Define Real Observable1

   Abstract                                  Define real observable 1  

Telling people that it all comes down to the Occam’s razor postulate of 1 is the truth (the simplest idea imaginable).
But they will never believe you unless you also point out that 1 is not just a squiggle on a piece of paper. To be meaningful and tangible we define real observable 1. Note the math then becomes rigorous(see below flow chart)  all the way down to the new pde because “real” (number) and “observable” (operator) are precisely definable. 

Then you point out the physics applications of the new pde which are awesome.

                           DEFINITION OF REAL OBSERVABLE 1 .   postulate real observable  1 . list-define math (sect.4)              . . z=zz algebraic definition of 1 is the small C limit of           . z=zz+C, δC=0,C<0  (1) needed to define real observable 1

    Plug left side z of eq.1 into right           |            Plug z=1+δz into eq.1 zz repeatedly and use δC=0 get   |   get δ(δz+δzδz)                                       Mandelbrot set  iteration                  | Im=Clifford Alg,real SR-> Cauchy seq subset generates real# 1        |                operator formalism  Fiegenbaum pt. subset from Clifford  alg|     so Dirac eq.(flat space) . | |   . new pde psi (curved space,fractal)                The (dr+dt)/ds=1 Hermitian operator on this psi defines observable1

 Note all we did here was to define real observable 1. But by writing out this definition we also inadvertently derived both real# math and physics from the postulate of 1. It is hard to exaggerate the importance of “figuring out” the actual origin of physics and (real#) mathematics. 


(A) Substitute z=1+δz into eq.1 and get δ(δz+δzδz)=0 (2) (B) Substitute the left side of z =zz+C back into the right side zz of eq.1 repeatedly and use δC=0 and get the Mandelbrot (1)set (fractal) iteration formula for some CM.

(A) So from eq.2 (δz-K)+δzδz=C (constant C and K) in general is a quadratic eq. with complex solution δz=dr+idt. Plug that back into eq.2 with K=δz to initialize to flat space and get δ(dr2+i(drdt+dtdr)-dt2)=0 since dr2-12dt2=ds2 is special relativity (Minkowski metric given 12=natural unit constant speed2=c2) invariance. The imaginary extremum is the Clifford algebra dr’dt’+dt’dr’=grdrgtdt+gtdtgrdr=0 since 2drdt=0 here for vacuum. gt is the Dirac gamma matrix. .Factor the real component and get 5 equations (eg.,e; dr+dt=ds, dr-dt=ds (3),etc.,dr-dt in IV quadrant so ds>0. (Complex unknown K for K=δz+δz’ (δz’)perturbation adds 2 degrees of freedom.)   We just derived special relativity here!

Square eq.3 to get +ds2=(dr+dt)2=(dr2+dt2)+drdt+dtdr implying dr2+dt2 =ds2 circle invariance at 45° since dr+dt and drdt+dtdr (cross term) are invariant. So dz=dseiq= dsei((sinqdr+cosqdt)/ds). Take the r derivative, define dr/ds=k, sinq=r, dz=psi and multiply both sides by ih and define momentum p=hk=mv to get the operator formalism prpsi=-ihdpsi/dr All three invariances imply the Dirac equation for e,v. Composite e,v is the GSW model. Composite 3e is baryons. We just derived quantum mechanics here!   

K not δz+δz’  The Clifford algebra small drdt area extremum is then the real# line drdt Mandelbulb Fiegenbaum pt. CM. Postulate 1 (eq.1) then requires a new frame of reference to give small fractal baseline δz’=CM/gamma=CM/m=rH=C. So K not δz+δz’ perturbation of flat space eq.3: (dr-δz’)+(dt+δz’)=ds=dr’+dt’ rotation since ds invariant. Defining krr=(dr/dr’)2=1/(1-rH/r)+., r=dr, in the Minkowski metric ds2=dr’2+dt’2+.,and using invariance drdt=dr’dt’=(sqrtkrr)dr(sqrtktt)dt, we obtain krr=1/ktt and thereby get 4D GR math. So the Fiegenbaum point neighborhood perturbation rotations q and Dirac equation give that new pde with that fractal rH (by 1040X scale change). The iteration of the new pde on the next higher fractal scale generates the Schwarzschild metric (i.e.,gravity): We just derived general relativity (GR) from quantum mechanics in one line!

Table Of Contents

       Postulate 1 as min(zz-z)>0 (so 1 is a real number) rewritten as                                

       z=zz+C (1.1.1) , δC=0,C<0 (1.1.2) (the rest is elementary algebra)


Section 1.1 Derivation of eq.1.1.4 Substitute z=1+δz into z-zz=C (eq.1.1.1) and get 1+δz=(1+δz)(1+δz)+C (1.1.3). Thus    δzδz+δz+C=0   (1.1.4) : Solving eq.1.4:  δz=[-1±sqrt(1-4C)]/2. So for noisy C>¼ in general  δz=dr+idt  (1.1.5)  (So we derived space-time.). eq.1.1.1, 1.1.2 holds only at CM so δ(idt)=0 (1.1.3, stability). So plug eqs.1.1.4, 1.1.5 into eq.1.1.2 to get the amazing equation:  δC =δ(dz)+δ(dzdz)=0 (1.1.6)                                                                                                                       Can add K in 1.1.4 δ(δzK)=0 in eq.1.1.6 to initialize to locally flat space. Results of eq.1.1.4:  δ(δz-K)=0  and eq.1.1.5    δ(δzδz)=δ[(dr+idt)(dr+idt)]= δ(dr2+i(drdt+dtdr)-dt2)=0          (1.1.7)   Then the Imaginary part of eq.1.1.7 is  δ(drdt+dtdr)=0 (1.1.8)         If dr,dt positive then drdt+dtdr=ds3=0 is a minimum. Alternatively if dr,dt is negative then drdt+dtdr=0 is maximum instead for dr-dt solutions. In fact all dr,dt sign cases imply a single invariant extremum:               drdt+dtdr=0   (our 1st invariant, sect.2.5)     (1.1.9)  Since we have two independent variables r,t in relδ(δzδz)=δ(dr2-dt2) =. δdr2-δdt2+δk2=δ(dr2-dt2+k2)=δ(dy2+dz2)=-Realδ(dz) still with two independent variables y,z because we can choose k2 randomly. Thus real eq.1.1.6 Real(δ(dzdz)+δ(dz)=0 becomes δ(dx2+dy2+dz2-dt2)=0 with 1.1.9 still holding(since δ(idt)=0). So we have 2+2=4 dimensions and dr2=dx2+dy2+dz2. Note in general dr,dt could be any two of these 4 independent variables implying eq.1.1.9 defines a Clifford algebra. Next factor the real part of eq.1.1.7 to get (our 4D universal 2nd invariant ds2=dr2-dt2): ds2=dr2-dt2=                                                                                                                                                                                                                     [[δ(dr+dt)](dr – dt))] + [(dr +dt)[d(dr – dt)]] =0. (1.1.10) Solve eq. 1.10 and get     (->±e)   dr+dt=sqrt2ds, dr-dt=sqrt2ds,    (1.1.11)          (->light cone v)    dr+dt=sqrt2ds, dr=-dt,             (1.1.12)               “        “              dr-dt=sqrt2ds,   dr=dt,          (1.1.13)     (->vacuum)            dr=dt,           dr=-dt)          (1.1.14) Equation 1.10 gives Special Relativity ds2=dr2-dt2 (note natural unit constant 12 (=c2) in front of the dt2) and eq.1.1.9 gives the Clifford algebra (sect.2.5). A third invariant in section 1.1.2 changes these operators into derivatives thereby making these Dirac equations. So 1.11-1.13 are (neutrino v &stable electron e) leptons. All the rest are composites* of e,v. >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 

*composites of e,v are Bosons (eg.,The SM: Z,W), and composite  3e baryons (eg.,proton), and Fiegenbaum Point fractal e cosmology (GR&gravity).  Excited states of e are u,T in appendix.   We finally do understand.      

 >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> Appendix z=0 details Rotate z by 45+45 Longer summary of section 2 Thus we rotate this (small C) δz through 2 quadrants: (so S=½±½=integer) from the 4 different axis’ max extreme (of 1.15 and 1.11) branch cuts and so get the 4 results: the Z, +-W, photon Bosons of the Standard Model (We have derived it!) with correct field equations (see A6). This is the mother of all reality checks! Also this rotates SR into GR (So we have derived GR too!!) and 1.1.11 into eq.2. This same rotation also breaks those two 2D degeneracies of eq.1.1.11 creating our 4D. Three eq.9 components in 2P3/2 at r=rH give the baryons (PartII) and the tauon and muon are S½ excited states of eq.2. That 1040X fractal electron scale difference of the rH (so next smaller rH gravity from our SR rotation) in the (eq.1.11) eq.2 is at the Fiegenbaum point where δC=0 (i.e., ground state) even for 4D Large fractal baseline (mixed and pure lepton) metric quantization states are in partIII. Mond is a special case.

Why Is This Important? So we derived both physics and math, the most rational thing we could be doing. That 4D implies we got not more and not less than the physical universe.  Also given the fractalness, astronomers are observing from the inside of what particle physicists are studying from the outside, that ONE thing (eq.1.2.7) we postulated. Try looking up at a starry night sky and contemplating that some time. This illustrates the infinite simplicity of this idea. So by knowing essentially nothing (i.e., ONE) you know everything!  We finally do understand (just postulate 1).

References (1) Penrose recently implied in a Utube video that all of physics might be extracted from the Mandelbrot set.  Here we have merely shown how to do that and why it is important. The Fiegenbaum pt neighborhood is a subset.

Table of Contents

      Postulate 1 


 PartI I     1U1 states

Part II    1U1U1 states 

Part III   Mixed State

 Rest       Miscellaneous


(1) Mandelbrot set derivation : Note that the Mandelbrot Set is derived uniquely from 1 {real set}. In that regard eq.1a is iteration zN+1=zNzN+C and eq.1beq.1b says that for some C that δC=0 allowing eigenvalues (giving eq.2AIA). Note if we solve eq.1a for noise C in δC=0 (1b) we get δ(C)=δ(zN+1-zNzN)=0 implying zN+1 is finite since ∞-∞ cannot equal 0 so that eq.1a,1b defines the Mandelbrot set {CM} since then z∞ cannot be infinity. Note also for limit 1,0 finite zN central limit →0(i.e.,small C) in eq.1a then z1=z=z=zz+C (eq.1). Eq.1 implies(1,0) corresponding to the dichotomy ‘set1 always with subset O’: so{1set,O}→(1,0) so OO=O→0∪0=0+0 in our whole number algebra.