Vol. 2 - The World of Mathematical Equations

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April, 2008 PROGRESS IN PHYSICS Volume 2Then if we interchange z 1 and z 3 we haveW(z 3 ; w)W(w; z 2 )W(z 1 ; w)W(w; z 4 ) == e ^t log[ (z 1 z 3)] Ae^t log(z 4 z 2) :From (67) and (68) as a choice of branch we haveW(z 3 ; w)W(w; z 2 )W(z 1 ; w)W(w; z 4 ) == RW(z 1 ; w)W(w; z 2 )W(z 3 ; w)W(w; z 4 ) ;(68)(69)where R = e i^t is the monodromy of the KZ equation. In(69) z 1 and z 3 denote two points on a closed curve such thatalong the direction of the curve the point z 1 is before the pointz 3 and in this case we choose a branch such that the angle ofz 3 z 1 minus the angle of z 1 z 3 is equal to .RemarkWe may use other representations of the productW(z 1 ; z 2 )W(z 3 ; z 4 ). For example we may use the followingrepresentation:W(z 1 ; w)W(w; z 2 )W(z 3 ; w)W(w; z 4 ) == e ^t log(z 1 z 3) e 2^t log(z 1 w) e 2^t log(z 3 w) Ae^t log(z 4 z 2) e 2^t log(z 4 w) e 2^t log(z 2 w) :(70)Then the interchange of z 1 and z 3 changes only z 1 z 3to z 3 z 1 . Thus the formula (69) holds. Similarly all otherrepresentations of W(z 1 ; z 2 )W(z 3 ; z 4 ) will give the sameresult.Now from (69) we can take a convention that the ordering(66) represents that the curve represented by W(z 1 ; z 2 )is up-crossing the curve represented by W(z 3 ; z 4 ) while (65)represents zero crossing of these two curves.Similarly from the dual KZ equation as a choice of branchwhich is consistent with the above formula we haveW(z 1 ; w)W(w; z 4 )W(z 3 ; w)W(w; z 2 ) == W(z 1 ; w)W(w; z 2 )W(z 3 ; w)W(w; z 4 )R 1 ;(71)where z 2 is before z 4 . We take a convention that the orderingin (71) represents that the curve represented by W(z 1 ; z 2 )is under-crossing the curve represented by W(z 3 ; z 4 ). Herealong the orientation of a closed curve the piece of curverepresented by W(z 1 ; z 2 ) is before the piece of curve representedby W(z 3 ; z 4 ). In this case since the angle of z 3 z 1minus the angle of z 1 z 3 is equal to we have that the angleof z 4 z 2 minus the angle of z 2 z 4 is also equal to and this gives the R 1 in this formula (71).From (69) and (71) we haveW(z 3 ; z 4 )W(z 1 ; z 2 ) = RW(z 1 ; z 2 )W(z 3 ; z 4 )R 1 ; (72)where z 1 and z 2 denote the end points of a curve which isbefore a curve with end points z 3 and z 4 . From (72) wesee that the algebraic structure of these quantum Wilson linesW(z; z0 ) is analogous to the quasi-triangular quantum group[66, 69].8 Computation of quantum Dirac-Wilson loopConsider again the quantum Wilson line W(z 1 ; z 4 ) given byW(z 1 ; z 4 ) = W(z 1 ; z 2 )W(z 2 ; z 4 ). Let us set z 1 = z 4 . In thiscase the quantum Wilson line forms a closed loop. Now in(61) with z 1 = z 4 we have that the quantities e ^t log(z 1 z 2 )and e^t log(z 1 z 2) which come from the two-side KZ equationscancel each other and from the multi-valued property ofthe log function we have:W(z 1 ; z 1 ) = R N A 14 ; N = 0;1;2; : : : (73)where R = e i^t is the monodromy of the KZ equation [69].RemarkIt is clear that if we use other representation of thequantum Wilson loop W(z 1 ; z 1 ) (such as the representationW(z 1 ; z 1 ) = W(z 1 ; w 1 )W(w 1 ; w 2 )W(w 2 ; z 1 )) then we willget the same result as (73).Remark For simplicity we shall drop the subscript of A 14in (73) and simply write A 14 = A.9 Winding number of Dirac-Wilson loop as quantizationWe have the equation (73) where the integer N is as a windingnumber. Then when the gauge group is U(1) we haveW(z 1 ; z 1 ) = R N U(1)A ; (74)where R U(1) denotes the monodromy of the KZ equation forU(1). We haveR N U(1) = e iN e2 0k0+g0 ; N = 0;1;2; : : : (75)where the constant e 0 denotes the bare electric charge (andg 0 = 0 for U(1) group). The winding number N is as thequantization property of photon.We show in the followingSection that the Dirac-Wilson loop W(z 1 ; z 1 ) with theabelian U(1) group is a model of the photon.10 Magnetic monopole is a photon with a specific frequencyWe see that the Dirac-Wilson loop is an exactly solvable nonlinearobservable. Thus we may regard it as a quantum solitonof the above gauge theory. In particular for the abelian gaugetheory with U(1) as gauge group we regard the Dirac-Wilsonloop as a quantum soliton of the electromagnetic field. Wenow want to show that this soliton has all the properties ofphoton and thus we may identify it with the photon.First we see that from (75) it has discrete energy levels oflight-quantum given byh := N e2 0k 0; N = 0; 1; 2; 3; : : : (76)Sze Kui Ng. New Approach to Quantum Electrodynamics 25
Volume 2 PROGRESS IN PHYSICS April, 2008where h is the Planck’s constant; denotes a frequency andthe constant k 0 > 0 is determined from this formula. Thisformula is from the monodromy R U(1) for the abelian gaugetheory. We see that the Planck’s constant h comes out fromthis winding property of the Dirac-Wilson loop. Then sincethis Dirac-Wilson loop is a loop we have that it has the polarizationproperty of light by the right hand rule along the loopand this polarization can also be regarded as the spin of photon.Now since this loop is a quantum soliton which behavesas a particle we have that this loop is a basic particle of theabove abelian gauge theory where the abelian gauge propertyis considered as the fundamental property of electromagneticfield. This shows that the Dirac-Wilson loop has properties ofphoton. We shall later show that from this loop model of photonwe can describe the absorption and emission of photon byan electron. This property of absorption and emission is consideredas a basic principle of the light-quantum hypothesisof Einstein [1]. From these properties of the Dirac-Wilsonloop we may identify it with the photon.On the other hand from Dirac’s analysis of the magneticmonopole we have that the property of magnetic monopolecomes from a closed line integral of vector potential of theelectromagnetic field which is similar to the Dirac-Wilsonloop [4]. Now from this Dirac-Wilson loop we can definethe magnetic charge q and the minimal magnetic charge q minwhich are given by:eq := enq min := n e e 0 n n me 0 k 0; n = 0; 1; 2; 3; : : : (77)where e := n e e 0 is as the observed electric charge for somepositive integer n e ; and q min := nme0k 0for some positive integern m and we write N = nn e n m ; n = 0; 1; 2; 3; : : : (byabsorbing the constant k 0 to e 2 0 we may let k 0 = 1).This shows that the Dirac-Wilson loop gives the propertyof magnetic monopole for some frequencies. Since this loopis a quantum soliton which behaves as a particle we have thatthis Dirac-Wilson loop may be identified with the magneticmonopole for some frequencies. Thus we have that photonmay be identified with the magnetic monopole for some frequencies.With this identification we have the following interestingconclusion: Both the energy quantization of electromagneticfield and the charge quantization property comefrom the same property of photon. Indeed we have:nh 1 := n n en m e 2 0k 0= eq ; n = 0; 1; 2; 3; : : : (78)where 1 denotes a frequency. This formula shows that theenergy quantization gives the charge quantization and thusthese two quantizations are from the same property of thephoton when photon is modelled by the Dirac-Wilson loopand identified with the magnetic monopole for some frequencies.We notice that between two energy levels neq min and(n + 1)eq min there are other energy levels which may be regardedas the excited states of a particle with charge ne.11 Nonlinear loop model of electronIn this Section let us also give a loop model to the electron.This loop model of electron is based on the above loop modelof the photon. From the loop model of photon we also constructan observable which gives mass to the electron and isthus a mass mechanism for the electron.Let W(z; z) denote a Dirac-Wilson loop which representsa photon. Let Z denotes the complex variable for electron in(1.1). We then consider the following observable:W(z; z)Z : (79)Since W(z; z) is solvable we have that this observableis also solvable where in solving W(z; z) the variable Z isfixed. We let this observable be identified with the electron.Then we consider the following observable:Z W(z; z)Z : (80)This observable is with a scalar factor Z Z where Z denotesthe complex conjugate of Z and we regard it as the massmechanism of the electron (79). For this observable we modelthe energy levels with specific frequencies of W(z; z) as themass levels of electron and the mass m of electron is the lowestenergy level h 1 with specific frequency 1 of W(z; z)and is given by:mc 2 = h 1 ; (81)where c denotes the constant of the speed of light and thefrequency 1 is given by (78). From this model of the massmechanism of electron we have that electron is with mass mwhile photon is with zero mass because there does not havesuch a mass mechanism Z W(z; z)Z for the photon. Fromthis definition of mass we have the following formula relatingthe observed electric charge e of electron, the magneticcharge q min of magnetic monopole and the mass m of electron:mc 2 = eq min = h 1 : (82)By using the nonlinear model W(z; z)Z to represent anelectron we can then describe the absorption and emission ofa photon by an electron where photon is as a parcel of energydescribed by the loop W(z; z), as follows. Let W(z; z)Zrepresents an electron and let W 1 (z 1 ; z 1 ) represents a photon.Then the observable W 1 (z 1 ; z 1 )W(z; z)Z represents anelectron having absorbed the photon W 1 (z 1 ; z 1 ). This propertyof absorption and emission is as a basic principle of thehypothesis of light-quantum stated by Einstein [1]. Let usquote the following paragraph from [1]:. . . First, the light-quantum was conceived of as a parcelof energy as far as the properties of pure radiation(no coupling to matter) are concerned. Second, Einsteinmade the assumption — he call it the heuristicprinciple — that also in its coupling to matter (that is,in emission and absorption), light is created or annihilatedin similar discrete parcels of energy. That, I26 Sze Kui Ng. New Approach to Quantum Electrodynamics
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Vol. 2 - The World of Mathematical Equations

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