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Planck Scale, Dirac Delta Function and Ultraviolet Divergence

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Planck Scale, Dirac Delta Function and Ultraviolet Divergence Planck scale, Dirac delta function and ultraviolet divergence Hua Zhang,a;1 Mingshun Yuanb aShenzhen Forms Syntron Information Co.,Ltd., Shenzhen 518000, P.R. China bFujian University of Technology,Mathematics and Physics Institute Fuzhou 350118, P.R. China E-mail: [email protected] Abstract: The Planck length is the minimum length which physical law do not fail. The Dirac delta function was created to deal with continuous range issue, and it is zero except for one point. Thus contradict the Planck length. Renormalization method is the usual way to deal with divergence difficulties. The authors proposed a new way to solve the problem of ultraviolet divergence and this method is self-consistent with the Planck length. For this purpose a redefine function δP in position representation was introduced to han- dle with the canonical quantization. The function δP tends to the Dirac delta function if the Planck length goes to zero. By logical deduction the authors obtains new commuta- tion/anticommutation relations and new Feynman propagators which are convergent. This article will deduce the new Feynman propagators for the Klein-Gordon field, the Dirac field and the Maxwell field. Through the new Feynman propagators, we can eliminate ultraviolet divergence. arXiv:1911.01789v5 [physics.gen-ph] 2 Jan 2020 1Corresponding author. Contents 1 Introduction2 1.1 History of renormalization2 1.2 The Planck scale & the Dirac delta function2 2 The renormalization function3 2.1 Rectangular coordinate system3 2.2 Cylindrical coordinate system4 2.3 Spherical coordinate system5 3 Quantization and propagator6 3.1 The Klein-Gordon field6 3.2 The Dirac field8 3.3 The Maxwell field9 4 Feynman loop diagram 10 4.1 Vacuum polarization 11 4.2 Electron self-energy 11 5 Symmetry 12 5.1 The Klein-Gordon field & the Dirac field 12 5.2 Gauge invariance 12 6 Conclusion 13 A Proof of convergence of loop diagram 13 A.1 Rectangular coordinate system 13 A.1.1 Vacuum polarization loop diagram 13 A.1.2 Fermion loop diagram of N photon external lines 15 A.1.3 Electron self-energy loop diagram 16 A.1.4 Arbitrary order case 17 A.2 Cylindrical coordinate system 17 A.2.1 Vacuum polarization loop diagram 17 A.2.2 Electron self-energy loop diagram 18 A.2.3 Arbitrary order case 18 B Calculate the masses of octet baryons from chiral perturbation theory 18 B.1 Rectangular coordinate system 18 B.2 Cylindrical coordinate system 20 B.3 The calculation formulas of mass splitting 20 – 1 – 1 Introduction 1.1 History of renormalization Infinities problem arose when QFT constructed in the late 1920s or in the early 1930s. Many physicists devote to this hard work. If this problem can not be solved, QFT will be abandoned. After World War II, Sinitiro Tomonaga, Julian Schwinger, and Richard Feynman[1–3] found the solution independently, which is called renormalization theory of QED, and it is the most successful theory up to now. The successful calculation of the anomalous magnetic moment of the electron, announce the success of renormalization method. In 1971,Kenneth G. Wilson raised renormalization group[4], which changing cutoff Λ of infinities will not impact the physical calculation results, for the interaction involve one or several momentum-dependent coupling constants. In 1972, Gerard ’t Hooft and Martinus J.G. Veltman proved that non-abelian gauge fields is renormalizable[5]. A criterion[6] for deciding which quantum field theories are renormalizable, that all infinities can be absorbed into a redefinition of a finite number of coupling constants and masses. The goal of this article is, infinities never arise in the procedure of calculation, we can achieve if not count infrared divergence. 1.2 The Planck scale & the Dirac delta function In 1912, Max Planck given the units of length[7], mass, time and temperature from the velocity c of propagation of light in a vacuum, and the constant of gravitation G, and the Planck constant h. In 1999, Peter J. Mohr and Barry N. Taylor published an article list fundamental physical constants[8], with 3 1=2 −35 lP = (~G=c ) = 1:6160(12) × 10 m (1.1) 5 1=2 −44 tP = lP =c = (~G=c ) = 5:3906(40) × 10 s (1.2) In 1927, in order to handle with continuous variables, Paul Dirac introduced a new generalized function[9] which named delta function δ(x). Delta function can be written as[10] 3 δij δ (x − x0) = lim (1.3) ∆Vi!0 ∆Vi Due to the limitation of Planck length, ∆Vi will not tend to zero. A new delta function can be defined as 3 δij δP (x − x0) = (1.4) ∆VP i ∆VP i is the limit case of volume element ∆Vi in Planck length. In different coordinate system, the volume element ∆Vi has different form, thus δP has different form. Our deducing base on the below hypothesis. i. In position representation, the Dirac delta function (not include Kronecker delta func- tion) in canonical commutation/anticommutation relations must redefine as δP function. δ ! δP (1.5) – 2 – ii. In canonical commutation/anticommutation relations, assume Z Z d3pe±ip:xf(p) ! d3pe±ip:x<3(p)f(p) (1.6) Where f(p) is the function of momentum p. For convenience, we call above item i: and item ii: renormalization transformations. On the action of renormalization transformations, δP function’s Fourier expansion is called renormalization function, denote as <3(p). Renormal- 3 3 ization function < (p) in rectangular coordinate system represent as <?(p), in spherical 3 3 coordinate system represent as <s(p). Below paragraphs will given the expressions of < (p) and δP . In this paper, the renormalization functions are introduced, and the explicit expression of the renormalization functions are given in rectangular and cylindrical coordinates. In Klein-Gordon field, Dirac field and electromagnetic field, new standard commutation, an- ticommutation relations and new Feynman propagators are proposed. The authors proved that the loop diagrams of vacuum polarization and electron self-energy are convergent in large momentum when introduced renormalization functions, and can be extend to arbitrary order case(for rectangular and cylindrical systems). In the appendix, the authors calculate the masses of octet baryons use renormalization functions(in rectangular and cylindrical system). 2 The renormalization function 2.1 Rectangular coordinate system In the Planck length limit, the particles have determinate place x2(x 0 − lP =2; x 0 + lP =2), 3 uncertainty ∆x = lP = ∆V . Wave function 3 Ψx0 (x) = δP (x − x 0) (2.1) In this article, if a symbol is added a subscript P or a superscript P , it represents the corresponding symbol under the renormalization transformations. ( 3 3 1=lP ; x2(x 0 − lP =2; x 0 + lP =2) δP (x − x 0) = (2.2) 0; x62(x 0 − lP =2; x 0 + lP =2) 3 For δP (x − x 0) its Fourier expansion 1 −ip:x0 3 Ψx0 (p) = p e <?(p) (2.3) (2π)3 With 3 lP px lP px lP py lP py lP pz lP pz <?(p) = [sin = ][sin = ][sin = ] (2.4) 2~ 2~ 2~ 2~ 2~ 2~ – 3 – l p l p l p l p Figure 1. Renormalization function <2 (p) = [sin P x = P x ][sin P y = P y ] ? 2 2 2 2 ~ ~ ~ 2 ~ , where x-axis and y-axis represent px and py, z-axis represent <?(p). 3 3 Compare with Fourier expansion of the delta funtion δ (x − x 0), we got a new item <?(p), When lP is infinitesimal, 3 3 lim δP (x − x 0) = δ (x − x 0) (2.5) lP !0 1 1 lim e−ip:x0 <3 (p) = e−ip:x0 (2.6) p 3 ? p 3 lP !0 (2π) (2π) 3 3 We call <?(p) as renormalization function in rectangular coordinate system, or call <?(p) as momentum distribution wave function in rectangular coordinate system. Compare the renormalization function with Fraunhofer moment aperture diffraction intensity distribution formula, they have similar expression. 2.2 Cylindrical coordinate system 3 πlP In the Planck length limit of cylindrical coordinate system ∆V = 4 , the delta function 8 4 3 < πl3 ; r < lP =2; jzj < lP =2 δP cylin(r; θ; z) = P (2.7) :0; r ≥ lP =2; jzj ≥ lP =2 – 4 – Corresponding Fourier expansion 1 −ip:x0 3 Ψx0 (p) = p e <cylin(p) (2.8) (2π)3 Where Z lP =2 Z 2π Z lP =2 3 4 i[(px cos θ+py sin θ)r+pzz]=~ <cylin(p) = 3 re drdθdz (2.9) πlP −lP =2 0 0 2 2 2 lP (px + py) lP pz lP pz = 0F1(2; − )[sin = ] (2.10) 16~2 2~ 2~ Where 0F1(a; z) is the generalized hypergeometric series (also see Figure 2./Figure 3./Fig- ure 4., and see [11, 12]), satisfy 2 2 2 1 n 2 2 2 lP (px + py) X (−1) lP (px + py) n F1(2; − ) = [ ] (2.11) 0 16 2 n!(n + 1)! 16 2 ~ n=0 ~ 2 2 2 lP (px + py) lim 0F1(2; − 2 ) = 1 (2.12) lP !0 16~ We have 3 lim <cylin(p) = 1 (2.13) lP !0 2.3 Spherical coordinate system 3 πlP In the Planck length limit of spherical coordinate system ∆V = 6 , the delta function 8 6 3 < πl3 ; r < lP =2 δP s(r; θ; ') = P (2.14) :0; r ≥ lP =2 Corresponding Fourier expansion 1 −ip:x0 3 Ψx0 (p) = p e <s(p) (2.15) (2π)3 Where Z 2π Z π Z lP =2 3 6 2 i(px sin ' cos θ+py sin ' sin θ+pz cos ')r=~ <s(p) = 3 r sin 'e drdϕdθ (2.16) πlP 0 0 0 q 2 2 Z π Z lP =2 r sin ' px + py 12 2 i(pz cos ')r=~ = 3 r sin 'e J0( )drd' (2.17) lP 0 0 ~ with the first kind of Bessel function of order 0(see [13]) q 2 2 r sin ' px + py Z 2π 1 i(px sin ' cos θ+py sin ' sin θ)r=~ J0( ) = e dθ (2.18) ~ 2π 0 – 5 – 1.0 0.8 0.6 0.4 0.2 -1000 -500 500 1000 -0.2 3 Figure 2.
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