Calculation of the Theoretical Mass Spectrum of Elementary Particles in Unitary Quantum Theory
Leo G. Sapogin^{1}, Yu. A. Ryabov^{2}
^{1}Department of Physics, Technical University (MADI), Moscow, Russia
^{2}Department of Mathematics, Technical University (MADI), Moscow, Russia
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To cite this article:
Leo G. Sapogin, Yu. A. Ryabov. Calculation of the Theoretical Mass Spectrum of Elementary Particles in Unitary Quantum Theory. International Journal of High Energy Physics. Special Issue: Symmetries in Relativity, Quantum Theory, and Unified Theories. Vol. 2, No. 41, 2015, pp. 7179. doi: 10.11648/j.ijhep.s.2015020401.16
Keywords: Unitary, Quantum, Wave Packet, Mass Spectrum, Elementary Particle
1. Introduction
In the standard quantum theory, a micro particle is described with the help of a wave function with a probabilistic interpretation. This does not follow from the strict mathematical formalism of the nonrelativistic quantum theory, but is simply postulated. A particle is represented as a point that is the source of a field, but can not be reduced to the field itself and nothing can be said about its "structure" except with these vague words.
This dualism is absolutely not satisfactory as the two substances have been introduced, that is, both the points and the fields. The points, that is the sources of a field, but not driven to the field. Presence of both points and fields at the same time is not satisfactory from general philosophical positions  razors of Ockama. Besides that, the presence of the points leads to nonconvergences, which are eliminated by various methods, including the introduction of a renormalization group that is declined by many mathematicians and physicists, for example, P.A.M. Dirac. Modern quantum field theory can not even formulate the problem of finding a mass spectrum. The original idea of Schroedinger was to represent a particle as a wave packet of de Broglie waves. As he wrote in one of his letters, he "was happy for three months" before British mathematician Darwin showed that the packet quickly and steadily dissipates and disappears. Then it turns out that this beautiful and unique idea to represent a particle as a portion of a field is not realizable in the context of wave packets of de Broglie waves. It was proved [14] by V.E. Lyamov and L.G Sapogin in 1968 that every wave packet constructed from de Broglie waves with the spectrum a(k) satisfying the condition of VinerPely (the condition for the existence of localized wave packets).
becomes blurred in every case. Later, de Broglie tried to save this idea by introducing nonlinearity for the rest of his life, but wasn't able to obtain significant results.
The trouble with the many previous field unification attempts was in trying to construct a particle model from classical de Broglie waves, whose dispersion is such that the wave packet becomes blurred and spreads out over the whole of space. The introduction of nonlinearity greatly complicated the task but did not lead to a proper solution of the problem.
There is a school in physics, going back to William Clifford, Albert Einstein, Erwin Schrödinger and Louis de Broglie, where a particle is represented as a cluster or packet of waves in a certain unified field. According to M. Jemer’s classification, this is a ‘unitary’ approach. The essence of this paradigm is clearly expressed by Albert Einstein’s own words:
"We could regard substance as those areas of space where a field is immense. From this point of view, a thrown stone is an area of immense field intensity moving at the stone’s speed. In such new physics there would be no place for substance and field, since field would be the only reality and the laws of movement would automatically ensue from the laws of field."
However, its realization appeared to be possible only in the context of the Unitary Quantum Theory (UQT) within last two decades. It is impressive, that the problem of mass spectrum has been reduced to exact analytical solution of a nonlinear integrodifferential equation [14]. In UQT the quantization of particles on masses appears as a subtle consequence of a balance between dispersion and nonlinearity, and the particle represents something like a very little waterball, the contour of which is the density of energy.
The Unitary Quantum Theory (UQT) represents a particle as a bunched field (cluster) or a packet of partial waves with linear dispersion [111,14]. Dispersion is chosen in such a way that the wave packet would periodically disappear and appear in movement, and the envelope of the process would coincide with the wave function. Based on this idea, the relativisticinvariant model of such unitary quantum field theory was built.
The principal nonlinear relativistic invariant equation is following [6,10,11]:
(1)
where =(ct,x); is the fourvelocity of the particle, matrices (32x32) satisfy the commutation relations
,
and is the metrical tensor. This fundamental equation of UQT describes, in our opinion, all properties of elementary particles. It is possible to derive from (1) the Dirac equation and also the relativistic invariant Hamilton – Jacoby equation [2,3,9,10]. We have succeeded in solving only the simplified scalar variant of eq. (1). However, the solution obtained has allowed to determine theoretically the elementary electrical charge and the finestructure constant with high precision (our theoretical value , the known experimental value [711]). Our efforts to find more complete solution of eq.(1) were unsuccessful. Note, our approach based on Unitary Quantum Theory has nothing in common with Standard Model of Elementary Particles.
Nevertheless, our idea to consider a particle as some moving wave packet which periodically disappears and appears in movement, has allowed to arrive to the conclusion [911] that such particle may be described by the common telegraph – type equation of second order. In onedimension case this equation is following:
(2)
Note, this equation would be relativistic invariant if the root would be placed in denominator.
Equation (2) is satisfied exactly by relativistic invariant solutions in the form of a standard planar quantummechanical wave and also in the form of disappearing and appearing wavepacket, viz.,
(3)
Or , (4)
where is an arbitrary function of its argument (xvt)
We will show that eq. (2) (considered in the case of 3dimension coordinate space ) allows, namely, to determine theoretically the mass spectrum of elementary particles. Such equation for the function is following:
(5)
(the symbol m is replaced by M). We will use the natural system of units and put, and will seek the solution of eq. (5) in the following form:
(6)
where is some function not depending on t. This function represents as hardened wave packet in coordinate space Substituting (6) in eq. (5) , we get
(7)
We will seek the solution of eq. (7) in form:
(8)
where
(8)
is the Legendre function, is the spherical harmonic and L, m are nonnegative integers L=0,1,2,3,…, besides Substituting (8) in eq. (7), we come to the following equation with respect to the function R(r)
(9)
The solution of this equation depends on parameter L and we obtain the family of solutions
of equation (5) depending on parameters L,m. It is natural to suppose that every solution of our equation (5) describes the amplitude of the partial world unitary potential determined by partial wave packet and the potential itself is represented by the quadrate of amplitude modulus, i.e.
(10)
Further, we consider the gradient of this potential as the tension of corresponding field (it is the custom in electrodynamics) of the partial wave packet and consider the quadrate of the tension as the density of the energy or of the wave packet’s mass distributed continuously in space. If we consider eq. (9) in some fixed spherical zone of radius, where the corresponding part of our hardened wave packet is placed, then it is natural to consider as the mass of this part of the partial wave packet, i.e. as the integral of density over given spherical zone. Such approach allows to replace the mass in (9) by integral
(11)
Where So, we will consider eq. (9) as the integrodifferential equation with respect to the function . For the sake of simplicity; we will use the following expression for M (after discarding the members which depend on and omitting index L):
(12)
We will use the following way to solve our integrodifferential eq. (9). Viz., at first, we rewrite this equation in form
(13)
At second, we substitute integral (12) for M and differentiate left and righthand sides with respect to r. We obtain
(13’)
At the third step, we set v=0 in (13’). The grounds are following. The solution of this equation depends on parameter v (the velocity of our particle). It is natural to suppose that the potential Ф describe processes which are continuous with respect to v (in any case, if v is less, than light velocity c), i.e. if and it is valid if Besides, we want to determine the inner (proper) characteristic of our wave packet not depending on the velocity of its movement. So, we set v=0 and obtain the differential equation for R(r) (after corresponding differentiation):
(14)
This equation possesses the analytical general solution (in addition to trivial constant solution):
(15)
where are arbitrary constants and J and Y are the Bessel functions. Since we seek the finite solution R(r) for and tending to zero for, we set and can set some positive value for . The calculations show the choice of these constants has influence only on the absolute value of the masses calculated below but the ratios of these masses remain the same. We have chosen the simplest values and have obtained following solution
(16)
where is the Bessel function of 1st type with imaginary argument, or
, (16’)
where is the modified Bessel function of 1^{st} type. Note, if v=0, then (13’) is reduced to following equation
where is some constant. The solution of this equation for coincides for with our solution (16’).
So, we obtain the following expression for the partial world unitary potential (taking into consideration (6, 8, 8’, 10) :
(17)
Now, we form considered as the tension of the world unitary field and form also the quadrate of its modulus considered as the mass density . We obtain:
(18)
The integrals of over all spherical space for different L=0,1,2 and is equal to required different masses of elementary particles, i.e.
(19)
Since do not depend on and the Legendre functions in expressions of may be integrated analytically, we calculated, at first, analytically (with help of Mathematica9) the integrals
(20)
and then calculated numerically (with the help of Mathematica9) the integrals
(21)
For example, we have obtained for L=0 и m=0 (with help of Mathematica9):
;
and
For L=1,m=1, we have obtained (with help of Mathematica9)
and
The calculations for small values of L are sufficiently simple. But for large L, the quantities are represented by long polynomials in r and with enormous numerical coefficients and the integration of these polynomials meets serious technical difficulties.
We consider the ensemble L+1 particles (masses) with given L and to be one family and we will use the notations for particles (masses) of the family with given L. We have calculated and analyzed in full the masses of 49 families (L=0…48, i.e. of 1225 particles. Our PC with 3GHz, RAM=32GB has required for these calculations nearly 3 weeks of computing time. All calculations were checked by Maple18.
We have compared our theoretical spectrum for 1225 masses with known experimental spectrum for elementary particles measured in MeV. The zeropoint for the matching of both spectra was required. We have taken for such matching the quotient of the muon mass to the electron mass. As we know, this quotient for observed muons and electrons is measured experimentally [15] with the most precision and is equal 206.768283(10). Each our calculated mass was divided consecutively by all other 1224 masses and the resulting quotients were compared with the mentioned number. It turned out that the quotient of our masses is equal to 206.7607796 (with relative divergence 0.0039%) and we have taken our mass equal to 0.2894982442536304* for zeropoint, i.e. for our electron mass. After, there were divided all other 1224 masses by and we have obtained our theoretical spectrum in electron masses which may be compared (after expressing in MeV) with known experimental masses. Here is the Table.1 with our masses for 33 cases of the well coincidence with well known experimental values (relative errors are less than 1% in 30 cases and between 1.3% and 1.8% in three cases):
(e – electron,  muon,  meson,protonetc.)
Theory  Experiment  Notation  Error%  
 0.51099906  0.51099906  e   
 105.6545640  105.658387 
 0.0036 
 135.8958708  134.9739 
 0.683 
 137.2902541  139.5675 
 1.62 
 541.7587460  548.86 
 1.29 
 894.0806293  891.8 
 0.25 
 936.3325942  938.2723  p  0.206 
 957.1290490  957.2 
 0.0083 
 1110.473414  1115.63 
 0.462 
 1224.151552  1233 
 0.71 
 1271.916682  1270 
 0.14 
 1331.705434  1321.32 
 0.78 
 1378,127355  1382.8 
 0.33 
 1524.617683  1520.1 
 0.29 
 1549.444919 

 0.28 
 1595.510637  1594 
 0.094 
 1601.282953  1600 
 0.08 
 1718.917400  1720 
 0.06 
 1774.917815  1774 
 0.051 
 1906.842877  1905 
 0.096 
 1965.115639  1950 
 0.77 
 2092.497779  2100 
 0.35 
 2195.695293  2190  N(2190)  0.25 
 2818.645188  2820 
 0.048 
 2954.549810  2980 
 0.85 
 3082.979571  3096 
 0.42 
 3543.664516  3556.3 
 0.35 
 3687.679612  3686.0 
 0.04 
 4496.650298  4415 
 1.84 
 5642.230394  5629.6 
 0.8 
 9499.927309  9460.32 
 0.41 
 10075.78271  10023.3 
 0.523 
 10533.15222  10580 
 0.442 
 131517  125000140000  Higgs  
 6962274  ?  Dzhan  ? 
Note, the ratio of our proton mass and our electron mass is equal 1832.355 with relative error 0.207% in comparison with well known experimental ratio 1836.152167. Our calculated spectrum containing 169 masses from muon to the heaviest mass approximates also others well known particles and, although the coincidences with experimental data are worse but quite acceptable (with relative divergences not more than several per cent). The mass values for negative m coincides with the mass valued for positive m (antiparticles?).
On the whole, this table shows the striking coincidence of our theoretical values with essential quantity of the known experimental masses and, by no means, such coincidence may be called occasional. The probability of such occasional coincidence is less .Note, the choice of the nominee for the electron’s mass is not unique and may be further calculations of families with L=60..100 would allow to obtain the better result. Our calculated theoretical spectrum contains also the values near to the masses of quarks. The experimental data for quarks are not so precise and are determined in an indirect way. We give the separate Table.2 with the calculated and experimental quark masses.
Theory  Experiment  Notation  
 5.003455873  37  down 
 2.75072130  1.53.0  up 
 94.4251568 
 strange 
 1271.9166 
 charm 
 4300.86662 
 beaty 
 179100 
 truth 
We have carried out also the series of calculation for L exceeding 48 including L=60. The ratio of maximal =0.0039443641689 to minimal =0.3909395521* is of order. The ratio of maximal to the mass =0.53046407191*of proton is equal 74400. This number does not contradict the known the experimental data.
Note, the radial function being the density mass as function of r, is equal zero always for r=0 and for all L, m, and, at first, increases very swiftly on the right from for r=0 and then very swiftly decreases. The plot of reminds for large L quasi deltafunction approaching to coordinates origin as L increases (very simplified analogy is shown on Fig.1).
Such theoretical model describes a particle as very small bubble in spacetime continuum cut by spherical harmonics. Curious, such model, namely, was considered by A. Poincare [12].
Certainly, we do not intend to assert that our results are adequate in full to the known experimental mass spectrum of elementary particles. The divergences are present. Our theoretical spectrum contains the large quantity (1053) of masses between electron mass and muon mass but such real particles are not observed till now. Our spectrum contains many light particles (L>48) with masses differing extremely little one from another. It may be supposed there is exists quasicontinuous distribution of lightest particles not affirmed till now by experiments. We suppose that this region of our calculated spectrum contains also the values corresponding to masses of all 6 neutrinos, and it will be possible to discover their theoretical masses after sufficiently precise experimental determination of their masses.
Our spectrum contains 169 particles from the muon to the heaviest particle but there is observed the large quantity of particles in this interval with short "lifetime" (so called "resonances") of order sec. These divergences require the further researches. With respect to light particles, it may be supposed there are exist some selection principles (not discovered till now theoretically) for such particles and these principles lead to essential decreasing of particles quantity between muons and electrons. We suppose that such principles arise theoretically from some relations between the tensors of different valences (ranks) and spherical functions for different L,m and leave this complicate problem for future researches. May be these light particles constitute the dark matter.It arise the question with respect to the particles with short "lifetime": may we take all these particles for elementary? Our Unitary Quantum Theory allows formulating the following criterion. "If the way which the particle (which we identify with appearing and disappearing wave packet) passes from the moment of its appearing to the moment of its destruction is much longer than de Broglie wave, then such particle may be called elementary". Have we reason to call "elementary" the particle with lifetime of order ? Let us point to following essential circumstance. Viz., if we will use the Schrödinger equation in spherical coordinates (relativisticnoninvariant) or Klein—Gordon equation (relativisticinvariant) instead our initial equation (5), then we will come to the same theoretical mass spectrum. Really, the mention Schrödinger equation is following:
(22)
where M is the particle’s mass. We will seek the solution of this equation in form of unitary wave
packet f: (23)
where is the function of coordinates and does not depend on the time. The function u is considered as the amplitude of the world unitary potential Ф. Substituting (23) in (22), we obtain (after simplification) following equation
(24)
This equation coincides with our equation (7) if we put instead . The further study described above remains without changes. Let us consider Klein—Gordon equation in spherical coordinates and in natural units system (c=1,)
(25)
where M is the particle’s mass. We will seek the solution
(26)
where is the function of coordinates not depending explicitly on t. Substituting (26) in (25), we obtain following equation after simplification:
(27)
This equation coincides in full with our equation (7) and we will come to the same results.
So, different initial equations (5), (22), (25) (the last is relativistic invariant and the other two are relativistic noninvariant) lead to the same theoretical mass spectrum. Note the following remarkable fact: the standard theory allowed to detect spectra by using always the quantum equations with outer potential and as corollaries to geometric relations between de Broglie wave length and characteristic dimension of potential function. The quantum equation of our theory do not contain the outer potential and describe a particle in empty free space; the mass quantization arises owing to the delicate balance of dispersion and nonlinearity which provides the stability of some wave packets number. It is the first case when spectra are detected by using the quantum equations without outer potential.
105.655  105.94  
120.31  121.826  
135.896  137.29  
153.827  159.796  
180.895  187.69  
219.639  221.135  
269.993  270.91  
318.997  335.848  
408.316  423.36  
529.951  531.566  
705.247  705.477  
936.333  957.129  
1524.62  1549.43  
2334.9  2557.69  
4315.87  4496.65  
10533.2  12941.1  
71060.4  87704.5 
106.241  108.291 
122.664  125.522 
142.287  144.326 
162.135  162.192 
192.661  192.917 
224.06  225.089 
276.443  280.151 
339.955  341.136 
423.429  432.83 
539.326  541.759 
730.141  738.98 
996.316  1110.47 
1595.51  1601.28 
2818.65  2906.6 
5642.23  6026.01 
16897.  18035.6 
131517.  179100. 
108.997  109.597 
125.71  127.187 
145.96  147.309 
165.33  172.249 
195.832  199.852 
231.432  231.656 
281.016  289.488 
342.52  349.235 
445.413  459.388 
560.236  571.51 
812.354  828.374 
1135.57  1137.9 
1718.92  1774.92 
2954.55  3082.98 
6570.85  6666.64 
18261.3  25000.7 
266419.  601983. 
110.133  112.784 
127.237  127.306 
147.698  149.62 
177.091  178.559 
203.297  205.588 
241.805  249.092 
300.299  301.848 
357.381  366.838 
461.593  472.253 
606.559  619.012 
866.997  894.081 
1224.15  1271.92 
1906.84  1965.1 
3545.66  3687.68 
7358.75  9219.36 
28935.4  33698.9 
1.20005x106  3.4545X106 
117.054  118.136 
131.445  133.013 
149.905  153.765 
178.758  180.585 
209.097  218.681 
252.972  253.184 
304.024  314.364 
373.402  402.126 
504.945  521.772 
672.537  686.757 
897.982  915.038 
1331.71  1378.13 
2092.5  2195.7 
3832.21  4300.87 
9499.93  10075.8 
36955.4  54518.8 
6.96227x107 
Here is the Table.3 with all our theoretical masses from the muon to the heaviest (MeV) .
In view of all said above, we are bold, nevertheless, to say that our results represent the substantial advancement on the way of solution for the extremely complicated theoretical problem of the mass spectrum for elementary particles and to underline that this advancement is owing to our Unitary Quantum Theory. We hope that further analysis with the help of exact equation (1) of our theory will allow to obtain more precise results.
We would like to propose the name "Dzhan—particle" for our heaviest particle in honour of the general Air Force RF cosmonaut V.A. Dzhanibekov. As we know, particles with mass of such order are observed in cosmic rays.
The authors are thankful to cosmonaut V.A. Dzhanibekov, to professors V.M. Dubovik (Dubna, JINR) and F.A.Gareev (Dubna, JINR), and to professor Yu.L.Ratis (Samara State University) for support of our work and fruitful discussions.
References