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 5 6. Propagation and Movement 6 7. Conservation Law and Balance Formula 8 8. Muon 8 9. Pion Positive 9 1. Pion Neutral Tau Proton 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, 218. 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 The unit of charge can be reset by linear variation of charge-unit Q CQ, Q : σ/c We will use it without detailed explanation. 2. Self-consistent Electrical-magnetic Fields Try equation for the free E-M field in mass-center frame (2.1) A = ia ν A ν /2 + cc. = J, Q e = 1 or with definition ν A ν = A = ie /σ A ν A ν /2 + cc. = J (A i ) := (V, A), (J i ) = (ρ, J), (J i ) = ( ρ, J) := ( i ) := ( t, x1, x2, x3 ) := ( i ) := ( t, x1, x2, x3 ) Q e is the absolute value of the charge of electron. It s deduced by using momentum to express e-current in a 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. The condition of invariant normalization for one particle is: (2.3) 1 = dt < A ν, A ν > except eternal particles, because the equation 2.1 is non-homogeneous, and A, A /( dt < A ν, A ν >) 1/2, A = na both describe the same field of one particle and both meet the equation Calculation of Recursive Re-substitution We can calculates the solution by recursive re-substitution 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 = Substituting into equation 2.1 with A = ia ν A ν /2 + cc., Q e = 1 ν A ν = 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 recursive resubstitution in the same form, 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 A >, 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 energy E k = ε ε t. With the matter number invariant normalization in space-time (2.3) then = 1 = dt < A ν, A ν > dt < A ν, J ν >= C = ε ε t /ε = e Ct dtε e Ct ε t is of cause the decrease of the crossing energy of all the decayed blocks.

4 4 WU SHENG-PING Figure 1. The function of j 1 Firstly 4. Solution 2 A = k 2 A is solved. Exactly, it s solved in spherical coordinate = 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 j l (x) is spherical Bessel function. Contrary to the well-known result: Θ = Pl m (cos θ), Φ = cos(α + mφ) R l = Nj l (kr) j 1 (x) = sin(x) x 2 j 1 () = cos x x x 2 j 1 (ax)j 1 (bx)dx = 1 δ(a b) a the functions j 1 (ax), j 1 (bx) are not orthogonal (why? CV), because a direct calculation shows that. 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 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 5 5. Electrons and Their Symmetries Some states of electrical field A are defined as the core of the electron, it s the initial function A i = V that is electrical, for the re-substitution to get the whole electron function: e + r : NR 1 (k e r)y 1,1 e iket 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. We use these symbols e to express the complete potential field A or the abstract particle. Define the notations of particle x e + xr := NR 1 (k x r)y 1,1 e ikxt By mainly the second rank of correction A 2, a static field, we have < e cµ i t e µ c > /2 + cc. = Q c, σ = 1, Q c = Q e We can use this to normalize the electron functions. < e cµ e µ c > /2 + cc. k e Q c, σ = 1 < e cµ i φ e µ c > /4 + cc. µ zc 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, σ = 1 By discussion in the section 13 the spin is The correction of the equation 2.1 is S z = µ z k e /Q e = 1/2 A n = A n 1 ( (ia i ia i )/2) u = A i (i t A i )( t (ia i ia i )/2) n 3 (ia i ia 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 = The increment of field energy ε on the coupling of e + r, e r ε e e 3 /σ k 1 e = s mainly between A 2 is

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 ε on the coupling of e + r, mainly between A 4 is ε x 1 2 e7 /σ k 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) = i( e) ν ( e) ν /2 + cc., Q e = 1 the properties of electrons are 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/2) Propagation and Movement Because field F is additive, the group of electrons are express by: (6.1) F = f i e i, < f i f i >= 1 i It s called propagation. The convolution is made only in space: f g = d 3 xf(t, x)g(t, y x) Each f i is normalized to 1. We always use f i e i, f i e i i i to express its abstract construction and the field. The following are stable propagation: particle electron photon neutino notation e + r γ r ν r structure e + r (e + r + e r ) (e + r + ) Define ς k,l,m (x) := R l (kr)y l,m, ς k = ς ± k (x) := ς k,1,±1(x) we can find easily it meets the following results, especially as the normalization is considered,

7 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 7 Theorem 6.1. R is any rotation. < R(ς ± k (x)), ς k (y x) >=, y O (ς k ς k ) = ( ς k ) ς k + ς k (ς k ) Proof. Calculate and use the identity ( ς k ) ς 1 = kς k ς 1 < (Nς 1 ) n, (Nς 1 ) n >=< Nς 1, Nς 1 >= 1 ς n 1 e int ς n 1 e in t δ(t r)/(4πr) = ς n 1 e int ς n 1 e in t F λi (ς n 1 ς n 1 1 4πr ) eiλixi δ(t r)/(4πr) = δ(t)/(4πr) δ(t r) The movement of the propagation is called Movement, ie. the third level wave: F = f f i e i Calculating the following coupling system e x : e x (A + A ) A = i e i, A = j e j Its field is B =: B + B dx e x A, B := dx e x A With the condition 2.1 the state of lowest energy is e x = (6.2) e x = e int ς N With the condition of charge hence < (B + B ) ν i t (B + B ) ν > /2 + cc. Q x, Q e = 1 (6.3) N (< A ν, A ν > + < A ν, A ν >)/Q x, Q e = 1 Consider the reaction e + r e + r e ipx e + r e ip x e + r to find the crossing part between e + r, e + r is zero. This result is crucial in calculating the the mass etc of particle. In fact, when the whole wave form is unclear, the first correction of interaction can be obtained by the current, and it s clear that the crossing part is zero. We always use t t < B, A >= t t dv (h(t) + h( t))a (h(t) + h( t))b

8 8 WU SHENG-PING with their Fourier expansions. The right is the crossing between initial state and final state, but the left is not. This is the reason why we can shift electrons from one side to the other in the reaction formula. The static MDM (magnetic dipole moment) of a group of electrons is r ( dx e x e µ j )/4 + cc., Q e = 1 j µ =< dx e x j e jµ r i dx e x j e µ j > /4 + cc. =< e jµ r i j e µ j > Q x k e /(4k x ) + cc. Make The harmonic wave has in Lorentz transform to prove. e ipx e(x) = e ip x e(x ) p ν p ν = p ν p ν = dt(e ipx e(x) e ip x e(x )), dx i = dt 7. Conservation Law and Balance Formula 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. Muon Generally, there are kinds of energy increments. Weak coupling W :< (e + r ) ν ( ) ν > Light coupling L :< (e + r ) ν (e r ) ν > Weak side coupling W s :< e x (e + r ) ν e x ( ) ν > < e x (e + r ) ν e ipx ( ) ν > Light side coupling Ls :< e x (e + r ) ν e x (e r ) ν > < e x (e + r ) ν e ipx (e r ) ν > Strong coupling S :< (e + r ) ν (e + r ) ν >

9 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 9 The core of µ is composed of µ + l : e µ (e + l ν l ) From the equation 13.1, 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 µ + l M + l ν l M + r = e M ( e r + ν l ) e µ e + l + e ip1x e M e r + e ip2x ν l e µ ν l + e ip1x e M ν l The outer waves e µ and e ip1x e M, e µ and e ip2x are coupling. The energy difference is kind of W s, the interacting field is between A 4. 2 < e µ (e + l )ν e ip1x (e r ) ν > 2 < (e + l )ν (e r ) ν > +cc. Sum up in spectrum of p 1 that with the emission positive: k e = 2 < (e µ (e + l k µ + k, A 4) ν ) e int (e r, A 4 ) ν /k e > 2 < e + l (e r ) ν ) > +cc. e The emission of decay is 1 = s 2k µ = ( 2)ε x k µ + k e = 2ε xk e k µ + k e [ s][1] The data in square bracket is experimental data. The decay of particle M is like a scattering with no energy emission The core of Pion positive is M + r + ν l 9. Pion Positive π + l : e πr e + r + e πl ( 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 + l µ l + ν r It s with balance formula e π e + r + e πr + e ip1x e µ ν r e πl e + r + e ip1x e µ + e ip2x ν r The the emission of energy, ie. the difference of cross energy, is kind of W ε x = s [( s][1]

10 1 WU SHENG-PING 1. Pion Neutral The core of Pion neutral is atom-like particle π : e πr (e + r + e + l ) + e πl (e r + ) It has mass approximately 4 64m e [4.2][1], zero spin and zero MDM. Its decay modes are π γ r + γ l The loss of energy is kind of L 1 2ε e = [ s][1] s The core of τ maybe that 11. Tau τ r : e τ (5e + r 5e + r e r ) 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 + l ν l e τ 5e + r + e ip1x e µ ν l + e ip2x ν l e τ 5e + r + e τ e r + e ip1x e µ e + l + e ip3x ν l The energy gap is kind of Ls 5 < e τ (e + r ) ν e ip1x e µ (e r ) ν > 5 < e τ (e + r ) ν e τ (e r ) ν > +cc. = 5 < e τ (e + r ) ν e ip1x (e r ) ν > 5 < e τ (e + r ) ν e τ (e r ) ν > +cc. 5ε e k τ /k e 1 = [ s; BR..17][1] s Depending on this kinds of particle including q n+ r := n(e + r e + r ) we can construct particles of great mass decaying without strong emission (light radiative), for example e L (qr n+ ) This series of particle include µ, τ and in fact almost all light radiative particles are of this kind, they are created in colliding. Another condition is possibly that, in the collision, the created light radioactive particle (qr n+ ) with different n is mixed to some rates as to the detector can t distinguish them. Because the channel width decides the channel branch rates, obviously the most experimental data violate this rule. So that the channels listing after the same name of a particle in fact belong to different particles. The core of Proton may be like 12. Proton p r : e + pr 4e + r e pr 3e r e pl 2e l 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

11 MATTER THEORY OF EXPANDED MAXWELL EQUATIONS 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 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 with velocity v of the system is M = 2 vt /2 v= = 2 vtr(t ij )/4 v= (13.1) M = tr(t ij ) =< A ν, A ν >= 2ε, Q e = 1 The General Theory of Relativity is (13.2) R ij 1 2 Rg ij = 8πGT ij /c 4 Firstly we redefine the unit second as S to simplify the equation 13.2 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 14. 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, 7521 (21) (URL: address: sunylock@139.com 8 Hanbei Rd, Tianmen, Hubei Province, The People s Republic of China. Postcode: 4317