Vol. 2 - The World of Mathematical Equations

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April, 2008 PROGRESS IN PHYSICS Volume 2believe, was Einstein’s one revolutionary contributionto physics. It upset all existing ideas about the interactionbetween light and matter. . .12 Photon with specific frequency carries electric andmagnetic chargesIn this loop model of photon we have that the observed electriccharge e := n e e 0 and the magnetic charge q min are carriedby the photon with some specific frequencies. Let ushere describe the physical effects from this property of photonthat photon with some specific frequency carries the electricand magnetic charge. From the nonlinear model of electronwe have that an electron W(z; z)Z also carries the electriccharge when a photon W(z; z) carrying the electric andmagnetic charge is absorbed to form the electron W(z; z)Z.This means that the electric charge of an electron is from theelectric charge carried by a photon. Then an interaction (asthe electric force) is formed between two electrons (with theelectric charges).On the other hand since photon carries the constant e 2 0 ofthe bare electric charge e 0 we have that between two photonsthere is an interaction which is similar to the electric forcebetween two electrons (with the electric charges). This interactionhowever may not be of the same magnitude as theelectric force with the magnitude e 2 since the photons maynot carry the frequency for giving the electric and magneticcharge. Then for stability such interaction between two photonstends to give repulsive effect to give the diffusion phenomenonamong photons.Similarly an electron W(z; z)Z also carries the magneticcharge when a photon W(z; z) carrying the electric and magneticcharge is absorbed to form the electron W(z; z)Z. Thismeans that the magnetic charge of an electron is from themagnetic charge carried by a photon. Then a closed-loop interaction(as the magnetic force) may be formed between twoelectrons (with the magnetic charges).On the other hand since photon carries the constant e 2 0 ofthe bare electric charge e 0 we have that between two photonsthere is an interaction which is similar to the magnetic forcebetween two electrons (with the magnetic charges). This interactionhowever may not be of the same magnitude as themagnetic force with the magnetic charge q min since the photonsmay not carry the frequency for giving the electric andmagnetic charge. Then for stability such interaction betweentwo photons tends to give repulsive effect to give the diffusionphenomenon among photons.13 Statistics of photons and electronsThe nonlinear model W(z; z)Z of an electron gives a relationbetween photon and electron where the photon is modelledby W(z; z) which is with a specific frequency for W(z; z)Zto be an electron, as described in the above Sections. Wewant to show that from this nonlinear model we may also derivethe required statistics of photons and electrons that photonsobey the Bose-Einstein statistics and electrons obey theFermi-Dirac statistics. We have that W(z; z) is as an operatoracting on Z. Let W 1 (z; z) be a photon. Then we have that thenonlinear model W 1 (z; z)W(z; z)Z represents that the photonW 1 (z; z) is absorbed by the electron W(z; z)Z to form anelectron W 1 (z; z)W(z; z)Z. Let W 2 (z; z) be another photon.The we have that the model W 1 (z; z)W 2 (z; z)W(z; z)Zagain represents an electron where we have:W 1 (z; z)W 2 (z; z)W(z; z)Z == W 2 (z; z)W 1 (z; z)W(z; z)Z :(83)More generally the modelQNn=1 W n (z; z)W(z; z)Z representsthat the photons W n (z; z); n = 1; 2; : : : ; N areabsorbed by the electron W(z; z)Z. This model shows thatidentical (but different) photons can appear identically and itshows that photons obey the Bose-Einstein statistics. Fromthe polarization of the Dirac-Wilson loop W(z; z) we mayassign spin 1 to a photon represented by W(z; z).Let us then consider statistics of electrons. The observableZ W(z; z)Z gives mass to the electron W(z; z)Z andthus this observable is as a scalar and thus is assigned withspin 0. As the observable W(z; z)Z is between W(z; z) andZ W(z; z)Z which are with spin 1 and 0 respectively we thusassign spin 1 2to the observable W(z; z)Z and thus electronrepresented by this observable W(z; z)Z is with spin 1 2 .Then let Z 1 and Z 2 be two independent complex variablesfor two electrons and let W 1 (z; z)Z 1 and W 2 (z; z)Z 2 representtwo electrons. Let W 3 (z; z) represents a photon. Thenthe model W 3 (z; z)(W 1 (z; z)Z 1 + W 2 (z; z)Z 2 ) means thattwo electrons are in the same state that the operator W 3 (z; z)is acted on the two electrons. However this model means thata photon W(z; z) is absorbed by two distinct electrons andthis is impossible. Thus the models W 3 (z; z)W 1 (z; z)Z 1 andW 3 (z; z)W 2 (z; z)Z 2 cannot both exist and this means thatelectrons obey Fermi-Dirac statistics.Thus this nonlinear loop model of photon and electrongives the required statistics of photons and electrons.14 Photon propagator and quantum photon propagatorLet us then investigate the quantum Wilson line W(z 0 ; z)with U(1) group where z 0 is fixed for the photon field. Wewant to show that this quantum Wilson line W(z 0 ; z) maybe regarded as the quantum photon propagator for a photonpropagating from z 0 to z.As we have shown in the above Section on computationof quantum Wilson line; to compute W(z 0 ; z) we need towrite W(z 0 ; z) in the form of two (connected) Wilson lines:W(z 0 ; z) = W(z 0 ; z 1 )W(z 1 ; z) for some z 1 point. Then weSze Kui Ng. New Approach to Quantum Electrodynamics 27
Volume 2 PROGRESS IN PHYSICS April, 2008have:W(z 0 ; z 1 )W(z 1 ; z) == e ^t log[(z 1 z 0 )] Ae^t log[(z z 1 )] ;(84)where ^t = e2 0k 0for the U(1) group (k 0 is a constant and wemay for simplicity let k 0 = 1) where the term e ^t log[(z z 0 )]is obtained by solving the first form of the dual form of the KZequation and the term e^t log[(z 0 z)] is obtained by solvingthe second form of the dual form of the KZ equation.Then we may write W(z 0 ; z) in the following form:W(z 0 ; z) = W(z 0 ; z 1 )W(z 1 ; z) = e ^t log (z 1 z0)(z z1) A : (85)Let us fix z 1 with z such that:jz 1 z 0 j=jz z 1 j r 1n 2 e(86)for some positive integer n e such that r 1 n 2 e; and we letz be a point on a path of connecting z 0 and z 1 and then aclosed loop is formed with z as the starting and ending point.(This loop can just be the photon-loop of the electron in thiselectromagnetic interaction by this photon propagator (85).)Then (85) has a factor e 2 0 log r 1n 2 which is the fundamentalesolution of the two dimensional Laplace equation and is analogousto the fundamental solution e2r (where e := e 0n e denotesthe observed (renormalized) electric charge and r denotesthe three dimensional distance) of the three dimensionalLaplace equation for the Coulomb’s law. Thus the operatorW(z 0 ; z) = W(z 0 ; z 1 )W(z 1 ; z) in (85) can be regardedas the quantum photon propagator propagating from z 0 to z.We remark that when there are many photons we may introducethe space variable x as a statistical variable via theLorentz metric ds 2 = dt 2 dx 2 to obtain the Coulomb’s lawe 2r from the fundamental solution e2 0 log n r12 as a statisticalelaw for electricity (We shall give such a space-time statisticslater).The quantum photon propagator (85) gives a repulsive effectsince it is analogous to the Coulomb’s law e2r . On theother hand we can reverse the sign of ^t such that this photonoperator can also give an attractive effect:W(z 0 ; z) = W(z 0 ; z 1 )W(z 1 ; z) = e^t log (z z 1)(z1 z0) A ; (87)where we fix z 1 with z 0 such that:jz z 1 jjz 1 z 0 j = r 1n 2 e(88)for some positive integer n e such that r 1 n 2 e; and we againlet z be a point on a path of connecting z 0 and z 1 and then aclosed loop is formed with z as the starting and ending point.(This loop again can just be the photon-loop of the electronin this electromagnetic interaction by this photon propagator(85).) Then (87) has a factor e 2 0 log r 1n 2 which is the fundamentalsolution of the two dimensional Laplace equation andeis analogous to the attractive fundamental solution e 2r of thethree dimensional Laplace equation for the Coulomb’s law.Thus the quantum photon propagator in (85), and in (87),can give repulsive or attractive effect between two points z 0and z for all z in the complex plane. These repulsive or attractiveeffects of the quantum photon propagator correspondto two charges of the same sign and of different sign respectively.On the other hand when z = z 0 the quantum Wilson lineW(z 0 ; z 0 ) in (85) which is the quantum photon propagatorbecomes a quantum Wilson loop W(z 0 ; z 0 ) which is identifiedas a photon, as shown in the above Sections.Let us then derive a form of photon propagator from thequantum photon propagator W(z 0 ; z). Let us choose a pathconnecting z 0 and z = z(s). We consider the following path:z(s) = z 1 + a 0 (s1 s)e i 1(s 1 s) ++ (s s 1 )e i 1(s 1 s);(89)where 1 > 0 is a parameter and z(s 0 ) = z 0 for some propertime s 0 ; and a 0 is some complex constant; and is a stepfunction given by (s) = 0 for s < 0, (s) = 1 for s 0. Thenon this path we have:W(z 0 ; z) == W(z 0 ; z 1 )W(z 1 ; z) = e^t log (z z 1)(z1 z0) A == e^t log a 0[(s s1)e i 1(s1 s) +(s 1 s)e i 1(s1 s) ](z1 z0) A == e^t0(90)log b[(s s 1)e i 1(s1 s) +(s 1 s)e i 1(s1 s) ] A == b (s s1 )e i^t 1(s 1 s) + (s 1 s)e i^t 1(s 1 s)Afor some complex constants b and b 0 . From this chosen ofthe path (89) we have that the quantum photon propagator isproportional to the following expression:12 1 (s s1 )e i 1(s s 1 ) + (s 1 s)e i 1(s s 1 )(91)where we define 1 = ^t 1 = e 2 0 1 > 0. We see that this isthe usual propagator of a particle x(s) of harmonic oscillatorwith mass-energy parameter 1 > 0 where x(s) satisfies thefollowing harmonic oscillator equation:d 2 xds 2 = 2 1x(s) : (92)We regard (91) as the propagator of a photon with massenergyparameter 1 . Fourier transforming (91) we have thefollowing form of photon propagator:ik 2 E 1; (93)28 Sze Kui Ng. New Approach to Quantum Electrodynamics
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