Vol. 2 - The World of Mathematical Equations

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April, 2008 PROGRESS IN PHYSICS Volume 2where we write ds 2 = a dx dx for some functions a by introducing the space-time variable x ; = 0; 1; 2; 3 withx 0 as the time variable; and g = g(s) g(s)a . Thus fromthe symmetry (227) we can derive General Relativity.Let us now determine the variable g(s). Let us considerg(s) = W(z 0 ; z(s)), a quantum Wilson line with U(1) groupwhere z 0 is fixed. When W(z 0 ; z(s)) is the classical Wilsonline then it is of path dependence and thus there is a difficultyto use it to define g(s) = W(z 0 ; z(s)). This is also thedifficulty of Weyl’s gauge theory of unifying gravitation andelectromagnetism. Then when W(z 0 ; z(s)) is the quantumWilson line because of the quantum nature of unspecificationof paths we have that g(s) = W(z 0 ; z(s)) is well definedwhere the whole path of connecting z 0 and z(s) is unspecified(except the two end points z 0 and z(s)).Thus for a given transformation s0 ! s and for any (continuousand piecewise smooth) path connecting z 0 and z(s)the resulting quantum Wilson line W0 (z0 ; z(s(s0 ))) is againof the form W(z 0 ; z(s)) = W(z 0 ; z(s(s0 ))). Let g 0 (s 0 ) == W0 (z0 ; z(s(s0 )))dsds0 . Then we have:g0 (s 0 )g 0 (s 0 )ds 02 == W0 (z0 ; z(s(s0 )))W 0 (z0 ; z(s(s0 )))(dsds0 ) 2 ds02 == W (z0 ; z(s))W(z 0 ; z(s))( dsds0 ) 2 ds02 == g(s) g(s)ds2 :(230)This shows that the quantum Wilson line W(z 0 ; z(s)) canbe the field variable for the gravity and thus can be the fieldvariable for quantum gravity since W(z 0 ; z(s)) is a quantumfield variable.Then we consider the operator W(z 0 ; z)W(z 0 ; z). Fromthis operator W(z 0 ; z)W(z 0 ; z) we can compute the operatorW (z0 ; z)W(z 0 ; z) which is as the absolute value of thisoperator. Thus this operator W(z 0 ; z)W(z 0 ; z) can be regardedas the quantum graviton propagator while the quantumWilson line W(z 0 ; z) is regarded as the quantum photonpropagator for the photon field propagating from z 0 toz. Let us then compute this quantum graviton propagatorW(z 0 ; z)W(z 0 ; z). We have the following formula:W(z; z 0 )W(z 0 ; z) == e ^t log[(z z 0)] Ae^t log[(z 0 z)] ;(231)where ^t = e2 0k 0for the U(1) group (k 0 > 0 is a constant andwe may let k 0 = 1) where the term e ^t log[(z z 0 )] is obtainedby 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 change the W(z; z 0 ) of W(z; z 0 )W(z 0 ; z) in(231) to the second factor W(z 0 ; z) of W(z; z 0 )W(z 0 ; z) byreversing the proper time direction of the path of connectingz and z 0 for W(z; z 0 ). This gives the graviton propagatorW(z 0 ; z)W(z 0 ; z). Then the reversing of the proper time directionof the path of connecting z and z 0 for W(z; z 0 ) alsogives the reversing of the first form of the dual form of theKZ equation to the second form of the dual form of the KZequation. Thus by solving the second form of dual form ofthe KZ equation we have that W(z 0 ; z)W(z 0 ; z) is given by:W(z 0 ; z)W(z 0 ; z) = e^t log[(z z 0 )] Ae^t log[(z z 0 )] == e 2^t log[(z z 0 )] A : (232)In (232) let us define the following constant G:G := 2^t = 2 e2 0k 0: (233)We regard this constant G as the gravitationalAconstant ofthe law of Newton’s gravitation and General Relativity. Wenotice that from the relation e 0 = z 1 2 1 e = n 1 ee wherethe renormalization number n e = z 1 2A is a very large numberwe have that the bare electric charge e 0 is a very smallnumber. Thus the gravitational constant G given by (233)agrees with the fact that the gravitational constant is a verysmall constant. This then gives a closed relationship betweenelectromagnetism and gravitation.We remark that since in (232) the factor G log r 1 == G log 1 r 1< 0 (where we define r 1 = jz z 0 j and r 1 isrestricted such that r 1 > 1) is the fundamental solution ofthe two dimensional Laplace equation we have that this factor(together with the factor e G log r 1 = eG log r1 1) is analogousto the fundamental solution G 1 r of the three dimensionalLaplace equation for the law of Newton’s gravitation.Thus the operator W(z 0 ; z)W(z 0 ; z) in (232) can be regardedas the graviton propagator which gives attractive effect whenr 1 > 1. Thus the graviton propagator (232) gives the sameattractive effect ofG 1 r for the law of Newton’s gravitation.On the other hand when r 1 1 we have that the factorG log r 1 = G log 1 r 1 0. In this case we may consider thatthis graviton propagator gives repulsive effect. This meansthat when two particles are very close to each other then thegravitational force can be from attractive to become repulsive.This repulsive effect is a modification of G 1 r for the law ofNewton’s gravitation for which the attractive force betweentwo particles tends to 1 when the distance between the twoparticles tends to 0.Then by multiplying two masses m 1 and m 2 (obtainedfrom the winding numbers of Wilson loops in (73) of two particlesto the graviton propagator (232) we have the followingformula:Gm 1 m 2 log 1 r 1: (234)From this formula (234) by introducing the space variablex as a statistical variable via the Lorentz metric: ds 2 =Sze Kui Ng. New Approach to Quantum Electrodynamics 45
Volume 2 PROGRESS IN PHYSICS April, 2008= dt 2 dx 2 we have the following statistical formula whichis the potential law of Newton’s gravitation:GM 1 M 21r ; (235)where M 1 and M 2 denotes the masses of two objects.We remark that the graviton propagator (232) is for matters.We may by symmetry find a propagator f(z 0 ; z) of thefollowing form:f(z 0 ; z) := e 2^t log[(z z 0 )] A : (236)When jz z 0 j > 1 this propagator f(z 0 ; z) gives repulsiveeffect between two particles and thus is for anti-matterparticles where by the term anti-matter we mean particleswith the repulsive effect (236). Then since jf(z 0 ; z)j ! 1asjz z 0 j ! 1 we have that two such anti-matter particlescan not physically exist. However in the following Section ondark energy and dark matter we shall show the possibility ofanother repulsive effect among gravitons.As similar to that the quantum Wilson loop W(z 0 ; z 0 ) isas the photon we have that the following double quantum Wilsonloop can be regarded as the graviton:W(z 0 ; z)W(z 0 ; z)W(z; z 0 )W(z; z 0 ) : (237)28 Dark energy and dark matterBy the method of computation of solutions of KZ equationsand the computation of the graviton propagator (232) we havethat (237) is given by:W(z 0 ; z)W(z 0 ; z)W(z; z 0 )W(z; z 0 ) == e 2^t log[(z z 0 )] A g e 2^t log[(z z 0 )] == R 2n A g ; n = 0;1;2;3; : : :(238)where A g denotes the initial operator for the graviton. Thusas similar to the quantization of energy of photons we havethe following quantization of energy of gravitons:h = 2e 2 0n; n = 0;1;2;3; : : : (239)As similar to that a photon with a specific frequency canbe as a magnetic monopole because of its loop nature we havethat the graviton (237) with a specific frequency can also beregarded as a magnetic monopole (which is similar to but differentfrom the magnetic monopole of the photon kind) becauseof its loop nature. (This means that the loop naturegives magnetic property.)Since we still can not directly observe the graviton in experimentsthe quantized energies (239) of gravitons can beidentified as dark energy. Then as similar to the constructionof electrons from photons we construct matter from gravitonsby the following formula:W(z 0 ; z)W(z 0 ; z)W(z; z 0 )W(z; z 0 )Z ; (240)where Z is a complex number as a state acted by the graviton.Similar to the mechanism of generating mass of electronwe have that the mechanism of generating the mass m d ofthese particles is given by the following formula:m d c 2 = 2e 2 0n d = Gn d = h d (241)for some integer n d and some frequency d .Since the graviton is not directly observable it is consistentto identify the quantized energies of gravitons as darkenergy and to identify the matters (240) constructed by gravitonsas dark matter.It is interesting to consider the quantum gravity effect betweentwo gravitons. When a graviton propagator is connectedto a graviton we have that this graviton propagatoris extended to contain a closed loop since the graviton isa closed loop. In this case as similar to the quantum photonpropagator this extended quantum graviton propagatorcan give attractive or repulsive effect. Then for stability theextended quantum graviton propagator tends to give the repulsiveeffect between the two gravitons. Thus the quantumgravity effect among gravitons can be repulsive whichgives the diffusion of gravitons and thus gives a diffusion phenomenonof dark energy. Furthermore for stability more andmore open-loop graviton propagators in the space form closedloops. Thus more and more gravitons are forming and the repulsiveeffect of gravitons gives the accelerating expansion ofthe universe [53–57].Let us then consider the quantum gravity effect betweentwo particles of dark matter. When a graviton propagator isconnected to two particles of dark matter not by connectingto the gravitons acting on the two particles of dark matter wehave that the graviton propagator gives only attractive effectbetween the two particles of dark matter. Thus as similar tothe gravitational force among the usual non-dark matters thegravitational force among dark matters are mainly attractive.Then when the graviton propagator is connected to two particlesof dark matter by connecting to the gravitons acting onthe two particles of dark matter then as the above case of twogravitons we have that the graviton propagator can give attractiveor repulsive effect between the two particles of darkmatter.29 ConclusionIn this paper a quantum loop model of photon is established.We show that this loop model is exactly solvable and thusmay be considered as a quantum soliton. We show that thisnonlinear model of photon has properties of photon and magneticmonopole and thus photon with some specific frequencymay be identified with the magnetic monopole. From the discretewinding numbers of this loop model we can derive the46 Sze Kui Ng. New Approach to Quantum Electrodynamics
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