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Научный журнал КубГАУ, №84(10), 2012 года
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УДК 531.9+539.12.01
МОДЕЛИРОВАНИЕ МЕТРИКИ АДРОНОВ НА ОСНОВЕ УРАВНЕНИЙ ЯНГА-МИЛЛСА
Трунев Александр Петрович к.ф.-м.н., Ph.D.
Директор, A&E Trounev IT Consulting, Торонто, Канада
В работе рассмотрена система уравнений Янга-Миллса в связи с уравнениями Эйнштейна и Максвелла. Сформулирована модель метрики, удовлетворяющая основным требованиям физики элементарных частиц и космологии
Ключевые слова: АДРОНЫ, МЕТРИКА, ПРОТОН, УРАВНЕНИЯ МАКСВЕЛЛА, УРАВНЕНИЯ ЭЙНШТЕЙНА, УРАВНЕНИЯ ЯНГА-МИЛЛСА
UDC 531.9+539.12.01
HADRONS METRICS SIMULATION ON THE YANG-MILLS EQUATIONS
Alexander Trunev Cand.Phys.-Math.Sci., Ph.D.
Director, A&E Trounev IT Consulting, Toronto, Canada
In this article we consider the Yang-Mills theory in connection with the Einstein and Maxwell equations. The model of a metric satisfying the basic requirements of particle physics and cosmology is proposed
Keywords: EINSTEIN EQUATIONS, HADRONS, MAXWELL EQUATIONS, METRICS, PROTON, YANG-MILLS EQUATIONS.
Introduction
The hypotheses on the global structure of space-time have been formulated in the famous Einstein's paper [1]. Einstein assumed that the universe is stationary, and the average density of matter and the total mass of the universe does not change over time. Friedman [2] has shown that the universe is expanding, it was confirmed by the astronomical data, and also served as a further recognition of general relativity. However, Einstein's theory of gravity was incompatible with Maxwell's equations, so there were numerous attempts to create a unified field theory [3] in the space of five dimensions [4-6]. At present, interest in the classical field theory models waned, and Einstein's equations themselves are primarily used for the solution of cosmological problems.
Due to the development of quantum theory, is a very relevant question of the structure of space-time scale of the proton, as modern lattice models of quantum chromodynamics (QCD) has been successfully used to predict the properties of hadrons [7]. The answer to the question of the structure of space-time can be obtained from the Yang-Mills theory [8-9], which is widely used in particle physics. Connection of the Yang-Mills equations to the Maxwell equations and
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Einstein equations as well was established in [10-12]. It has been shown that on 4-manifolds with conformal connection [13], the system of the Yang-Mills equations split into Einstein's equations, Maxwell's equations and the equations of motion of matter. So the necessary prerequisites for a unified field theory have been creating. In this paper we formulated a model of hadrons metric satisfying the basic requirements of particle physics and cosmology.
The basic equations of the model of the cosmological scale
Consider the example of a purely temporary solution of the Yang-Mills
equations in the space of torsion-free [11]. We define a metric space as
y = hijWW = -dt2 + a 2(t )(dxl)2 + /32(t)(dx2)2 + c 2(t )(dx3)2 (1)
Here lij = h is the metric tensor of the Minkowski space of signature (- + + +),
(W = dt, (W = a(t)dx1, (W = b(t)dx2, w4 = c(t)dx3 .
Yang-Mills equations can be reduced in this case to Einstein's equations, Maxwell's equations, and equations of motion of matter accordingly, we have
bn = -
11 3
f ••
•Л
a b c
b
22
a b c
f ■■
a
a
a ab схсЛ — + ^— +
ьab ac Ьсл — + — + — ab ac bc
f о
b
33
b44
3
V
f ■
ab ac
1
+ — 6
6- Л
b c bc
— + — +^
ab +b+bc Л
ab b bc
V f ■ ■
V
1 r
b c bc
+ — 6
a c ac — + — + —
V a c ac у
a c bc c
+ — +
ac bc c
+
6
a b a&b
— + — +
a
b ab
b12bc = M, b34bc = N
(2)
1
1
b22 + (b11 + b22 ) = S2 a
b33 + (b11 + b33 ) b ~ S3
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b44 + (b11 + b44 ) - S4
С
S 2 + S 2
f h • Л
b + c
b c
2b12 + 2b34 b11
a
V a
+ b22 b11
Л f Л/3
+ bo
33
У
ab
aft
\
+ b22 + b33
+ b
44
ac
V ac
+ b22 + b44
S 3 + S 3
b a c ^ — + —
V a c
- 2b12 - 2b34 - bn
b ^ b33 b11
+ b
22
ь ab
ab
\
+ b22 + b33
+ b
44
У
" bc bc
+ b33 + b44
S 4 + S 4
f • h \
a b
a b
2b - 2b
12 34
b
11
c
V c
\
+ b44 b11
+ b
22
ac
V ac
+ b22 + b44
+ b
rbc
33
bc
+ b33 + b44
Here bij + bji - 2(hbij )Лу — Tij is the energy-momentum tensor of matter;
M, N - the parameters characterizing the electromagnetic field. Note that the Einstein equations in this notation have the form:
bj + bi + bhij — Rj (3)
b — hbij; Ry is Ricci tensor. Einstein equations (3) can be reduced to the first four equations (2) in the case of the metric (1). In contrast to the standard Einstein equations, they do not contain dimensional parameters characterizing the interaction of the gravitational field with the distribution of matter. This is due to the fact that the quantities in equations (2) - (3) and described by the classical Yang-Mills equations are geometric quantities, like the Ricci tensor in the right-hand side of equation (3).
Some exact solutions and numerical model
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In [11] found a particular solution of equations (2) in elementary functions
1
a
2 PQ n t + 10 2 лг2
—~-; b = c -------; M2 + N
t0 + t
b
5
11
6(t0 +t)
2Q
b — b
2 , ^2 ^33
12Q4
b
1
(4)
44
6(t 0 + t У
Here P, Q, t0 are the arbitrary constants.
The solution (4) describes a singular time-a process in which the energy density
increases at t ® -t0. Among such processes in our universe, we can specify a hypothetical explosion - Big Bang. In this case, the characteristic density of baryonic matter decreases with time, whereas the density of electromagnetic energy remains constant. Note that in the model (2), these densities are diagonal
and off-diagonal components of the tensor bij, respectively.
We point out one particular solution of the system (2)
l4
a - a0 exp(4t), p- B, c - C; M2 + N2 = B2C2 —;
b __b -H. b = b =4
b11 b22 о ; b33 b44 ^
3 6
y - _dt2 + a^e24(dxl)2 + B2(dx2)2 + C2(dx3)2.
(5)
Here ao, B, C,4 are the arbitrary constants. The solution (5) gives us an
example of a space of constant negative curvature R - hRj - _24. In this case, the density of baryonic matter and electromagnetic energy density remains constant over time.
Solutions (4-5) were used to adjust the numerical model for the system (2). Explore different modes of transition of the solution (4) to a solution of the type
(5) - Fig. 1. We found that if in the initial time we set b11 - c°nst, then in the
next moment, all the diagonal tensor components bij tend to constant values, and the components of the metric tensor are increasing exponentially - Fig. 1.
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Fig. 1. The exact solution (4) (top) and the transition from the solution (4) to a
solution in which all diagonal components btj tends to a constant (bottom): on the left components of the metric tensor, on the right the accuracy of the solution are shown.
However, only on the solution of the type (5), the density of electromagnetic energy remains constant over time. In all other cases, the density varies
according to the equations: b 12 Pc = M, b34Pc = N.
This disappearance of the electromagnetic field at a large scale is against the astronomical observations, so let's mark solutions (5), as the hypothetical scenario that describes the universe at a large scale.
An alternative scenario is the type (4), but it contradicts the experiments in high-energy physics. Indeed, a decrease in the density of baryonic matter means, among other things, that the proton is unstable. This follows from the fact that the density of the model of baryonic matter and proton density are linked by the fact that the Einstein equation (3) contains only the geometric parameters that describe the density distribution at any scale. However, experimental evaluation
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-э-э
of the proton lifetime show that the half-life of a proton is more than 10 years, i.e. greatly exceeds the lifetime of the universe (about 1010 years).
For this reason, we should discard the other cosmological models with decreasing density of matter. In this sense, Einstein's hypothesis [1] on the stationary distribution of the density of matter in the universe is correct, however, this hypothesis does not imply that the metric is also stationary. We obtained the solution (5) combines the properties of Einstein's model [1], and Friedman’s model [2] as well, describes the universe as a time-dependent metric, and with a constant density distribution of baryonic matter and electromagnetic field.
Model of the proton scale
Note that the Einstein equations in the form (3) are universal, i.e. describe the metric in any scale, because their solution depends only on the initial conditions. Choosing these conditions across the proton, we create model that describes the metric of hadrons. Let us consider the metric of the proton, and other elementary particles. In [12] obtained all the solutions of the Yang-Mills equations in the case of a centrally symmetric metric. A particular case of the centrally symmetric metric is
y = hijWW
d 2o
Ю
= -KG
dt2 + e 2vdr2 + dO2 + G2(O)dj2
(6)
Here K = const is the Gaussian curvature of the quadratic form
dO2 + s2(O)dj2, function n = n(r, t) is determined by solution of the Yang-Mills equations. As we address the fundamental geometric structure of the observable universe, then we are interested, first of all, the periodic solutions, which form a lattice. This requirement stems from the obvious fact that all points of space should be equal to each other, but each observer can reproduce
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all the observed variety of phenomena. This is only possible if the basis of the space is a periodic structure.
Among all the solutions of the Yang-Mills equations, obtained in [12] in the case of the metric (6), there is one, which is expressed in terms of Weierstrass elliptic function. In this case, the Yang-Mills equations simplified to the form:
A„ = 2( A A — k), e"= At, t = t ± r + 10 A = 3/Т2МГ/УТ2; g 2, g 3),
1 k 1 k
b11 =—b22 = ^ A —~ , b33 = b44 = ^ A —~ , b12 = b21 = 0.
3 6 6 3
(7)
Here g2, g3are the invariants of the Weierstrass function, and g2 =k2Vf2
70 is a free parameter related to the choice of origin. Metric corresponding to this solution, apparently, can describe the fundamental structure of the universe and, in particular, the structure of elementary particles. The solution of (7) and the corresponding metrics are characterized by two periods, which, obviously, should be related to the parameters of elementary particles.
Note that the sum of the diagonal components of the energy-momentum tensor
in this case is Tu = 2 A . Therefore, setting the characteristic density of matter and space-time scale, we can obtain the general solution, which describes the metric at any scale. We show that the metric (5) described by equations (7) as well. To do this, we consider the solution of the first equation (7) near constant level of density of matter, given by Eq. A2 = k2.
Introduce a new function and variables according to the formulas
f = A — k, t = t + r
22
Suppose that J << k , then the equations of the model (7) and the solution are of the form
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ftt = kf, e = ft
f = f) exp( Vkt) + fi exp(-Vkt), (8)
e' = fokk - frke~rkt -—- f)4ke^kt In general, as it is known, the corresponding metric is reduced to [12, 14]
y = - dt2 + ch 2 (Vkt +10) dr2 + dO2 + cos2 (VkO + 60) dj (9)
Consider the universe at the moment of its existence in a small neighborhood of the coordinate system fixed to the solar system, wheret » t. It is known that the observed average density of matter in the universe is a small quantity compared
to the baryon density, so Vk << 1. Obviously, under these conditions, to match the metrics (5) and (9) it will be enough to put k = l.
We thus proved that the metric of the observable universe is associated with a metric of the periodic lattice, given by the Weierstrass function (7). Note that
although the metric (5) and (9) are similar, but the tensors btj are not similar. Indeed, the metric (5) is compatible with zero electromagnetic field that describes the components of the tensor b12 = M / bc, b34 = N / Pc . Whereas
in the metric (7) electromagnetic field is zero, since b12 = 0 .
Therefore, it is necessary to explain the mechanism of the electromagnetic field in the initial lattice (7), which does not contain the electromagnetic field. For this we note that the Yang-Mills field in the linear case is divided into a set of independent electromagnetic fields [15]. Consequently, the electromagnetic field generated at a low energy density, starting from the atomic nuclei and atoms. For its occurrence does not require any additional sources other than the Yang-Mills field and the lattes. In this case the electromagnetic fields have a wavelength multiple of the lattes scale. These waves travel at a constant speed (light) from any source along the lattes in accordance with Maxwell's equations. Thus, the quantization of the electromagnetic waves is a consequence of the
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presence of the lattice in the metric (7), the period of which is evident in all of the observed phenomena.
Metric model of elementary particles
Suppose that g2 = л/12", g3 = 1, then the half-periods of the Weierstrass
function defined asW = 1.33003, w2 = 0.66501 +1.61260i . Calculation of half-periods and the construction of appropriate 3D images of the Weierstrass function and its first derivative module carried out using the Wolfram Mathematica 9.0 [16]. Consider the metric lattice is formed at the specified parameters - Fig. 2.
Figure 2: Metric parameters
=
At
and Tii/ 2 =A in the case
g2 = 3/12, g3 = 1.
As follows from Fig. 2 data, the peaks of the Weierstrass function fuse to form solid walls, stretching along the lines t ± r = const of the
period20)13l\2 » 6.09. These characteristics describe convergent (sign +) or diverging (minus sign) spherical waves. It is easy to see that the solutions of the first equation (7) are symmetric with respect to the change t ® -t, however, the metric (5) is asymmetric with respect to time reversal. Therefore, although the microscopic events are reversible in time, macroscopic events are known to be irreversible, which is due, in particular, increasing of entropy. It can be http://ej.kubagro.ru/2012/10/pdf/68.pdf
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assumed that the emission of spherical waves is the main process in the expanding universe, and then the determination of the metric lattes should
assume that7 = t - r + ^ .
From classical electrodynamics it is known that in the process of electromagnetic wave generation an electrical charge acts as source. In this model, it is most natural to assume that the electromagnetic waves are produced in the interaction of the charge with the lattice. According to present science a charge of hadrons occurs as the sum of fractional charges of quarks. But the origin of the electric charge of the quarks themselves also requires explanation.
In the lattice model can be defined lattice defect type bubble. Lattice with a single bubble is described by the model (7), and the corresponding metric has the form (6). In the bubble we put A2 = к2, while in the outer region the solution given in the form (7), therefore we have,
A2 = к2, = 0, It < t0
A = Vi2p(t/Vl2,gi,g2), = AT
t
> t
0
(10)
On the borders of the bubble are continuous function A and its first derivative,
t
=t
0
(11)
к = V12p(r0/3/i2, g1, g 2), A,= 0,
In the particular case of the lattice in Fig. 2, with the invariants given in the formg2 = Vh2, g3 = 1, we find the first zero and the corresponding parameter
of the metric 70 = 30449983 к = 2Л°38°34 . The corresponding lattice is shown in Fig. 3.
Similarly, the solution is constructed for the other roots of the second equation (10). All of these roots only effect on the size of the bubble, whereas the value к does not change. Bubble moves at the speed of light along the characteristics t - r = const _ Fig. 3. To the outside observer, the bubble is still a homogeneous particle with a finite length.
Any bubble can be turned inside out, just reversed inequalities in (10) - Fig. 4.
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Figure 3: Metric parameters e - |At and Tn /2 - A in the presence of a homogeneous bubble in the case g2 - Vf2, g3 - 1.
Figure 4: Metric parameters e - |At and Tu 12 - A in the presence of an inhomogeneous bubble in the case g2 - Vf2, g3 - 1.
In this case, we can extend the definition of the metric in the bubble outer region, using the solution (8), valid for a slight deviation of the metric from the metric bubble, for which the equality A2 - a2 is true. Then the metric outer space coincides with the metric of the universe, and the metric inside the bubble will be part of the lattice. To the outside observer in our universe this bubble looks like a complex, multi-layered particle with finite scale.
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Finally, the third type of particles can be formed as a combination of the first two, plunging the homogeneous bubble of the type shown in Fig. 3 in another bubble, as shown in Fig. 4. The result is a bubble, a restricted shell of finite thickness - Fig. 5. Our next task is to show that such a particle, under certain conditions, can be catching, for example, in a magnetic trap, i.e. it has an electric charge.
Figure 5: Metric parameters homogeneous bubble in
e
and Tu / 2 - A in the presence of a
homogeneous outer space in the case
g2 - Vl2, g3 - 1.
First, we note that the particle of the type shown in Fig. 4-5, can be expanded at any rate, not just at the speed of light in our universe. Indeed, the solution build in the inner region depends on a combination of variables ? — t - r . The default is that the unit of speed is the speed of light. In reality, however, this rate depends on the speed of the external borders, which can be chosen by anyone, including an arbitrarily small. Hence, we find that there may be a spherical particle, which expand in sync with the space of the universe. Therefore, from the point of view of the outside observer they seem having static form like protons.
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Second, we note that the cavity inside the particle may contain electromagnetic field, because the parameters in the cavity can be described by equations (2). In the special case of a static electromagnetic field solution of this problem can be written as (5). But the static electromagnetic field in the cavity can be limited only if the cavity is in the external static field or cavity itself contains an electrical charge or charge distribution. It would be logical to attribute these charges to baryonic matter, which is also contained in the cavity, according to (5). Thus, the model developed above allows us to describe the structure of the proton and other hadrons, and to explain the origin of the electric charge.
Finally, we note that the model of particle metrics development above can be combined with the model of the glueball [17], which has been successfully used to predict hadron masses [18].
References
1. Einstein, Albert. Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie (Cosmological Considerations in the General Theory of Relativity)// Koniglich Preussische Akademie der Wissenschaften, 1917.
2. Friedman, A. Uber die Krummung des Raumes// Zeitschrift fur Physik 10 (1): 377386, 1922 (English translation in: Friedman, A. On the curvature of space/ General Relativity and Gravitation 31 (12): 1991-2000).
3. Einstein A. Unified field theory of gravitation and electricity//Session Report of Prussian Acad. Sci, 414-419, 1925.
4. Kaluza, Theodor. Zum Unitatsproblem in der Physik. Sitzungsber, Preuss. Akad. Wiss. Berlin. (Math. Phys.) 1921: 966-972.
5. Einstein A., Bergmann P. Generalization of Kaluza’s Theory of Electricity// Ann. Math., ser. 2, 1938, 39, 683-701.
6. Einstein A., Pau1i W. On the non-existence of regular stationary solutions of relativistic field equations// The Annals of Mathematics., 1943, v. 44 (2), p. 131-137.
7. S. Durr, Z. Fodor, J. Frison et all. Ab Initio Determination of Light Hadron Masses// Science, 21 November 2008: Vol. 322, no. 5905 pp. 1224-1227.
8. Yang C. N., Mills R. L. Conservation of Isotopic Spin and Isotopic Gauge Invariance// Phys. Rev. 96: 191-195. 1954.
9. H. Fritzsch, M. Gell-Mann, H. Leutwyler. Advantages of the color octet gluon picture// Phys. Lett. B 47 (1973) 365.
10. Krivonosov LN, Luk’yanov VA. Connection of the Yang-Mills theory with the Einstein equations// Proceedings of the universities, Mathematics 2009, № 9, p. 69-74 (in Russian).
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11. Krivonosov LN, Luk’yanov VA. Connection of Young-Mills Equations with Einstein and Maxwells Equations // Journal of Siberian Federal University, Mathematics & Physics, 2009, 2 (4), 432-448 (in Russian).
12. Krivonosov LN, Luk’yanov VA. The Full Decision of Young-Mills Equations for the Central-Symmetric Metrics / / Journal of Siberian Federal University, Mathematics & Physics, 2011, 4 (3), 350-362 (in Russian).
13. Cartan, Elie, Espaces a connexion affine, projective et conforme// Acta Math. 48: 142, 1926.
14. AZ Petrov. New methods in general relativity. - Moscow: Nauka, 1966.
15. Bryce S. DeWitt. Dynamical Theory of Groups and Fields. - Gordon and Breach, NY, 1966.
16. Wolfram Mathematica 9.0/ http://www.wolfram.com/mathematica/
17. V. Dzhunushaliev. Scalar model of the glueball// Hadronic J. Suppl. 19, 185 (2004); http://arxiv.org/pdf/hep-ph/0312289v4.pdf
18. Trunev AP Simulation of hadron masses and atomic nuclei excited states in the gluon condensate model // Polythematic network electronic scientific journal of the Kuban State Agrarian University (The Journal KubGAU) [electronic resource]. - Krasnodar KubGAU, 2012. - 07(81). P. 545 - 554. - http://ej.kubagro.ru/2012/07/pdf/40.pdf
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