MATTER THEORY OF EXPANDED MAXWELL EQUATIONS

Transcription

1 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS WU SHENG-PING Abstract. This article try to unified the four basic forces by Maxwell equations, the only experimental theory. Self-consistent Maxwell equation with the e-current from matter current is proposed, and is solved to four kinds of electrons and the structures of particles. The static properties and decay and scattering are reasoned, all meet experimental data. The equation of general relativity sheerly with electromagnetic field is discussed as the base of this theory. In the end the conformation elementarily between this theory and QED and weak theory is discussed. Contents 1. Unit Dimension of sch 1 2. Self-consistent Electrical-magnetic Fields 2 3. Calculation of Recursive Re-substitution 3 4. Solution 4 5. Electrons and Their Symmetries 4 6. Propagation and Movement 6 7. Conservation Law and Balance Formula 7 8. Effect 8 9. Muon Pion Pion Neutral Tau Proton Neutron Great Unification Conclusion 11 References Unit Dimension of sch A rebuilding of units and physical dimensions is needed. Time s is fundamental. We can define: The unit of time: s (second) The unit of length: cs (c is the velocity of light) The unit of energy: /s (h is Plank constant) Date: Mar 27, Key words and phrases. Maxwell equations, Decay, Electron function,recursive resubstitution. 1

2 2 WU SHENG-PING The unit dielectric constant ɛ is [ɛ] = The unit of magnetic permeability µ is then [µ] = We can define the unit of Q (charge) as [Q]2 [E][L] = [Q]2 c [E][T ]2 [Q] 2 [L] = c[q] 2 cɛ = cµ = 1 [Q] = [H] = [Q]/[L] 2 = [cd] = [E] Then : C = ( ) 1/2 C is charge SI unit Coulomb. For convenience we can define new base units by unit-free constants c = 1, = 1, [Q] = then all physical unit are power of second s n, the units are reduced. Define UnitiveElectricalCharge : σ = σ = C 64e e /σ = e/σ = /64 Define s Cs =: κ : m e = 1 We always use it. 2. Self-consistent Electrical-magnetic Fields Try so-called expanded Maxwell equation for the free E-M field in mass-center frame (2.1) A = ia ν A ν /2 + cc. = J, σ = 1 or with definition A = ie /σ A ν A ν /2 + cc. = J, e = 1 ν A ν = 0 (A i ) := (V, A), (J i ) = (ρ, J), (J i ) = ( ρ, J) := ( i ) := ( t, x1, x2, x3 ) := ( i ) := ( t, x1, x2, x3 ) It s deduced by using momentum to express e-current in a electron: the mass and charge have the same movement in electron. The equation 2.1 have symmetry CP T, cc.p T The energy of the field A are (2.2) ε := dv (E E + H H )/2 = < A ν 2 A ν > /2

3 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 3 It s also valid with Lorentz gauge. As a convention the time-variant part is neglected. With invariant normalization for one particle, the condition is found: (2.3) 1 = e /σ dt < A ν, A ν >, e = 1 by the non-homogeneous equation Calculation of Recursive Re-substitution We can calculates the solution by recursive re-substitution (RRS) for the two sides of the equation. For the equation ˆP B = ˆP B make the algorithm (It s approximate, the exact solution needs a rate in the resubstitution) ˆP ( B k + B n+1 ) = ˆP B k k n k n One can write down a function initially and correct it by re-substitution. Here is the initial state V = V i e ikt, A i = V, µ µ A ν i = 0 Substituting into equation 2.1 and then we an find hence ν J ν = 0 ν A ν = 0 We calls the fields correction A n with n degrees of A i the n degrees correction. The dynamic process in the mass-center frame also can be solved by RRS, with harmonic (or others) initial wave. The second degree correction (dependence) is < ia ν f A fν + cc. 1/( ) ia ν i A iν + cc. > /4 The normalization is on < A ν A ν >, the static mass of wave. The decay to a stable state is calculated in isolated system. It s a process the EM energy ε t = ε ε f transfer to mechanical (kinetic) energy E k = ε 0 ε t. With the matter number invariant normalization in space-time (2.3) then = 1 = 0 0 e /σ dt < A ν, A ν >, e = 1 e /σ dt < A ν, J ν > /2 = C = e /σ ε 0 ε t /ε 0 = e Ct 0 dte /σ ε 0 e Ct ε 0 ε t is of cause the decrease of the crossing energy of all the decayed blocks.

4 4 WU SHENG-PING 4. Solution Firstly 2 A = k 2 A is solved. Exactly, it s solved in spherical coordinate 0 = r 2 ( 2 f k 2 f) = k 2 r 2 f + (r 2 f r ) r + 1 sin θ (sin θf θ) θ + 1 sin 2 θ (f φ) φ Its solution is f = RΘΦ = R l Y lm Θ = Pl m (cos θ), Φ = cos(α + mφ) R l = Nj l (kr) j l (r) is spherical Bessel function. and Then by its Fourier form j 1 (r) = sin(r) r 2 cos r =: NJ(x) r j 1 (0) = 0 with normalization of F (x) 0 r 2 j 1 (ar)j 1 (br)dr = 1 δ(a b) 2a2 F (x) := NR 1 (r)y 1,1 (θ, φ) < F (k x), F (k x) >= δ(k) F (x) F (x) = δ(x) < F (k x) F (x) F (k x) F (x) >= 1/k 2 1k 2 2k 2 3 The solution of l = 1, m = 1, Q = e /σ is calculated or tested for electron, V = NR 1 (kr)y 1, 1 e ikt 5. Electrons and Their Symmetries Some states of electrical field A are defined as the core of the electron, which s the initial function A i = V that is electrical, for the re-substitution to get the whole electron function e: : NR 1 (k e r)y 1,1 e iket e r e + r e + l : NR 1 (k e r)y 1, 1 e iket : NR 1 (k e r)y 1, 1 e iket : NR 1 (k e r)y 1,1 e iket r, l is the direction of the spin. Normalize the electron function with charge and mass Then < e µ i t e µ > /2 + cc. = 1/e /σ, e = 1 < e µ e µ > /4 + cc. = 1, σ = 1 k e = 1

5 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 5 Figure 1. The function of j 1 The magnetic dipole moment of electron is calculated as the first rank of proximation r A/4 + cc. µ z =< A i i φ A i > /4 + cc. = Q e 2k e By the discussion in the section 2 the spin is Define It meets the following results S z = µ z k e /Q e = 1/2 ς k,l,m (x) := R l (kr)y l,m, ς k = ς ± k (x) := ς k,1,±1(x) ς n 1 (x)e int ς n 1 (x)e in t δ(t r)/(4πr) = ς n 1 (x)e int ς n 1 (x)e in t The correction of the equation 2.1 is (5.1) A n = A n 1 i (A i A i )/2 u, k = 1 = (A i (i t A i ))((i t (A i A i )/2) n 3 ((i (A i A i )/2) u = δ(t r)/(4πr) The convolution is made in 4-d space. The function of e + r is decoupled with e + l < (e + r ) ν + (e + l )ν, (e + r ) ν + (e + l ) ν > /2 < (e + r ) ν, (e + r ) ν > /2 < (e + l )ν, (e + l ) ν > /2 = 0 is The increment of field energy e /σ ε on the coupling of e + r, e r mainly between A 2 e /σ ε e 2e 3 /σ m 1 e = s

6 6 WU SHENG-PING This value of increments on the coupling of electrons are ε e e + r e r e + l e + r e r e + l The increment of field energy e /σ ε on the coupling of e + r, mainly between A 4 is e /σ ε x 1 4 e7 /σ m 1 e s The calculations referenced to latter theorems. This value of increments on the coupling of electrons are ε x e + r e r e + l e + r e r e + l Because ( e) = ie /σ ( e) ν ( e) ν /2 + cc. the properties of electrons are The propagation: e + r e r e + l e + r e r e + l Q S r r l l l l r r µ B r l l r l r r l ε(m) The following are stable propagation: 6. Propagation and Movement A := f e i, i f e = d 3 yf(x y)e(y) particle electron photon neutino notation e + r γ r ν r structure e + r (e + r + e r ) (e + r + ) The movement of the propagation is called Movement, ie. the third level wave: f f i e ij Calculating the following coupling system of particle x A := e x (A + A ), < e x e x >= 1 A + := c n c e c A := c n c e c

7 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 7 c, c differ in charge, spin and being negative or not. Make RRS with the initial state e x : NR 1 (k x )Y e ikxt to get the whole function e x. With the normalization conditions of charge (6.1) < e x ( i e i ) ν i t e x ( i e i ) ν >= Q x hence k x cc n 2 cc / Q x < e x ( i e i ) ν e x ( i e i ) ν > /2 k x The static MDM (magnetic dipole moment) of a group of electrons is µ x = r (e x i e i )/4 + cc. Use the two degree static correction of A i as in 5.1 e 2 /σ < e iµ φ e µ i > k ei/(4k x ) + cc. i The harmonic wave has in Lorentz transform e ipx e(x) = e ip x e(x ) p ν p ν = p ν p ν = 0 Consider the reaction Because 7. Conservation Law and Balance Formula A 1i A 2i A 1f A 2f t t(< A 1, A 2 > + < A 2, A 1 >) = t t(< A 2, A 1 > + < A 1, A 2 >) with their Fourier expansions, hence equivalently with the same energy emission A 1i + A 2f A 1f + A 2i No matter in E-M fields level or in movement (the third) level, the conservation law is conservation of momentum and conservation of angular momentum. A balance formula for a reaction is the equivalent formula in positive matter, ie. after all negative terms is shifted to the other side of the reaction formula. Balance formula is suitable for the analysis of the energy transition of decay. The invariance of electron itself in reaction is also a conservation law.

8 8 WU SHENG-PING 8. Effect Generally, there are kinds of EM energy increments i.e. EM effect Weak coupling W : < (e + r + )ν (e + r + ) ν > /2 < (e + r ) ν (e + r ) ν > /2 < ( )ν ( ) ν > /2 Light coupling L : < (e + r + e r ) ν (e + r + e r ) ν > /2 < (e + r ) ν (e + r ) ν > /2 < (e r ) ν (e r ) ν > /2 Weak side coupling W s : < (e x (e + r + ))ν (e x (e + r + )) ν > /2 < (e + r ) ν (e + r ) ν > /2 < ( )ν ( )ν > /2 Light side coupling Ls : < (e x (e + r + e r )) ν (e x (e + r + e r )) ν > /2 < (e + r ) ν (e + r ) ν > /2 < (e r ) ν (e r ) ν > /2 Strong coupling is S : < (e + r ) ν (e + r ) ν > /2 The core of muon is 9. Muon µ + l : A i = e µ (e + l e r e + l ) Make RRS on e µ like 5.1 to get the solution. These e are complete electron functions. From the equation 15.1, µ is approximately with mass 3m e /e /σ = 3 64m e [3.2][1] (The data in bracket is experimental by the referenced lab), spin S e (electron spin), MDM µ B k e /k µ. The main channel of decay with balance formula µ + l e r + ν l ν l e µ e + l + e ip1x e r + e ip3x ν l e µ ν l + e ip2x ν l The main EM effect is kind of W s < (e µ (e + l + e r )) ν (e µ (e + l + e r )) ν > /2 + < (e + l )ν (e + l )ν > /2+ < (e r ) ν (e r ) ν > /2 The neutrino coupling in e ipx ν l is still coupled, and in e x ν l, decoupled. In fact calculate the equation 6.1 to find k µ exactly 2ε xm e 1 = k µ s /e /σ [ s][1]

9 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 9 The core of pion is 10. Pion π + r : e π e + r + e π (e + r + ) It s approximately with mass 4 64m e [4.2][1], spin S e and MDM µ B k e /k π +. Decay Channels: π + r µ l + ν r e π e + r + e π + e ip1x e µ ν r e ip1x e µ ν r + e ip2x ν r The main EM effect is kind of W 1 ε x = s /e /σ [( s][1] 11. Pion Neutral The core of pion neutral is like a atom π 0 : (e + r + e + l, e r + ) It has mass approximately 4 64m e [4.2][1], zero spin (?) and zero MDM. Its decay modes are π 0 γ r + γ l The loss of energy is kind of L 1 2ε e = s /e /σ [ s][1] The core of tauon maybe that 12. Tau τ + l : e τ (5e + r 5 e + l ) Its mass approximately 51 64m e [54][1], spin S e, MDM µ B k e /k τ. It has decay mode τ + l µ + l + ν r ν r e τ 5e + r + e ip3x ν r + e ip1x e µ ν l e τ 5e + r + e τ e r + e ip1x e µ e + l + e ip2x ν r The main EM effect is kind of Ls < (e τ (5e + r + e r )) ν (e τ (5e + r + e r )) ν > /2 + < 5(e + r ) ν 5(e + r ) ν > /2+ < (e r ) ν (e r ) ν > /2 18ε e 1 = k τ /m e s /e /σ 1 = s 0.17 /e /σ [ s; BR. 0.17][1] Depending on this kinds of particle including q n+ r := n(e + r ) we can construct particles of great mass decaying without strong effect (light radiative), for example e x (q n e) This series of particle include µ, τ and in fact almost all light radiative particles are of this kind, they are created in colliding.

10 10 WU SHENG-PING 13. Proton The core of proton may be like p r : e p ( 4e + r 3e r 2 ) The mass is 29 64m e [29][1] that s very close to the real mass. calculated as 3µ N, spin is S e. The proton thus designed is eternal. The MDM is 14. Neutron Neutron is the atom of a proton and a electron and a neutrino, n = (p + r, ν l, ) Muon circles around proton with m µ ωr 2 = 1 The effect is between their (proton and muon) magnetic fields of A 2 (gross current) m ν ω 2 r ε e k e /(k p r 2 ) m ν = 1 2 e6 /σ k e The EM energy emitted by neutrino is approximately e6+4+2 /σ k 3 e/(k 2 pe 2 /σ ) = s /e /σ 15. Great Unification The mechanic feature of the electromagnet fields is T is stress-energy tensor, T ij = F k i F kj g ij F µν F µν /4 T ij = mu i u j, u = dx/ds T 00 is quantum expression of the energy, by Lorentz transform it s easy to get the quantum expression of momentum. The observed mass in mass-center frame is M = dv T 00 (15.1) M =< A ν, A ν > /2 = ε The General Theory of Relativity is (15.2) R ij 1 2 Rg ij = 8πGT ij /c 4 Firstly we redefine the unit second as S to simplify the equation 15.2 R ij 1 2 Rg ij = T ij We observe that the co-variant curvature is R ij = F ikf k j + g ij F µνf µν /8

11 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS Conclusion Fortunately, this model explained all the effects in the known world: strong, weak and electromagnetic effects, and even subclassify them further if not being to add new ones. In this model the only field is electromagnetic field, and this stands for the philosophical with the point of that unified world is from an unique source, all depend on a simple hypothesis: the current of matter in a system can be devised to analysis the e-charge current. Except electron function my description of particles in fact is compatible with QED elementarily, but my theory isn t compatible to the theory of quarks. In fact, The electron function is a good promotion for the experimental model of proton that went up very early. Underlining my calculations a fact is that the electron has the same phase (electron resonance), which the BIG BANG theory would explain, all electrons are generated in the same time and place, the same source. References [1] K. Nakamura et al. (Particle Data Group), JPG 37, (2010) (URL: address: sunylock@139.com 8 Hanbei Rd, Tianmen, Hubei Province, The People s Republic of China. Postcode: