VIEWS: 8 PAGES: 11 POSTED ON: 12/3/2011 Public Domain
New Generally Covariant First Order pde Here we show the consequences of a new generally covariant partial differential equation generalization of the Dirac equation: (/x) -=0 (1) Note the new 00=1-rH/r , rr =1/oo diagonalized coefficients giving the general covariance. Note substitution of rH=2e2/mec2 gives the standard Dirac equation result /x+i=0 as r gets large See http://davidmaker.com for derivation of this new pde Next we Solve /x-=0 (1) Solve equation 1 directly (see similar methods of solving Schrodinger equation using separation of variables =R(r)yL,m½(,), spherical harmonics (eg.,S, P states), etc., given oo=1-rH/r=1-2e2/rmec2). For example there is stability at rrH, since the metric time component oo (=1-rH/r) is zero so clocks slow down and we get (baryon) stability. Also at this radius rrH 1) 2P3/2 states fill ahead of the 2P½ as noted in Alberto (Alberto, 1998), in the hard shell ultra-relativistic approximation giving his L-S coupling. Thus the 2P3/2 state(and its sp2 hybrid) for this new electron Dirac equation gives an azimuthal trifolium, 3 lobe shape, so this ONE charge e (color not needed to guarantee this) spends 1/3 of its time in each lobe (1/3e fractionally charged lobes). 2) Lobe structure is locked into the center of mass (asymptotic freedom), can‟t leave, just as with the Schrodinger equation spherical harmonics (here as Dirac half integer spherical harmonics). 3) There are six 2P states (corresponding to the 6 flavors) (Halzen, quarks); the P wave scattering produces jets; giving us all the main properties of quarks! (strong force) from first principles (eq.1). Note that the 2S½ state solution is twice the energy of the 2P3/2 of equation 1 (Alberto, 1998). 4) We can identify the 2S½ then with the tauon and the 1S½ with the muon respectively with the 2P3/2 at r=rH then the baryons. 5) From equ.1 the small electron proper mass and these large baryon 2P3/2 , lepton 2S½ state energy eigenvalues imply ultrarelativistic electron motion. There is straight line ultrarelativistic electron motion for the S state (since no rotation) so Fitzgerald contraction rH(1-v2/c2) is to Fermi/2000, a “point”. But 2P states (L=1) are rotational and so even for the same degree of ultrarelativistic motion have the small Fitzgerald contraction= rHcos(1-cos2)d=½rH=1/2 Fermi, not a point. This explains why leptons are “point” particles and baryons are not. 6) Frobenius series numerical solution for rrH of equation 1 done in ch.19 of http://davidmaker.com. For example the J=1 state 2P solution for rrH is mass eigenvalue 2mP forming the two 2P z direction lobes giving the properties of the deuteron and thereby deriving the core concept of nuclear physics. 7). Again the rH boundary provides a hard shell potential producing a Van der Waals equation of state thereby implying the liquid equation of state observed in 100GeV gold-gold collisions at BNL. 8) The eq.1 energy E=1/√oo =1/√ (1-rH/r) has singularities in it and so when its associated potential is substituted into the S matrix we find the W and Z as resonances. This new pde can be used to derive the weak interaction. 1 rH / 2r 3 / 8rH / r .. 1 1 Energy 2 For r>>rH oo rH 1 r 9) The first term is the mass energy, the second term is the Coulomb potential (energy) and the third term gives a small addition to the Coulomb potential that allows us to drop the higher order diagrams and renormalization in QED. This third term then provides the Lamb shift. (i.e.,Energy perturbation expectation value of hydrogen atom 2,0,0 state). The precision of the old renormalization QED is still maintained. Summary: Note these are ONE step derivations from eq. 1 of these important results, not added assumptions as in the mainstream approach. Note these successes are due to rH not being zero; thereby keeping this theory generally covariant if nonzero forces are present. Comparison with Standard Method The alternative approach is to set rH=0 in equation 1. Setting rH=0 removes the general covariance when nonzero force is present, which is a mistake that people have compensated for by fudging in adhoc pathologies such as many gauges, renormalization and counter-terms, higher dimensions, 19 free parameters that you can adjust any way you want, turning theoretical physics into a veritable junk pile. The result is a confusion that has stopped the progress of theoretical physics for the past 30 years. Conclusion This new generally covariant generalization of the Dirac equation gives many results that are otherwise merely assumptions added to the old Dirac equation theory. It solves the many left over problems the old Dirac equation cannot help us solve, gets new valid physics beyond the standard model and thereby gets theoretical physics moving again. Backup Slide A Equation 1 Is Generally Covariant By Construction New generally covariant (Dirac) PDE The spherical symmetry background metric coefficient (44) 00=1-rH/r can be inserted into a Dirac equation by starting with 4 4 ds dx dx 2 (2) 1 1 which is a generally covariant expression. In the spherically symmetric case one can diagonalize to ds 2 11dx12 22 dx2 33dx3 44 dx4 2 2 2 Define px from px = dx/ds and define from: d p x i dx ( h 1, mo=1) , linearize to get the Dirac matrices and multiply both sides of the resulting equation by /ds and get d d d d or 11 1 1 22 2 2 33 3 3 i 44 4 4 dx dx dx dx (/x)- = 0 (1) Note only multiplications, redefining, and at the end, the standard Dirac equation linearization was used to modify equation 1. Thus we have not compromised the general covariance of equation 2 in deriving equation 1. Thus equation 1 by construction is generally covariant. Backup Slide B How Equivalence Principle Can Be Applied to E&M Recall that the electrostatic force Eq=F=ma so E(q/m)=a. Thus there are different accelerations „a‟ for different charges „q‟ in an ambient electrostatic field „E‟. In contrast with gravity there is a single acceleration for two different masses as Galileo discovered in his tower of Pisa experiment. Thus gravity (mass) obeys the equivalence principle and so (in the standard result) the metric formalism gij can apply to gravity. Single Charge e Approach Allows Use of Metric Coefficients gij Note that E&M can also obey the equivalence principle but in only one case: if there is a single e and Dirac particle me in Eq=ma and therefore (to get the correct geodesics): oo = goo=1-2e2/rmec2 =1-rH/r (with rr=1/oo) and so then trivially all charges will have the same acceleration in the same E field. This then allows us to insert this metric gij formalism into the standard Dirac equation derivation instead of the usual Minkowski flat space- time gijs (below). Thus by making E&M obey the equivalence principle you force it to have ONE nonzero mass with charge. Thus you force a unified field theory on theoretical physics. Backup Slide C Summary of Applications Fractalness: Note dr2=rrdr‟2 observer (i.e.,dr‟) near rH (most likely position) also sees a huge selfsimilar universe dr‟2/(1-rH/r‟)=dr2=dr‟2/(1-rH/rH)=dr‟2/(0)= and his own high frequency dirac zitterbewegung of equation 1: (/2)2 =(1/dT)2 =1/dt‟2(1-rH/r‟) = 1/(dt‟2(1-rH/rH)) = 1/0 = . Solve equation 1 for r>rH, r=rH and r<rH to get the physics results. For example: rrH Frobenius series solution for near rrH gives baryon properties: as a spherical harmonic 2P3/2 without any of the QCD postulates and free parameters r<rH Note fractalness so that observer r<rH in huge universe: cosmology, radial coordinate expansion due to next higher fractal scale implies gravity, and eq 1 written between two fractal scales implies the left handed Dirac doublet core of Standard Model. r>rH QED (without the higher order diagrams)