Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 261) Bled, Slovenia, June 23-July 1, 2018 12 On Triple-periodic Electrical Charge Distribution as a Model of Physical Vacuum and Fundamental Particles E.G. Dmitrieff * Irkutsk State University, Russia Abstract. In this study we consider triple-periodical electrical charge distributions with the pattern similar to the Weaire-Phelan structure. According to it, the space is splitted to opposite-charged cells separated with electrically neutral border. Possible configurations obtained as results of exchanges of these cells appear to have properties that can be corresponded to the quantum numbers of known fundamental particles. We find it promising to use models of this kind, aiming to infer the axioms and constants of the Standard Model from the emergent geometrical properties of the distribution. Povzetek. Prispevek obravnava trojne periodične porazdelitve električnih nabojev, ki imajo vzorec podoben Weaire-Phelan strukturam. V modelu je prostor razdeljen na celice z nasprotnimi naboji, ki jih loci elektricno nevtralna meja. Konfiguracije, ki sledijo z izmenjavo teh celic, imajo lastnosti, ki jih avtor poveze s kvantnimi stevili kvarkov in leptonov. Avtor meni, da ti modeli omogocijo izpeljavo privzetkov in konstant standardnega modela. Keywords: Particle model,Weaire-Phelan tessellation 12.1 Introduction The spin-charge-family theory presented in [1], [2], [8], [9] offers reasonable explanations for the phenomena of the Standard Model of the fundamental particles. Originating from Clifford algebra, it comes to the binary internal degrees of freedom, explaining properties of existing fundamental particles and predicting existence of extra fermion families. In turn, we reproduce particle properties starting with binary code model. As we have shown in [7], Boolean models designed for fundamental particles can reproduce most of their properties, including charges (electrical, color, weak and hyper-charge), lepton- and baryon numbers, fermion flavor and family membership, and boson spin magnitude. The particles are represented as combinations or codes of symbols carrying one of two possible values, so these models are binary. * E-mail: eliadmitrieff@gmail.com 262 E.G. Dmitrieff Developing these models, we started with well-known linear codes, that consist of binary digits (bits) with usual values either 1 or 0. Then, in order to reduce the amount of information carried by the code, we abandoned the linear structure in favor of spatial one. Also we have symmetrized and normalized the values carried by bits, using + 6 and — 6 instead of 1 and 0. These values could be directly interpreted as electrical charge in units of electron charge e. Using spatial combination of eight symbols of this kind, we managed to represent all known fundamental particles. Also, analyzing unused combinations, we proposed existence of new scalar particle forming the vacuum condensate. It could be represented by this combination that is repeated periodically, filling the space as a tessellation. Since the tessellation can be chiral, the space filled with small alternating charged regions, comparing to simple empty space, has an advantage of offering possible explanation for difference between right- and left-handed particles in respect of the vacuum. Different particle codes, substituting vacuum codes in the tessellation, violate the periodicity with different ways. We suppose that it may be used to infer associated rest energies (masses) instead of postulating them. Treating vacuum expectation value as Coulomb potential between neighboring opposite-charged "bits" [11], we estimated that the distance between them should be on scale of« 10-21 m. Being inspired by idea of vacuum domains [3], we suppose that the interpretation of these "bits" as domains can explain the problem of their observations absence. As asserted originally by Zeldovich with co-authors, the vacuum domains should appear as consequence of symmetry break in the phase transition. In our models, they do exist but have the correlation radius on sub-particle scale instead of cosmological one. This should happen in case the 2-order phase transition is not yet complete but just approaches its critical point. Having a model with some spatial distribution of charged bits, or vacuum domains, we recognize that it is necessary to find out the pattern of this distribution which is consistent with other observable properties of vacuum and particles, including their symmetry, mass spectrum, propagation, interactions and so on. After checking simple (NaCl-like) and volume-centered (CsCl-like) cubic lattices, we found out that the A15 (Nb3 Sn-like) lattice, or Weaire-Phelan structure, has some advantages allowing it to be the possible vacuum- and particle model. 12.2 Overview of the original Weaire-Phelan tessellation The original Weaire-Phelan structure is described in [12]. It is a foam of equal-volumed cells separated by thin walls. Among other structures, having the same cell volume, this one has the minimal (known at the present time) inter-cell wall area, so it is a candidate solution for the Kelvin problem [14]. There is evidence of self-assembling of this tessellation driven by minimization of the surface energy [13]. Cells forming the Weaire-Phelan structure have almost flat faces and just slightly curved edges, thus they can be closely approximated by irregular polyhe- 12 On Triple-periodic Electrical Charge Distribution as a Model... 263 dra. It is nesessary to use two kinds of them - dodecahedra (D) and tetrakaideca-hedra (T)J. Unlike dodecahedra, the tetrakaidecahedra have three possible orientations in respect of the three Cartesian axes. The cells of both kinds can be included in the tessellation in two ways, so they became chiral. Eight cells, differing in kind, chirality, and orientation, form one translation unit. These translation units, in turn, form simple cubic grid. Assuming the size of translation unit to be I = 4A in each dimension (where A is a scale factor, and 4 is used to get most of coordinates integer), we get the unit volume Vu = I3 = 64A3, and cell volume Vc = 8 Vu = 8A3 (remembering that all cells are equal-volumed). Having the coordinate axes perpendicular to the hexagonal faces of the tetrakaidecahedra, and associating the origin with the center of one of dodecahedra, one can obtain coordinates of the centers of all other cells: D TX Ty (0,0,0) (0,2,1) (1,0,2) (2,1,0) (2,2,2) (0,2,3) (3,0,2) (2,3,0) These coordinates are expressed in units of À and derermined up to 4À, meaning that one can obtain coordinates of each cell by adding of "even" vector Ve = (4nx, 4ny, 4nz) À, n G Z. (12.1) Further, we omit the scale factor À where it shouldn't cause misunderstanding. Here we chose the R and L mark of the chirality by the arbitrary choice. There are four symmetry axis C3 defined by equations ±x = ±y = ±z. Since the structure does not possess reflection symmetry, it is chiral, so there are two mirror-reflected structures. For instance, after reflecting in the plane x = y the chirality is reversed and the coordinates are changed as the following: D Tx Ty Tz (0,0,0) (2,0,1) (0,1,2) (1,2,0) (2,2,2) (2,0,3) (0,3,2) (3,2,0) After performing the shift (move) of the whole infinite structure with the "odd" vector Vo = VE + (±2, ±2, ±2)À, (12.2) 1 The dodecahedron is a pyrithohedron with twelve equal pentagonal faces, possessing terrahedral symmetry Th, and the tetrakaidecahedron is truncated hexagonal trapezohe-dron posessing rotoreflextion symmetry C3h, with two hexagonal faces, four large and eight small pentagonal faces. Tz 264 E.G. Dmitrieff i.e. for the half-size of the translation unit (2%/3A), in direction of C3 axis, we get the original structure again: D Tx Ty Tz (2,2,2) (2,0,3) (2,3,0) (1,2,0) (0,0,0) (o,2,l) (2,1,0) (l,0,2) Thus, the structure possesses global PS symmetry, where P is parity (particularly, exchange of any two coordinates) and S is shift along C3 axis on the half of translation unit size (or, generally, on the odd vector). It also means that despite of mirror asymmetry of each finite part, there is only one infinite Weaire-Phelan structure, which is either right- or left-handed depending on the choice of origin. It can be also considered as two overlapped chiral structures consisting of the same elements but shifted in respect of each other with the odd vector (12.2). 12.3 Dual-charged Weaire-Phelan structure To use the Weaire-Phelan structure as a spatial version of binary-code model, we need to assume that each cell carries electrical charge with magnitude of 6. Since the space containing no particles is electrically neutral, the counts of positive and negative cells in any volume > I3 should be the same. Any change of cell charge, that can be from + 6 to — 1, or back, would cause the total electric charge to change on ± 1. Thus, all the particles in this model will have discrete charges with step of 3, that is according to experiments. So the existence of particles with charges, for instance, of ± 2, is impossible. In general, the charge inside cells can be distributed being determined by physical law acting on this scale, for instance: • all the charge can be concentrated in cell centers, in point-size sub-particles (partons or rishons); • the charge can be distributed smoothly inside cells around their centers, falling to zero on the inter-cell borders; • the charges of opposite sign can be concentrated on both sides of the walls between opposite-charged cells, and also can be smoothly distributed along them. In the following subsections we consider these simplified assumptions of the charge distribution. We assume that the basic "vacuum" alteration of charged cells in the tessellation should fulfill the following requirements: • each translation unit should be electrically neutral, and • cells with opposite chirality should also be opposite-charged. 12 On Triple-periodic Electrical Charge Distribution as a Model... 265 So, we assume positive charge of cells of one chirality and negative for another. However, at this stage we do not recognize any natural rule that would define the absolute chirality. So, there are 23 = 8 choices of Tt charges and also 2 choices of D. We make this choice as shown in the following table: Cell type Charge Coordinates Plane Dr - (0,0,0) x + y + z = 4n Tri + (0,2,1), (1,0,2), (2,1,0) x + y + z = 4n + 3 Dl + (2,2,2) x + y + z = 4n + 2 Tli - (0,2,3), (3,0,2), (2,3,0) x + y + z = 4n + 1 In the last column of the table we show the equations of planes that contain all the cell centers of particular type. Making the choice of charge sign for Tt, we break the symmetry between C3 axis, so one of them becomes dedicated. Also, making this choice for D cell charge breaks symmetry between opposite handednesses. So there are two possible dual-charged Weaire-Phelan structures. That corresponds to the principal possibility of physical vacuum with reversed chirality. 12.4 Cell Centers approximation Here we abstract from the details of spatial distribution of the electric charge, and suppose it is just concentrated somewhere in the vicinity of the cell centers. We do so to simplify the charge calculation, replacing the integration of the charge density in the volume of interest with counting the number of centers of positive and negative cells falling into it. Since the coordinates of the cell centers are integers (i.e., proportional to the scale factor A), they can lay on the certain planes only, between which, in this approximation, there is nothing. 12.4.1 CPS symmetry The following set of grids (Fig. 12.1) illustrates the placement of positive and negative cells' centers, as black and white circles, respectively, in the cubic translation unit of size 4 x 4 x 4 starting with its left bottom front corner from the origin of reference frame. Centers of D-cells are marked with double-border. The first grid is the cross-section for plane z = 0, the second one is for plane z = 1 and so on. The plane z = 4 is the same as z = 0 due to the periodicity. Considering the translation unit cube that is shifted with the "even" vector (2n + 1 )(2,2,2), for instance (-2, -2, -2), i.e. performing S operation, we get the scheme on the Fig. 12.2 (the first grid is plane z = — 2 and so on). After reflecting in the plane x = y (P operation) we get the scheme on the Fig. 12.3. One can ensure that this shift operation (S) followed by reflection (P) has the same result as the charge inversion (C). So these three operations being applied 266 E.G. Dmitrieff ; = 0 4 = 0 f. —( 1 1—tj p—1 —( —f - 0 1 2 3 4 x Fig. 12.1. The placement positive- and negative-charged cell centers in the translation unit 2 3 y r-< ^33J _-4. Due to the CPS symmetry, we also have T rr 9 > 1 qenv(Z< ^3J _+ 4; T rr 9 > 1 q env (Z > J _+ 4 for the environment of the negative-charged small T-triangle at Z _ 9/%/3. (12.6) (12.7) 272 E.G. Dmitrieff 12.4.6 Handedness change as Exchange of D triangles Although D-sublattice has no influence on the total charge of the small T+triangle's environment qenv (12.5), exchanges in it can redistribute the electric charge between rear (Z < —3) and front (Z > —3) half-spaces because it is asymmetric in respect of the plane (Z = 3/%/3). We examine such exchanges whether they can be used to represent the particle's handedness that also does not influence on its charge. Following the model that assumes the charge is located closely to the cell centers, we must conclude that D triangles just before and after small triangle at Z = 3/%/3 have the charges 2 1 Iq2 = qD(Z = —: )=+ f 1 (12.8) lq4 = qD(Z = ^ )=- 2, and in case they exchange, the charge of 1 will redistribute from rear half-space to the front one4. However, considering the case when the charge of cells is not concentrated in their centers, being instead distributed on radii comparable to the inter-centers distance (k 2... %/5 k 2.236), we recognize that the charge of cells in plane Z = 2/%/3 would not reside just before the plane Z = 3/%/3. It is so because the offset between these planes is significantly less than the inter-center distance: 1/%/3 k 0.577 < 2 (Fig. 12.8), and is comparable with the distribution radius. That is why one should assert that some part of q2 would reside after the plane Z = 3/%/3, and some part of q4 would, in turn, reside before it (see Fig. 12.8). To be the representation of the reversed handedness, the D-exchange operation should redistribute only the half of the charge (12.8). This requirement is fulfilled in case a quarter of the charge of each D-cell in triangles is distributed on the other side of the plane Z = 3/%/3 that is located at the 1/V3 of its center. So we can use this condition to obtain more realistic rule of the charge distribution rather than simple charged point in the cell center. Now we use the halved values of (12.8), that are equal to qenv by the magnitude, so they can effectively compensate them: "D'(i=" >=- 4, (12.9) In this case, 3 3 2 11 (Z < —) = qlnvIZ < —) + qD*(Z = —) = -4 + 1 = 0; I3 f f 4 1 1 (12.10) 'lZ> "/! ] = q^nv|Z>- )+qD*(Z="/3 }=-4-1=-2, 4 Such an exchange also can be considered as a rotation of a spatial hexagon containing all six cell centers of the both D-triangles, with the angle of 60° in any direction q q 12 On Triple-periodic Electrical Charge Distribution as a Model... 273 Fig. 12.8. Visually overlapping cells of two D triangles with Z = 2/v5 and Z = 4/Vl in polyhedral approximation and after exchange between D-triangles at Z = 2/%/3 and 4/%/3 they would turn into 3 3 2 1 1 1 WZ < -3 = qT™(Z < 733 + qD*(Z = -) = — 1— 1 = — 2; (1211) 3 3 4 11 i1211) qenv(Z > ^) = q!nv(Z > ^) + = ^) = — 4 + 4 = 0. So we can use them to represent weak isospin and weak hypercharge for "down" particles: 31 Tdown = qenv(Z < -3) = 1 £qi + ^2) (12.12) 31 YWown/2 = Iqa + qenv(Z > -3) = Iqs + 2(^4 + ^qs) (12.13) At Z = 9/V3, the small T-triangle and its neighborhood are inverted in respect to Z = 3/%/3 due to the CPS symmetry, so original qenv(Z < ) = qTnv(Z < -3) + qD*(Z = ) = + 4 — 4 = 0; 9 > T „ 9 . D*,r 10 , 11 1 q env (Z > -3) = qlnv (Z > -) + qD*(Z = ) = + 4 + 4 = + 2 would turn after D-exchange into (12.14) 9 9 8 1 1 1 qenv(Z < —) = q!nv(Z < ^) + q°*(Z = —) = + 4 + 4 = + ^ 9 9 10 1 1 qenv(Z > -3) = q!nv(Z > -3)+ qD*(Z = -3) = +1 - 4 = 0, (12.15) 274 E.G. Dmitrieff and both the values qenv(Z < -73) and £q9+qenv(Z > -93) would, again, coincide with weak isospin T3 and weak hypercharge YW/2 for "up" fermions, respectively: 9 1 T3up = qenv(Z < —) = 2C^qz + Zq8) (12.16) YWP/2 = Iq9 + qenv(Z> —) = Iq9 + 1 (Iqio + Iqn) (12.17) 1 So the exchange between D triangles (or, that is the same, rotation of the distorted D hexagon) can be used as a model representing switching between two handednesses. 12.4.7 Down fermions as Inversions in small T+triangle Inverting charges of cells it the small T+triangle q3, namely of q (2,1,0), q (2,1,0) and q(2,1,0) in (x,y,z) reference frame5, one can get eight possible cases (Table 12.2). The total electric charge Q, that changes with steps of ± 1 according to the count of inverted cells, coincides with the electric charge of eight "down"6 fermions. T3 := q«3) q310 q302 q321 £q3 q(>3) YW 2 : q(>3) Q symbol + + + +1/2 0 0 - + + dR1 + — + > +1/6 —1/3 —1/3 dR2 2 + + — > © nx 00 2 x@ h dR3 + — — uf1 0 — + + So —1/6 —1/2 —2/3 —2/3 uf2 uf3 — — — §0 —1/2 —1 —1 1- Table 12.2. Eight cases of inversions in the small T-triangle at Z = 3/V^ with original (unchanged) D-triangles at Z = 2/\/3 and 4/\/3, associated with weak-uncharged "down" fermions The original unchanged state with Q = 0 is the vacuum state, so it takes place of the left-handed anti-neutrino, that, according to experiments, does not exist. In 5 In the (£,, Z,u) reference frame they are q(V2, V3, 0), q(-—, V3, -—), q(--j, V3, -g)). 6 We consider anti-"up" fermions as "down" ones, and vice versa. The "up" particles as well as "up" (anti-"down") anti-particles have the electric charge greater by 1 then the charge of corresponding "down" particles or anti-particles. 12 On Triple-periodic Electrical Charge Distribution as a Model... 275 T3 := q(<3) q310 q302 q321 £q3 q (>3) YW 2 :_ q(>3) Q symbol + + + +1/2 +1/2 0 Vr - + + dL1 + - + s* +1/6 +1/6 —1/3 dL2 2x °9) YW/2 := q(>9) Q + + + •I +1/2 + 1 +1 1+ - + + uR1 + — + •2 +1/6 +2/3 +2/3 uR2 1 v • 2 *©• © + + — •s 2 ^ * > uR3 + — — ® dL1 0 — + + —1/6 +1/2 +1/3 +1/3 dL2 dL3 — — — o°o —1/2 0 0 ED + + + •I +1/2 +1/2 +1 1+ — + + uL1 1 2 X®»% + + + + •s •S +1/6 ) +1/6 +2/3 uL2 uL3 + — — @ dR1 +1/2 — + + —1/6 0 —1/6 +1/3 dR2 dc3 dR — — — —1/2 —1/2 0 VL Table 12.4. Eight cases of inversions in the small T-triangle at Z = 9/v^3, repeated twice with original and exchanged D-triangles at Z = 8/V3 and 10/associated with "up" fermions Since D cell has 6 equal-charged and also 6 opposite-charged neighbors, the inversion does not affect the area (AS = 0) and AEo = 0. (12.19) In contrast, among 14 neighbors of T cell six ones are equal-charged but there are eight opposite-charged ones. Both opposite-charged neighbors that become equal-charged ones in an inversion, are separated with the hexagonal faces with area S6. So AEt = 2ApS6. (12.20) Assuming the energy density for wall between equal-charged cells is greater than for opposite-charged ones, Ap > 0 and AE > 0. In case of inversions of two neighboring cells, there is an additional effect caused by their common face. In case two neighbor cells exchange their charge (thus, they are D and T touching each other with large pentagonal face S5L or two T touching each other 12 On Triple-periodic Electrical Charge Distribution as a Model... 277 with small pentagonal face S5s or hexagonal one S6) the common face remains separating opposite-charged cells, instead of being turned into separating equal-charged cells, so energy effect is negative: AEq^T = -2ApS5L, AET ^T5 =-2ApS5s, (12.21) AET ^T6 = -2ApS6. In case of two neighbor cells inverting in the same direction, the additional effect of the common face is opposite, i.e. positive: AEQ^T = 2ApS5L, AET ^T5 = 2ApS5s, (12.22) AEt ^t6 = 2ApS6. Note that numerical values of the faces' areas (in units of A2) are such that S5L is almost equal to the arithmetic mean of S6 and S5s: S5L S5s S6 S6 + S5s — 2S5L Now we can build the simple hierarchical seesaw model of mass based on addition and subtraction of energy effects. • Since D exchanges have AE = 0, massless particles like photon and neutrino must be associated with D-only exchanges. • Following our 8-bit model [7], associate W+ boson with five defects combination shown on Fig. 12.9W. Note that it is colorless and has correct electric charge Q = +1. The affected area of these defects is 1.77477, 1.15338, 2.41260, so 0.0164. (12.23) Fig. 12.9. Models of W+ and Z0 bosons in polyhedral approximation 278 E.G. Dmitrieff ASW = 6 (S6 + S5L) = 25.12422. (12.24) Using experimental value of mW = 80.385GeV we get Ap = "mW « 3.1995 GeV/A2. (12.25) AS W • Following the same way, we associate Z0 boson with neutral six T defect configuration shown on Fig. 12.9Z. Using the same Ap value, we get mZ = 12ApS6 « 92.629 GeV. (12.26) • The Higgs boson having, accordingly to 8-bit model, the defects structure similar to Z boson but with one additional D defect pair (Fig. 12.9H), must have one of D cells isolated the same way as W has, to get the appropriate mass: mH = Ap(12S6 + 6S5L) « 126.699 GeV. (12.27) Fig. 12.10. Models of y photon and H0 bosons in polyhedral approximation • For charged lepton we suppose the structure of small-T-triangle inversion combined with eight inversions of D cells providing the compensation (Fig.12.11).This mechanism does not follow the pattern used in 8-bit model for fermion families representation7, but it offers effective mass reduction below GeV scale. mt = Ap(6S6 — 12S5s + 6S5L) « 0.315 GeV. (12.28) • The zero-charged compensating "frame" consisting from D cells could be associated with massless neutrino (Fig.12.11v). • Although the exchange between two or more stacked T cells has the positive energetic effect, its magnitude does not depend on the stack length, and originates just from the non-compensated ends of the stack that has the color charge due to their asymmetry. So it can be associated with the gluon thread terminated with quarks. 7 the latter involves additional T-D exchange. 12 On Triple-periodic Electrical Charge Distribution as a Model... 279 Fig. 12.11. Models of charged lepton with compensated mass, and massless neutrino in polyhedral approximation 12.6 Analytical approximation of charge distribution In addition to the Polyhedral and Cell-Center approximations we consider an approximation of the structure by the triple-periodical analytical function of electrical charge density distribution. The electrical charge of the cell concentrated at its center (x0,y0, z0) can be expressed analytically using the 5-function: q = 6 6(x — x0,y — y0,z — z0) dx dy dz (12.29) The delta function can be considered as the spherically-symmetrical Gaussian distribution with zero deviation: §(x,y,z) = lim 6(x,y,z, c); cf —^0 5(x,y,z, c) = (c\/2n)3 x2+y2+z2 2a2 (12.30) (12.31) As we have shown in section 12.4.6, the model explaining the weak isospin T3 = 0 for right-handed fermions and T3 = ±1/2 for left-handed ones by the charge exchange between D-triangles at Z = 2/V3 and 4/V3, requires one quarter of the charge of each D cell to reside behind the section plane located at the distance of 1/\/3 from the cell center: -1/V3 bco +CO p(x,y,z) dx dy dz = 4 (12.32) x=—co —co z=—co Assuming charge density p(x, y,z) to be the Gaussian distribution (12.31), and solving the equation 1 -1/V3 (^v/2n)3 x2 +y 2 + z2 1 2 a2 dx dy dz = — » 4 (12.33) x=—CO —CO z=—CO e e e 280 E.G. Dmitrieff numerically, we found ct « 0.87377. Soliton model To construct the charge distribution in the analytical form, we can use, instead of each cell, some spherical-symmetrical function, which decreases quite rapidly on distance from its center, i.e. soliton. We consider the soliton function as normalized error function Pi = ± 6 exp 6CTv2n (x - Xi)2 + (y - yi)2 + (z - Zi)2 2CT2 (12.34) representing positive or negative charged cell with the center at (xt,yt, zi). The charge density in the particular point is calculated as a sum of contributions of all the cells in the model: P = X Pi (12.35) i One can manage the position and charge of each individual cell, so this model should be flexible. On another hand, it requires extensive computation to calculate each point. Triple-periodic trigonometric function Since the most interesting application of this model is to represent the only one or several defects being surrounded by the "pure" vacuum, we looked for the periodic function that has the same symmetry as the dual-charged Weaire-Phelan structure considered above. It is intended to represent the pure vacuum avoiding calculating of plenty periodically allocated solitons. At first, we consider the real function that has zero surface close to the Schwartz P minimal surface [15]. Po = cos x + cos y + cos z, (12.36) or, equivalently, p0 = ^ cos xi. (12.37) i xn yn zn Po = cos — + cos — + cos —, (12.38) 2À 2À 2À It has minimum in points (2nnx, 2nny,2nnz) = 2n(nx,ny,nz) and maximum in n(2nx + 1,2ny + 1,2nz + 1 ) since ^ = - sin xi = 0 ^ xi = nni, (12.39) oxi and dx? =- cos xi. (12.40) The last equation also means that Apo = -Po, (12.41) 12 On Triple-periodic Electrical Charge Distribution as a Model... 281 so po is eigenfunction of the Laplasian, with eigenvalue —1. The translation unit with nx = ny = nz = 0 is a cube with xt G [-n; n]. So, p0 has one minimum in (0,0,0) and one maximum in n (2,2,2). As the second step, we consider the surface p0 = 0. Its saddle points are the same with the T cell center points. So we can add the function with extremals at these points, namely at centers of D cells: pxz = 1 sin y (1 — cos x)(1 + cos z) (12.42) Pyx = 1 sin z(1 — cos y)(1 + cos x) (12.43) Pzy = 1 siny (1 - cosz)(1 + cosy) (12.44) pxy = 1 sinz(1 - cosx)(1 + cosy) (12.45) pyz = 1 sin y(1 - cos y)(1 + cos z) (12.46) pzx = 1 siny (1 - cosz)(1 + cosx) (12.47) PR = Pxy + Pyz + Pzx (12.48) PL = Pyx + Pzy + Pxz (12.49) We construct right and left vacuum electric charge density as P0R = Po + PR (12.50) P0L = Po + PL. (12.51) Note that Pxz (12.42) and other Ptj can be rewritten in the following way: Pxz = 1 (sin y + sin y cos z - sin y cos x - sin y cos x cos z), (12.52) so pr and pl can be represented as sums of four functions listed below, which accumulate summands of four particular types, that occur in (12.42). Introducing "Schwartz P"- like distribution Pe = cos(x - 9) + cos(y - 9) + cos(z - 9), (12.53) right and left gyroid-like distributions Gr = cos x sin y + cos y sin z + cos z sin x, (12.54) Gl = cos x sin z + cos y sin x + cos z sin y, (12.55) and "layers-with-holes" distribution H = cos x sin y cos z + cos y sin z cos x + cos z sin x cos y, (12.56) 282 E.G. Dmitrieff we can express pR through them: Por = 4 [p«/2 + Gl - Gr - H] - 3Po. (12.57) Since G and H are also eigenfunctions of the Laplasian A: AG = -2G; AH = -3H, (12.58) one can find the scalar electric potential: divgrad cpoR = A^or = 4npoR, (12.59) 1 1 9or = Po - 77— 12n 16n Pn/2 + ^Gl - 1Gr - (12.60) Combining triple-periodical trigonometric equation for the vacuum state with doubled opposite-charged soliton located in particular cell centers one can obtain a model representing one or more particles surrounded by the vacuum. 12.7 Discussion 12.7.1 Two-dimension model Consider the surface of zero potential (12.60): 11