Generally Covariant Generalization of the Dirac equation that does not require gauges

Postulate 1 . z=zz (algebraic definition of 1)

 Dirac in 1928 made his equation(1) flat space(2). But space is not in general flat, there are forces. So over the past 100 years people have had to try to make up for that mistake by adding adhoc convoluted gauge force after gauge force until theoretical physics became a mass of confusion, a train wreck, a junk pile. So all they can do for ever and ever is to rearrange that junk pile with zero actual progress in fundamental theoretical physics being made,.. forever. By the way note that Newpde (3)

References 

Click on pink titles below for backup physics:

  Postulate 1 

Introduction

 Part I     1U1 states

Part II    1U1U1 states 

Part III   Mixed State

 Rest       Miscellaneous

Summary:

Start with an: Occam’s razor optimized (i.e., δC=0, ||C||=noise)
       POSTULATE OF 1                   So                                                                                   
        z=zz (1) is the algebraic definition of 1,o. So add constant C (i.e., z’=z’z’-C, δC=0) (2))    Solve eqs.1,2 for z’.       (Use this z’=1+δz to find δz=psi in the Newpde.) So just postulate1               

This is an Occam’s razor optimized (δC=0, z=zz not z=zzzzz) postulate of 1. You aren’t going to do any better than “optimize” because of real noiseC 

  Solve eq.1,2 for z’ with iteration and successive approximation.

Iteration Plug z’=1+δz into eq.2 get                                δz+δzδz=C      (3) The eq.3 quadratic eq. solution  for realC<<-¼ is   δz=dr+idt   (4)                                               imbedding real line C in the complex plane. Substitute z’ on right (eq.2) into left z’z’ repeatedly& get z_N+1=z_Nz_N+C. δC=0 requires us to reject the Cs for which δC=δ(z_N+1-z_Nz_N)=δ(∞-∞) NOT 0. Initial  z=z_o=0 is then Mandelbrot set C_M (fig2).  Eq1 solution is 1,0 soinitial  z=z_o=0 gets the Mandelbrot set C_M (fig1) out to some ||C|| distance from C=0.  C is found from δC=(dC_M/d(||C||))d||C||=0 extremum giving the Fiegenbaum point ||C_M|| =   ||-1.40115..|| global max given this  ||C_M|| is largest in fig.1.  The intersection of real C<-¼ line with C_M is at the (dense C_M) Fiegenbaum pt ‘providing at least tiny real line C=C_M displacements’ dr’dt’=|D|²<<1). Successive approximation z=1+dz=0 so δz=-1. Thus from eq.3 δ(δz+δzδz)=0= ddz(1)+2(ddz)(dz)=δδ(1)+2(δδz(-1+D)= (δδz(-1+2D)= ½δ(δzδz)+δδzD~½δ(dzdz)=0=  Therefore plug eq.4 in here: δ(dz+dzdz)~½δ(dzdz)=½δ[dr²-dt²+i(drdt+dtdr)]=0  (5) 

 real and Imaginary->Dirac equation->spin½ lepton                                         gⁱ are the Dirac matrices here. The positive scalar drdt in eq.7, (fig.3, quadrant I) implies the eq.5 non infinite extremum imaginary drdt+dtdr=0= gⁱdrg^jdt+ g^jdtgⁱdr= (gⁱg^j+g^jgⁱ)drdt so Clifford algebra (gⁱg^j+g^jgⁱ)=0, i≠j. Also factor eq.5 Minkowski real  δ(dr²-dt²)=  δ[(dr+dt)(dr-dt)] =0=[[δ(dr+dt)](dr-dt)]+[(dr+dt)[δ(dr-dt)]]=0 get

(+-e)    dr+dt=ds, dr-dt=ds  =ds₁     for    +ds-> I, IV quadrants       (7)   (light cone v) dr+dt=ds, dr=-dt,   for   +ds->   II quadrant             (8)       “        “             dr-dt=ds,  dr=dt,      “         “    III quadrant            (9)   (->vacuum)       dr=dt,       dr=-dt      so dt=0=dr (So eigenvalues of dt, dr=0 in eq.11) (10) 

We square eq.7  ds₁²=(dr+dt)(dr+dt) =[dr²+dt²] +(drdt+dtdr) =ds²+ds₃=ds₁². Since ds₃ (is max or min) and ds₁² (from eq.6) are invariant then so is Circle ds²=dr²+dt² =ds₁²-ds_3. Note this separate ds is a minimum at 45deg and so Circle=dz=dse^iA=dse^i(dA+Ao)= dse^{i((cosAdr+sinAdt)/(ds)+Ao)},  A_o=45deg. We define k=dr/ds,  w=dt/ds, sinA=r, cosA=t. dse^i45deg=ds’. So take the ordinary dr derivative (since flat space) of ‘Circle’  so , ). Cancel that e^i45deg coefficient then multiply both sides of eq.11 by h and define dz=psi, p=hk. then implies (Hermitian) operator hk observables formalism(QM,  also appendix B3). Eqs 7-11 and eq.6 Clifford algebra imply 2D Dirac equations for e,v.     z=1 small C’ in eq.2 must Lorentz gamma boost Reldz to small scales Explicitly D=C’≈Reldz‘>> Rel(dz’dz’) in eq.6. Thus strongly nonuniform Reldz’=C_M/m=C_M/m₁=C’=r_H defining mass m₁ and charge C_M. So for A_o=45deg then perturbed uniform C eq.7 reads:      (dr-dz’)+(dt+dz’) =dr’+dt’=ds           (12)                                          since ds invariant. Define k_rr=(dr/dr’)²=(dr/(dr-(C_M/m₁)))²=1/(1-r_H/r)² =A₁/(1-r_H/r) +A₂/(1-r_H/r)²  r=dr,N=-1. The partial fractions A_I term can be split off from RN and so k_rr=1/[1-((C_M/m₁)r))] (13)     So we have: ds²=k_rrdr’² +k_oodt’² +..(14).   From eq.6 dr’dt’=sqrtk_rrdr’sqrtk_oodt’=drdt so  k_rr=1/k_oo  (15) So given this 2D perturbation we get curved space 4D. For 4D our eq.3 Clifford algebra then implies (g^xsqrtk_xxdx+g^ysqrtk_yydy+g^zsqrtk_zzdz+g^tsqrtk_ttidt)²=k_xxdx²+k_yydy²+k_zzdz²–k_ttdt²=ds². Multiplying the bracketed term by 1/ds & δz=psi so eq 11 implies 4D Newpde   gⁱ(sqrtk_ii)dpsi/dx_i=(w/c)psi  for e,v                                       (16) (covariant derivative still ordinary since psi (complex) scalar). Therefore postulate1>Newpde  

 

For spherical symmetry curved space k_oo, k_rr:    (g^xsqrt(k_xx)dx+g^ysqrt(k_yy)dy+g^zsqrt(k_zz)dz+g^tsqrt(k_tt)idt)²= k_xxdx²+k_yydy²+k_zzdz²–k_ttdt²=ds²
In contrast k_xx=k_yy=k_zz=k_tt=1 is flat space, Minkowski,  in the ordinary 1928 Dirac equation. The difference between the ordinary Dirac equation and the Newpde is that we didn’t just drop that k_mn. There actually are forces after all!

To make up for this mistake we have as result for a 100 years been adding gauge force after gauge force creating a adhoc convoluted mess, a train wreck, a junk pile which we are stuck in forever.

Figures:

Figure 1 solves the puzzle of physics and real#math (just postulate 1) and figure 5 explains why. 

(Ockam’s razor motivated) Postulate 1 <->Physics   :

z=

Mandelbrot Set Plug in the left side (of eq.1) z into the right side zz repeatedly and use  δC=0 and get the Mandelbrot set iteration formula.                                                                                               The Mandelbrot set C_M is (and from the postulated  δC_M=0)_, z_N+1=z_Nz_N+C_M (since  δ(z’-zz)=δ(z_N+1-z_Nz_N)= δ(∞-∞)=0). (the z=zz) z=0=z_o. Fiegenbaum point C_M smallest real line Mandelbulb on next smaller (baseline) scale. Mandelbulb areas (drdt) for smallest Clifford algebra extremum drdt.

fig.4

Boost  m₁=tau+mu+m_e=3e from S=½ to L=1 at stable r=r_H, 2P_3/2 with m₁/2=m_p reduced mass. See above sect.2 of above postulate 1 file for some more applications. See PartI, partII, PartIII for the rest.      

Comparison and contrast with mainstream concerns:

The origin of the many postulates of quantum mechanics is considered to be a mystery by one and all. 

Not really:  1) Quantum mechanics arises from a circle.(see below)

 The connection between general relativity and quantum mechanics is stated over and over again to be the greatest unsolved problem in physics.

Not really:  2) General Relativity(GR) comes out of quantum mechanics in 1 step.(see below)

A century has been devoted to the origin of high energy particle physics by thousands of people using billions of dollars worth of particle accelerators. It is supposed to be quarks, gluons, qcd SU(3) gauge forces, require many free parameters,..and so on and so forth.

Not really:  3) (high energy 3e composite) Particle physics is just a ortho and para state (see below)

The latest version “Standard electroweak Model” is now encumbered by pages of Lagrangian density terms, and so literally hundreds of postulates.

Not really: 4) Rotate e,v to get the Standard electroweak Model (it’s merely composite e,v). (see below)

Expansion of the above answers: 

(1)Plug z=1+δz into  z=zz+C, δC=0 (eq.1) and get δ(δz+δzδz)=0 which splits into  Im Clifford Algebra and Real Minkowski metric components which imply that circle ds²=dr²+dt² so δz=dse^iA. Write in angle ‘A’ radian measure dependence on r,t, take the r partial derivative and rearrange and you get a Hermitian operator and so “observables” formalism and then the Dirac equation of quantum mechanics (given that Clifford algebra). See sect.1 of above postulate 1 file.

(2)  Note square roots of k_oo=1-r_H/r=1/k_rr in the new pde (so same k_ij math functionality as in the Schwarzschild metric g_ij but coming from QM instead.).  So squaring (actually iterating) the New pde on the next higher(cosmological)  fractal scale, where r_H=2GM/c², gets the Schwarzchild metric gij of GR, a one step derivation. General relativity comes out of quantum mechanics in one simple step!

(3) The two positrons in 2P_3/2 on the r=r_H shell are moving rapidly, the central electron is not moving so much at the center, and this New pde 2P_3/2 state is stable because of h/e flux quantization(PartII) and r=r_H motion. So a Clebsch Gordon two body analysis with Paschen Back effect excitation, in a straightforward way, leads to a ortho(s,c,b) and para(t) states with ground state(u/d), the proton. Use Frobenius series solutions (Ch.9) individually for each of the s,c,b,t associated Nucleonic doublets (eg.,P,N for u/d) to get the particle multiplets. See PartII also.

4) The 4 axis (New pde) e,v rotations (giving consecutively the 4 Bosons: Zo,W+,W-,photon) at r=rH on the r,t plane are implied by eq.7 (also see appendix A). We even get Maxwell’s and Proca’s equations in these iterative rotations through the New pde e,v solutions which means taking a first derivative at each 45deg. rotation.  

Conclusion: Ultimate Okcam’s razor postulate 1 implies math AND physics. We finally figured it out.