 , while the other half of these vector fields, namely the vector fields represented by the matrices iγ[1,2,3] , γ0 , γ0 γ[1,2,3] are tangent to hyperspheres in the Euclidean space 4 R8 . Thus, the geometric structure which would be interesting from the point of view of physical applications of the algebra of linear vector fields A∇ with the structure of the matrix algebra M4 (C) is closely related to the geometry of the Euclidean spaces R8 and 4 R8 . More precisely, in this case A∇ ({4{6} R8 }) is the algebra of linear vector fields of a system of the Euclidean spaces {34 R8 , R8 } or of an equivalent system of semi-Euclidean spaces {4{6} R8 }. Moreover, since γ5 · γα + γα · γ5 = 0, (3.17) wherein α = 0, 1, 2, 3, γ5 = iγ0 γ1 γ2 γ3 , then, in a 4-dimensional vector space hγ5 γ0 , γ5 γ1 , γ5 γ2 , γ5 γ3 iR = t∗ γ5 γ0 + x∗ γ5 γ1 + y ∗ γ5 γ2 + z ∗ γ5 γ3 ,
(3.18)
a quadratic metric form s∗2 = −t∗2 + x∗2 + y ∗2 + z ∗2 . 12
(3.19)
is induced by the square (t∗ γ5 γ0 + x∗ γ5 γ1 + y ∗ γ5 γ2 + z ∗ γ5 γ3 )2 = (−t∗2 + x∗2 + y ∗2 + z ∗2 )E,
(3.20)
where E is the identity matrix. At the same time, in a 4-dimensional vector space hγ0 , γ1 , γ2 , γ3 iR = tγ0 + xγ1 + yγ2 + zγ3 , (3.21) a quadratic metric form s 2 = t 2 − x2 − y 2 − z 2 ,
(3.22)
(tγ0 + xγ1 + yγ2 + zγ3 )2 = (t2 − x2 − y 2 − z 2 )E.
(3.23)
is induced by the square
The sum of these two squares induces a doublet (a Finsler product) of the Minkowski spaces with the metric form S 2 = t2 − x2 − y 2 − z 2 − t∗2 + x∗2 + y ∗2 + z ∗2 .
(3.24)
Then, if we pass to the dual representation: τ0 = γ0 + γ5 γ0 , τ2 = γ2 + γ5 γ2 ,
τ1 = γ1 + γ5 γ1 , τ3 = γ3 + γ5 γ3 ,
(3.25)
τ0∗ = γ0 − γ5 γ0 , τ2∗ = γ5 γ2 − γ2 ,
τ1∗ = γ5 γ1 − γ1 , τ3∗ = γ5 γ3 − γ3 ,
(3.26)
then the sum of four squares induces a metric form of the hyperbolic type, namely: (T τ0 + T ∗ τ0∗ )2 + (Xτ1 + X ∗ τ1∗ )2 + (Y τ2 + Y ∗ τ2∗ )2 + (Zτ3 + Z ∗ τ3∗ )2 = T T ∗ + XX ∗ + Y Y ∗ + ZZ ∗ = t2 − x2 − y 2 − z 2 − t∗2 + x∗2 + y ∗2 + z ∗2 . (3.27) As an example of a space which does not allow building an algebra of linear vector fields (due to the lack of tangent vector fields on the evendimensional spheres), we can consider the Euclidean space R3 . Here we should note that the system of the semi-Euclidean space {3{1} R3 }, which corresponds to the system of equations x21 + x22 = const,
x22 + x23 = const,
x21 + x23 = const,
(3.28)
already allows building an algebra of linear vector fields, its Lie algebra being isomorphic to the algebra of su(2). 13
4
The dynamics of the algebra of vector fields
As a simple example, a dynamic representation of linear vector fields we turn to the local algebra of linear vector fields with the structure of the algebra of complex numbers. Since, by Cauchy-Riemann equations on coordinate functions x0i (x) harmonic functions condition imposed: ∂ 2 x0i ∂ 2 x0i + = 0, ∂x21 ∂x22
(4.1)
the setting to the plane (x, y) potential u(x, y), which satisfies the differential equation: ∂ 2 u(x, y) ∂ 2 u(x, y) + = 0, (4.2) ∂x2 ∂y 2 we will be treated as a dynamic representation of the local algebra of linear vector fields with the structure of the algebra of complex numbers. Similarly, in the equations of Cauchy-Riemann local algebra of linear vector fields with the structure of the matrix algebra M4 (C) has an 8-dimensional manifold with cylindrical coordinates (zi , ϕi )4 dynamic view as potential u(z, ϕ), which satisfies the system of equations: ∂ 2 u(z, ϕ) ∂ 2 u(z, ϕ) + = 0. ∂zi2 ∂ϕ2i
(4.3)
However, to revert to the original coordinates (xj )8 , then the system would have to have the same ∆i =
∂ 2 u(x) ∂ 2 u(x) + = 0, ∂x22i−1 ∂x22i
(4.4)
or an equivalent type of ∆1 + ∆2 + ∆3 + ∆4 ∆1 − ∆2 + ∆3 − ∆4 ∆1 + ∆2 − ∆3 − ∆4 ∆1 − ∆2 − ∆3 + ∆4
= 0, = 0, = 0, = 0,
(4.5)
which implies that the potential for u(x) should be imposed a harmonic function condition in Euclidean space and pseudo-Euclidean spaces with metric signature (4, 4). It should also be noted that although we have local (differential) the conditions to be satisfied by the potential, but we do not have global (integral) conditions that from a physical point of view would have 14
been more interesting. Let us try, however, to guess the vacuum solution of the Laplace equation. First of all, note that since the function f (x+iy) p = ln ρ+iϕ, where i is the generator of the algebra of complex numbers, ρ = x2 + y 2 , ϕ = arctan xy , in the sense of performing analytic Cauchy-Riemann conditions, and the function f (x + jy) = ln ρ + jϕ, where j isqthe generator of the algebra of p double numbers, ρ = x2 − y 2 , ϕ = ln x+y , in the sense of performx−y ing analytic hyperbolic Cauchy-Riemann conditions, the functions of ln ρ and ϕ are harmonic or Euclidean metric, metric or pseudo-Euclidean plane. Furthermore, it should be noted thatif the Cartesianq(linear) coordinates p x+y 0 0 2 2 (x , y ), defined in the polar modified ln x − y , ln x−y identities using {(x0 , 0), (0, y 0 )} = ϕ = 0)}, the coordinates {(ln ρ, p associated with the y 2 2 cylindrical manifold X = ln x + y , Y = arctan x by the relation x0 + y 0 = X,
x0 − y 0 = Y,
(4.6)
then we can talk about the two coordinate systems of the same cylindrical manifold R × S 1 , homeomorphic to the plane punctured. Thus, since the functions of: 1 ln(x21 + x22 ), 2 1 Y = ln(x23 + x24 ), 2 1 Z = ln(x25 + x26 ), 2 1 T = ln(x27 + x28 ), 2
x2 , x1 x4 ϕY = Y ∗ = arctan , x3 x 6 ϕZ = Z ∗ = arctan , x5 x8 ϕT = T ∗ = arctan x7
ϕX = X ∗ = arctan
X=
(4.7) (4.8) (4.9) (4.10)
satisfy the equations of Cauchy-Riemann, each of these functions will satisfy the system of equations of Laplace, and the values of these functions are equal to the corresponding coordinates of the linear subspace hτ0 , τ1 , τ2 , τ3 , τ0∗ , τ1∗ , τ2∗ , τ3∗ iR algebra M4 (C), from which the induced quadratic metric form S 2 = t2 − x2 − y 2 − z 2 − t∗2 + x∗2 + y ∗2 + z ∗2
(4.11)
space 4 R8 , containing a doublet Minkowski spaces, namely 3 R4 and 1 R4 .
15
Thus, a change of variables: T = t + t∗ , X = x∗ + x, Y = y ∗ + y, Z = z ∗ + z,
T ∗ = t − t∗ , X ∗ = x∗ − x, Y ∗ = y ∗ − y, Z ∗ = z ∗ − z,
(4.12)
this quadratic form is given to the hyperbolic form: S 2 = T T ∗ + XX ∗ + Y Y ∗ + ZZ ∗ ,
(4.13)
where the coordinates of the (X, X ∗ , Y, Y ∗ , Z, Z ∗ , T, T ∗ ) are isotropic coordinates space 4 R8 , and the scalar product associated with this hyperbolic form, is as follows: 2ab = aT bT ∗ + aT ∗ bT + aX bX ∗ + aX ∗ bX + aY bY ∗ + aY ∗ bY + aZ bZ ∗ + aZ ∗ bZ . (4.14) Couple isotropic coordinates pseudo-Euclidean plane is the simplest system of coordinates corresponding cylinder R × S 1 or torus S 1 × S 1 , obtained by compactification of the plane as a result of folding respectively one or two isotropic coordinate. However, full compactification (compactification of two isotropic coordinate) of all pseudo-space planes 1 R4 turns it into a toroidal manifold S 3 × S 1 . Indeed, since the space of 1 R4 is given by the set of pseudo-planes, such as 1 R4 , where the bar code coordinate x0∗ is the index set of tsentalnosimmetrichnyh direct Euclidean space (x∗ , y ∗ , z ∗ ), ie, he runs the projective plane RP 2 , then we have the topological equivalence of 1 R4 ∼ = 2 1 1 2 1 1 ∼ RP × R2 , and the compactified space R4 homeomorphic RP × S × S = S 3 × S 1 . Note here that as a result of shrinking the space 1 R4 , it turns into a cone isotropic two adjoining spheres S 3 . Similarly, we can be compactified Minkowski space and the (t, x, y, z). In turn, as 4 R8 ∼ = RP 3 × RP 3 ×1 R2 , the result is shrinking doublet Minkowski spaces we obtain the product of spheres RP 3 ×RP 3 ×S 1 ×S 1 ∼ = S 4 ×S 4 , while the isotropic cone of Euclidean 4 space R8 compactified in two contiguous space RP 3 × RP 3 × S 1 . From the point of view of mathematical modeling of physical phenomena, space 4 R8 is of particular interest. Indeed, suppose that in this space given a dynamic flow of a continuous medium, the vector field v of velocities of particles whose divergence-free everywhere. However, the vector field v can have topological features, the flow line of which is compactified (in the simplest case to a circle) on the corresponding toroidal manifold, which occurs when v2 = 0. In addition, even if the line integral along the way: Z S = v∗ dl (4.15) L
16
the scalar product covectorial field v∗ velocities of particles of the medium without the vector field features a differential path element dl, traversed feature vector field in the space of 4 R8 , measured in this way the angular measure of the topological features of proper rotation caused by the action of the flow of particles. Then the line integral can be compared with the action of a material point in the interval of the world line of the Minkowski space 3 R4 , obtained by orthogonal projection of the trajectory of the 4 R8 to 3 R4 . On the other hand, if in the space of 4 R8 Minkowski subspace 3 R4 is not defined globally (by one basis), but only locally (by the reference bundle of 4dimensional manifold), we can define the covariant derivative of a vector field tangent to the path specified in the 4 R8 along the lines of the corresponding global manifold as the orthogonal projection of the derivative of this field on a given manifold. In addition, let covector particles flow velocity field potential, ie, v∗ = ∇u. Then, demanding implementation of the variational equation Z δ ∇2 ud8 x = 0, (4.16) providing for uniform flow, we obtain a necessary condition in the form of the differential equation ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u ∂ 2u − 2 − 2 − 2 = ∗2 − ∗2 − ∗2 − ∗2 . ∂t2 ∂x ∂y ∂z ∂t ∂x ∂y ∂z
(4.17)
We should not however forget that each pair of coordinates (t, t∗ ), (x, x∗ ), (y, y ∗ ), (z, z ∗ ) is a cylindrical coordinate system or toroidal manifold, and therefore the equation 4.17 should be imposed condition of periodicity of the potential gradient ∇u(T, X, Y, Z, T ∗ , X ∗ , Y ∗ , Z ∗ ) on the isotropic coordinates with an asterisk or all isotropic coordinates. We note here that in isotropic coordinates, the equation 4.17 takes the form: ∇2 u ∇2 u ∇2 u ∇2 u + + + = 0. ∂T ∂T ∗ ∂X∂X ∗ ∂Y ∂Y ∗ ∂Z∂Z ∗
(4.18)
In turn, a stronger requirement of stability (equilibrium) of the flow, which amounts to a locally extremal line integral along an arbitrary path: Z δS = δ ∇udl = 0, (4.19) L
leads to the necessary condition, which consists in the fact that the surface level of the potential function u(x) should be minimal in the space of 4 R8 , 17
that is, they must have zero mean curvature. In other words, the variational equation 4.19 has a solution given by the differential equation d ? n(x) = 0,
(4.20)
∇u(x) where n(x) = |∇u(x)| , and the asterisk denotes the Hodge operator. However, if we are interested in the dynamics characteristics of the vector field, we should refer to the same variational equation 4.19, but on the condition that varies in space trajectory characteristics 4 R8 , and the gradient of the potential role played by the external field. Let us now consider a classical limit when from the standpoint of global observer the part of isotropic coordinates are fixed
t − t∗ = 0,
x + x∗ = 0,
y + y ∗ = 0,
z + z∗ = 0
(4.21)
therefore are independent only the isotropic coordinates T ∗ = t − t∗ = 2t, Y = y − y ∗ = 2y,
X = x − x∗ = 2x, Z = z − z ∗ = 2z
(4.22)
which are related by metric of Minkowski space. Then in space (T ∗ , X, Y, Z) the feature dynamics is given by variational equation Z Z δ m Gds + q Ads = 0, (4.23) S
S
where S — the world especially of feature, m, q — characteristics of feature, as well as the gravitational and electromagnetic vector fields are the projection of the gradient vector field ∇u on the subspace corresponding to ∂u ∂u ∂u ∂u ∗ , , , , G(T , X, Y, Z) = ∂t ∂x ∂y ∂z (4.24) ∂u ∂u ∂u ∂u 0 0 0 0 A(t , x , y , z ) = , , , ∂t∗ ∂x∗ ∂y ∗ ∂z ∗ Let us now look at the space and time through the eyes of a local observer. ∗ Let in the space 4 R8 are such coordinate functions of t0 , x0 , y 0 , z 0 that |vv∗ | = 0 ∇y 0 ∇t0 ∇t ∇x0 ∇z 0 , and tetrad of , , , is a local orthonormal covector |∇t0 | |∇t0 | |∇x0 | |∇y 0 | |∇z 0 | basis of a pseudo-Rieman manifold (t0 , x0 , y 0 , z 0 ), the metric that is induced from the 4 R8 on the condition that the time coordinate t0 parameterized by own time of local observer τ , which is measured by a isotropic coordinate of 18
the plane (t, t0 ), and the spatial coordinates parameterized by own length l of the local observer, which is measured by a other isotropic coordinate of the plane (t, t0 ). Then the ratio of own length to own time is a variable equal to the absolute speed of light. So, when measured in a plane (t, t∗ ) deviation ∇t0 ∇t l −2ϕ of |∇t and 0 | from |∇t| in the hyperbolic angle ϕ, we obtain the ratio τ = e 0 0 0 0 2 2ϕ 2 −2ϕ in the manifold (t , x , y , z ) is induced metric is ds = e dt − e dx2 − e−2ϕ dy 2 − e−2ϕ dz 2 , which is due to nearly equal e2ϕ ≈ 1 + 2ϕ corresponds to the field limit of the gravitational potential ϕ. Thus, the potential function u0 (x1 , . . . , x8 ) = tϕ =
e−ϕ ∗ eϕ (T + X + Y + Z) + (T + X ∗ + Y ∗ + Z ∗ ) (4.25) 2 2
are encouraged to take as a vacuum without considering the potential of local (field) changes its shape, and the parameter ϕ — be held responsible for global (evolutionary) change the shape of the vacuum potential. Note also that the formal potential of the vacuum can be represented by a complex analytic function u¯0 (x1 , . . . , x8 ) =
e−ϕ ∗ eϕ (T + X + Y + Z) + i (T + X ∗ + Y ∗ + Z ∗ ). (4.26) 2 2
In this case, we simply assume that the u0 (x1 , . . . , x8 ) = Re u¯0 (x1 , . . . , x8 ) + Im u¯0 (x1 , . . . , x8 ). However, if in addition to linear harmonic functions to consider harmonic functions formed a linear combination of the real and imaginary components of the complex linear fractional functions, they can be treated as a particle-like solutions of the Laplace equation. For example, the function 1 , (4.27) u¯(x1 , . . . , x8 ) = u¯0 + ρ + iρ∗ p p wherein ρ = x2 + y 2 + z 2 , ρ∗ = x∗2 + y ∗2 + z ∗2 , with the proviso that x = T − X + Y − Z, x∗ = T ∗ − X ∗ + Y ∗ − Z ∗ , y = T + X − Y − Z, y ∗ = T ∗ + X ∗ − Y ∗ − Z ∗ , z = T − X − Y + Z, z ∗ = T ∗ − X ∗ − Y ∗ + Z ∗ ,
(4.28)
can be considered an extension of the complex Newtonian potential of a material point. On the other hand, let the radius of the sphere q ρ = x21 + x22 + x23 + x24 + x25 + x26 + x27 + x28 , 19
the pseudo-radius R=
q
x21 + x22 + x23 + x24 − x25 − x26 − x27 − x28 ,
and hyperbolic angle ϕ = ln Rρ defines a complex potential function u¯(x1 , . . . , x8 ) = ln R + iϕ.
(4.29)
Then, we believe that we are dealing with an evolving universe vacuum potential, where the parameter evolution τ coincides with the hyperbolic angle ϕ. And since it is implicitly assumed that the radius of the sphere of ρ is a constant, independent of the evolution parameter τ , then τ = ln
1 . R
(4.30)
We note here that the surface level of the potential function u(x) = ln R is homeomorphic proivedeniyu S 3 × R4 . In conclusion, it should be noted that all these physical analogy require further study, including by expanding the space of partial solutions of our system of equations Laplace. It is also necessary to pay attention to the transformation of a complex vector ψ1 = ξx + iξt∗ , ψ = ξ + iξ , 2 y z (4.31) ψ= ∗ ψ = ξ + iξ 3 x t, ψ4 = ξy∗ + iξz∗ , where (ψ ∗ γ0 , ψ) = ξ 2 = ξt2 − ξx2 − ξy2 − ξz2 − ξt2∗ + ξx2∗ + ξy2∗ + ξz2∗ = 1, −1, 0, (4.32) which induces an isometric transformations of a real linear space 4 R8 . We note here that under Lorentz transformations of Minkowski space (t, x, y, z) we have bispinor representation realized in the algebra of vector fields tangent space 4 R8 (which is equivalent to the algebra of tangent vector fields of {2 R4 , R4 } isomorphic sl2 (C)), while relatively rotations of Euclidean space (x, y, z) is reduced to a representation of the spinor representation, implemented in the algebra of vector fields tangent space R4 isomorphic su(2). Thereby bispinor ψ has the meaning complexified direction vector congruence of straight parallel trajectories of features, and in combination with the 20
rotation angle (phase action) features from it is possible to construct the wave function bispinor congruence trajectories features Ψ = ψeiS , where S — action features, depending on its trajectory. Moreover, if we consider the vector ξ as a vector-valued functions ξ(t, x, y, z) with domain in Minkowski space, which delivers extremum of the functional Z ξ(x) · ∇ξ(x)d4 x, (4.33) and hence is a solution of the variational equation: Z δ ξ · ∂t ξ t + ∂x ξ x + ∂y ξ y + ∂z ξ z d4 x = 0,
(4.34)
where ξ t , ξ x , ξ y , ξ z — values of the corresponding tangent vector fields of space 4 R8 to point ξ, namely: 4 R8 ξ, : ξ t = γ0 ξ, ξ x = γ1 ξ, ξ y = γ2 ξ, ξ z = γ3 ξ, where γ0 , γ1 , γ2 , γ3 — real Dirac gamma matrices, then a complex vector function ψ(t, x, y, z) satisfies the Dirac equation. On the other hand, since the features are compact geometric objects (circle, 8-shaped twisted circle, tori, etc.) lying on the compactified isotropic cone of Euclidean space 4 R8 , then it makes sense to investigate symmetries of these compact geometry objects for compliance with global gauge symmetries of the so-called Standard Model of elementary particles. We note here that an arbitrary circle is associated with an arbitrary complex number of unit length, two-dimensional torus (the product of two circles) is associated with a pair of unit complex numbers, three-dimensional torus — with the triple unit complex numbers, and proper rotation of the unit circle z = eiϕ can be presented as action of group U (1). However, if the rotation of a iϕ1 iϕ2 pair of unit circles z1 = e , z2 = e representation of the action of the orthogonal group SOR (2), then an arbitrary composition of rotation of the pair of circles and their own rotations represented by the action of unimodular diagonal all the matrix elements which are of modulus unity, iα iαmatrices, 1 2 ie diag e , e , under the condition α1 + α2 = 2πk, the unitary transformation belongs to the group SU (2). At the same time, an arbitrary com position of the rotations of three unit circle z1 = eiϕ1 , z2 = eiϕ2 , z3 = eiϕ3 , represented by theaction of SOR (3), and their own rotations represented by the action of diag eiα1 , eiα2 , eiα3 , where α1 + α2 + α3 = 2πk, belongs to the group of unitary transformations SU (3). Thus, given identically n2 − 1 =
(n − 1)n (n − 1)n + (n − 1) + , 2 2
(4.35)
meaning equal dimensions groups SU (n) and the group consisting of elements of the three products SU (n), where the first element and the third 21
belongs SOR (n), and the second element belongs to a group to the group diag eiα1 , . . . , eiαn , under the condition α1 + · · · + αn = 2πk, it can be concluded, that all these rotation circle, 2-torus and 3-torus form a group U (1) × SU (2) × SU (3), which coincides with the group of unbroken symmetry model, which describes both the strong, weak and electromagnetic interactions. Thus, a group of unbroken symmetry describes the symmetry of the circle, 2-torus and 3-torus lying on the compactified isotropic cone of the space 1 R4 , that is, on the two touching spheres S 3 . In general, it should be understood that the isotropic coordinates (T, X, Y, Z) of Minkowski space (t, x, y, z) as well as the isotropic coordinates (T ∗ , X ∗ , Y ∗ , Z ∗ ) of space (t∗ , x∗ , y ∗ , z ∗ ), wherein t=T x=T y=T z=T
+X +Y −X +Y +X −Y −X −Y
+ Z, − Z, − Z, + Z,
x∗ = T ∗ + X ∗ + Y ∗ + Z ∗ , x∗ = T ∗ − X ∗ + Y ∗ − Z ∗ , y∗ = T ∗ + X ∗ − Y ∗ − Z ∗, z∗ = T ∗ − X ∗ − Y ∗ + Z ∗,
(4.36)
compactified, but as a result of the evolution of the vacuum potential (due to the presence in it of multipliers eϕ e−ϕ ) diameter of one compactified isotropic cone increases, the other decreases. At the same time, the vacuum potential is transformed not only globally — resulting in the formation and evolution of its global minimal surfaces level, but also locally — or as a result of the formation and motion of its local minimal surfaces level without features (fields), or as a result formation and motion of its local minimal surfaces level with singularities (particles), or as a result of random fluctuations in the dynamic flow of a continuous medium.
References [1] Tyshkevich R.I., Fedenko A.S., Linear Algebra and Analytic Geometry (in Russian), Vyisheyshaya Shkola, Minsk 1976 [2] Gelfand I.M., Lectures on Linear Algebra, Dover Publications, New York 1998 [3] Vinberg E.B., A Course in Algebra, Graduate Studies in Mathematics, Vol. 56, AMS 2003 [4] Gorbuzov V.N., Mathematical Analysis: Field Theory (in Russian), Grodno State University, Grodno 2000. [5] Efimov N.V., Rozendorn E.R., Linear Algebra and Multidimensional Geometry, Mir, Moscow 1975. 22
[6] Rosenfeld B.A., Non-Euclidean Geometries, Gostekhizdat, Moscow 1955. [7] Novikov S.P., Taimanov I.A., Modern Geometric Structures and Fields, Graduate Studies in Mathematics, vol. 71, AMS 2006. [8] Hestenes D., Space-Time Algebra, Gordon & Breach, New York 1966. [9] Kassandrov V.V., Algebraic Structure of Space-Time and Algebrodynamics (in Russian), People Fried. Univ. Press, Moscow 1992. [10] Rumer Y.B., Fet A.I., The theory of unitary symmetry (in Russian), Nauka, Moscow 1970. [11] Pavlov D.G., Generalization of Scalar Product Axioms, Hypercomplex Numbers in geometry and Physics, 1 (2004) 5–18 [12] Siparov S.V., On the Anisotropic Geometrodynamics (in Russian), Hypercomplex Numbers in Geometry and Physics 10 (2008) 64–75 Igor Bayak Khimvolokno Technology Complex JSC Grodno Azot 100 Kosmonavtov, Grodno 230013, Belarus E-mail address:
[email protected]
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