Vol. 2 - The World of Mathematical Equations

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April, 2008 PROGRESS IN PHYSICS Volume 20RSections from (202) we have the following space-time statisticswith the two dimensional subspace:e 4(2) 4R1d 3 k(kE 2 2k E (p E q E )x+(p E q E )2 x k 2 = 2 4) 2= e4(2) 4R10 dx Rd3 k(k 2 2k(p E q E ;0)x+(p E q E ) 2 x ; 2 4) 2(203)where the statistical factor 1(2) 3 of three dimensional spacehas been introduced to give the factor 1(2) 4 of the four dimensionalspace-time statistics; and we let k = (k E ; k), k 2 == k 2 E k 2 and since the photon energy parameter 0 is afree parameter we can write 2 0 = k 2 + 2 4 for some 4 .Then a delta function concentrating at 0 of a one dimensionalmomentum variable is multiplied to the integrand in(203) and the three dimensional energy-momentum integralin (203) is changed to a four dimensional energy-momentumintegral by taking the corresponding one more momentum integral.From this we get a four dimensional space-time statisticswith the usual four dimensional momentum integral and withthe statistical factor 1(2) 4 . After this additional momentumintegral we then get (203) as a four dimensional space-timestatistics with the two dimensional momentum variable.Then to get a four dimensional space-time statistics withthe three dimensional momentum variable a delta functionconcentrating at 0 of another one dimensional momentumvariable is multiplied to (203) and the two dimensional momentumvariable of (203) is extended to the correspondingthree dimensional momentum variable. From this we thenget a four dimensional space-time statistics with the three dimensionalmomentum variable.Then we have that (203) is equal to:e 4 i 3 2 (2 3(2) 4 2(2) ) R1 dx0: ((p E q E ) 2 x(1 x) 2 4) 1 (204)2Then since the photon mass-energy parameter 4 is a freeparameter for space-time statistics we can write 4 in the followingform: 2 4 = (p q) 2 x(1 x) ; (205)where pq denotes a two dimensional momentum vector.Then we let p q = (p E q E ; p q). Then we have:(p E q E ) 2 x(1 x) 2 4 == (p E q E ) 2 x(1 x) (p q) 2 x(1 x) == (p q) 2 x(1 x) :Then we have that (204) is equal to:e 4 i 3 2 (2 3(2) 4 2(2) ) R1 dx0 ((p q) 2 x(1 x)) 1 2= e4(206)i 3 2 (2 3(2) 4 2(2)) 1= (207)((p q) 2 ) 1 2 : 4((p q) 2 ) 2 1=e 4 i16((p q) 2 ) 1 2 = e 2 iThus we have the following potential:V seagull (p q) =e 2 i4((p q) 2 ) 1 2: (208)This potential (208) is as the seagull vertex potential.We notice that (208) is a new effect for electron-electronor electron-positron scattering. Recent experiments on the decayof positronium show that the experimental orthopositroniumdecay rate is significantly larger than that computed fromthe conventional QED theory [33–52]. In the following Section24 to Section 26 we show that this discrepancy can beremedied with this new effect (208).24 Reformulating the Bethe-Salpeter equationTo compute the orthopositronium decay rate let us first findout the ground state wave function of the positronium. Tothis end we shall use the Bethe-Salpeter equation. It is wellknown that the conventional Bethe-Salpeter equation is withdifficulties such as the relative time and relative energy problemwhich leads to the existence of nonphysical solutionsin the conventional Bethe-Salpeter equation [7–32]. Fromthe above QED theory let us reformulate the Bethe-Salpeterequation to get a new form of the Bethe-Salpeter equation.We shall see that this new form of the Bethe-Salpeter equationresolves the basic difficulties of the Bethe-Salpeter equationsuch as the relative time and relative energy problem.Let us first consider the propagator of electron. Sinceelectron is a spin- 2 1 particle its statistical propagator is of theform i p m . Thus before the space-time statistics the spin-12 form of electron propagator is of the form p iE ! which canibe obtained from the electron propagatorp 2 E!2 by the factorization:p 2 E ! 2 = (p E !)(p E + !). Then we considerthe following product which is from two propagators of twospin- 1 2 particles:[p E1 ! 1 ][p E2 ! 2 ] == p E1 p E2 ! 1 p E2 ! 2 p E1 + ! 1 ! 2 =:=: p 2 E ! 2 b;(209)where we define p 2 E = p E1 p E2 and ! 2 b := ! 1 p E2 + ! 2 p E1! 1 ! 2 . Then since ! 1 and ! 2 are free mass-energy parameterswe have that ! b is also a free mass-energy parameter withthe requirement that it is to be a positive parameter.Then we introduce the following reformulated relativisticequation of Bethe-Salpeter type for two particles with spin- 1 2 : 0 (p E ; ! b ) =i 2 0[p E1 ! 1 ][p E2 ! 2 ] Rie 2 0 (q E ;! b )dq E((p E q E ) 2 2 0) ; (210)where we use the photon propagator ikE2 (which is of the2 0effect of Coulomb potential) for the interaction of these twoSze Kui Ng. New Approach to Quantum Electrodynamics 41
Volume 2 PROGRESS IN PHYSICS April, 2008particles and we write the proper energy kE 2 of this potentialin the form k 2 E = (p Eq E ) 2 =; and 0 2 3 2 (3 2 3is as the coupling(3))= 2 2 3 (3 2 3 ) @ 2(3)2R1@(0) 2 0 dx hp 2 021 x p 2i3 2== 2 2 3 (3 2 3 )R @ 22 2 1 dt(3) @(0) 2 1 t= 2 2 3 (3 2 3 ) 1 dt0)R 2(3) 1 tZ(p 2 2 30)t p 2== 2 3 2 (3 2 3 ) 1 1(3) (p 2 2 x 2 2 3dx =0id 3 q(p q) 2 0(q) ; (211) = 2 12Then let us choose 0 such that 0 = 2 0Rh(p) = 0 2ip 2 0 2 i(p q) +i24((p q)12 ) 1(p 2 0) 2 2 : (212)(213)0Rh 2i 10Rh(p) = 0 ip 2 2 4((p q) 2 ) 1 02i(q)d 3 q +3 q[q 2 +2xpq+xp 2 (1 x)=0] 2 3 + 0 ip 2 2 ((p q) 2 ) + i4((p q) 2 ) 1 1(1 x)dx=[+x(1 x)p 2 (1 x)0] 2 3 2dx= [+xp 2 0][(1 2 x)(xp 2 0)] 2 3 2parameter. We shall later also introduce the seagull vertexterm for the potential of binding.Let us then introduce the space-time statistics. Since wehave the seagull vertex term for the potential of binding whichis of the form of a circle in a two dimensional space from theabove Section on the seagull vertex potential we see that theappropriate space-time statistics is with the two dimensionalspace. Thus with this space-time statistics from (210) wehave the following reformulated relativistic Bethe-Salpeterequation: 0 (p) =0p 2 2 0where we let the free parameters ! b and 0 be such thatp 2 = p 2 E p 2 with ! b 2 = p 2 + 0 2 for some constant 0 2 == a 1 > 0 2 where a is as the radius of the binding system; and(p q) 2 = (p E q E ) 2 (p q) 2 with 2 0 = (p q) 2 . Wenotice that the potentiali(p q) 2 of binding is now of the usual(relativistic) Coulomb potential type. In (211) the constant e 2in (210) has been absorbed into the parameter 0 in (211).We see that in this reformulated Bethe-Salpeter equationthe relative time and relative energy problem of the conventionalBethe-Salpeter equations is resolved [7–32]. Thus thisreformulated Bethe-Salpeter equation will be free of abnormalsolutions.Let us then solve (211) for the relativistic bound states ofparticles. We show that the ground state solution 0 (p) canbe exactly solved and is of the following form:We have: 0 (p) =1 1((p q) 2 ) (q 2 =0) 2 2==(2+1 1)!(2 1)!(1 1)!R10(2+1 1)!(2 1)!(1 1)!R10= 2R10iR dThus we have:(1 x)dx[x(p q) 2 +(1 x)(q 2 2 0) 2 ] 3 =(1 x)dx[q 2 +2xpq+xp 2 (1 x) 2 0] 3 =(1 x)dx[q 2 +2xpq+xp 2 (1 x) 2 0] 3 :3 q((p q) 2 )(q 2 =0) 2 2R= i2R10 (1 x)dx d= i 2 2 3 2 (3 3 2 )(3)R10= 2 3 2 (3 3 2 )(3)R10@ 22R1@(0) 2 0 dx hxp 2 02 i3 21 x =(p 2 2 0)t p 23 2= 0(p 2 2 0) : (214) 2 .From thisvalue of 0 we see that the BS equation (211) holds. Thus theground state solution is of the form (212). We see that whenp E = 0 and ! 2 b = p 2 + 2 0 then this ground state gives thewell known nonrelativistic ground state of the form 1(p 2 + 2 0) 2of binding system such as the hydrogen atom.25 Bethe-Salpeter equation with seagull vertex potentialLet us then introduce the following reformulated relativisticBethe-Salpeter equation which is also with the seagull vertexpotential of binding:(q)d3 q ;(215)where a factor e 2 of both the Coulomb-type potential andthe seagull vertex potential is absorbed to the coupling constant0 .Let us solve (215) for the relativistic bound states of particles.We write the ground state solution in the followingform:(p) = 0 (p) + 1 (p) ; (216)where 0 (p) is the ground state of the BS equation when theinteraction potential only consists of the Coulomb-type potential.Let us then determine the 1 (p).From (215) by comparing the coefficients of the j ; j == 0; 1 on both sides of BS equation we have the followingequation for 1 (p):(q)d 3 q :(217)This is a nonhomogeneous linear Fredholm integralequation. We can find its solution by perturbation. As afirst order approximation we have the following approximationof 1 (p):42 Sze Kui Ng. New Approach to Quantum Electrodynamics
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