Discrete Phase Space, Relativistic Quantum Electrodynamics, and a Non-Singular Coulomb Potential
NNoname manuscript No. (will be inserted by the editor)
Discrete Phase Space, Relativistic QuantumElectrodynamics, and a Non-Singular CoulombPotential
Anadijiban Das · Rupak Chatterjee · Ting Yu
Received: date / Accepted: date
Abstract
This paper deals with the relativistic, quantized electromagneticand Dirac field equations in the arena of discrete phase space and continuoustime. The mathematical formulation involves partial difference equations . Inthe consequent relativistic quantum electrodynamics, the corresponding Feyn-man diagrams and S -matrix elements are derived. In the special case ofelectron-electron scattering (Møller scattering), the explicit second order ele-ment (cid:104) f | S | i (cid:105) is deduced. Moreover, assuming the slow motions for two exter-nal electrons, the approximation of (cid:104) f | S | i (cid:105) yields a divergence-free Coulombpotential. Keywords
Discrete phase space · partial difference equations · quantumelectrodynamics · divergence-free Coulomb potential PACS · · · · Anadijiban DasDepartment of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A 1S6,CanadaE-mail: [email protected] ChatterjeeCenter for Quantum Science and EngineeringDepartment of Physics, Stevens Institute of Technology, Castle Point on the Hudson, Hobo-ken, NJ 07030, USAE-mail: [email protected] YuCenter for Quantum Science and EngineeringDepartment of Physics, Stevens Institute of Technology, Castle Point on the Hudson, Hobo-ken, NJ 07030, USAE-mail: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] A p r Anadijiban Das et al.
Partial difference equations have been studied [1] for many years to investigateproblems of mathematical physics [2,3,4,5,6]. Quantum mechanics has beenexactly represented in phase space continuum with the usual time variable[7]. In recent years, an exact representation of quantum mechanics has beenintroduced in the discrete phase space and continuous time arena [8,9,10,11].This representation involves a characteristic length l >
0. Contrary to theusual expectations, this representation is exactly relativistic.Furthermore, this representation can be elevated to the second quantizationof free fields, interacting fields, and the new S -matrix theory [12,13,14].In this paper, we discuss a new discrete phase space-continuous time for-mulation of quantum electrodynamics, the S -matrix, and the correspond-ing Feynman prescriptions [14]. We specifically concentrate on second orderelectron-electron scattering (or Møller scattering). Furthermore, in the lowmomenta approximation of two external electrons, we derive a new Coulombpotential which is devoid of any singularity whatsoever. We hope that a futureprecise experiment can verify the validity of this new Coulomb potential. There exists a characteristic length l > (cid:126) = c = l = 1 and express all physical quantities as di-mensionless numbers. Greek indices take values from { , , , } whereas theRoman indices take values from { , , } . Einstein’s summation convention isadopted in both cases. We denote the flat space-time metric by η µν with thecorresponding diagonal matrix [ η µν ] := diag [1 , , , − n , x ) ≡ ( n , n , n , t ) ∈ N × R , n j ∈ N for j ∈ , , x ≡ t ∈ R .Let a function f from N × R into R or C be denoted as f ( n , t ) = f ( n , n , n , t ).The various partial difference operators and the partial differential operators are shown below [12,13]: ∆ j f ( n , t ) := f ( ..., n j + 1 , ..., t ) − f ( ..., n j , ..., t ) (1a) ∆ (cid:48) j f ( n , t ) := f ( ..., n j , ..., t ) − f ( ..., n j − , ..., t ) (1b) ∆ j f ( n , t ) := 1 √ (cid:104)(cid:112) n j + 1 f ( ..., n j + 1 , ..., t ) − √ n j f ( ..., n j − , ..., t ) (cid:105) (1c) ∂ t f ( n , t ) := ∂∂t [ f ( n , t )] (1d) itle Suppressed Due to Excessive Length 3 We denote Hermite polynomials and some useful properties by the following[12,13]: H n j ( k j ) := ( − n j e ( k j ) d n j ( dk j ) n j [ e − ( k j ) ] (2a) ξ n j ( k j ) := ( i ) n j e − ( k j ) / H n j ( k j )( π ) / ( n j / (cid:112) ( n j )! (2b) (cid:90) R (cid:89) j =1 (cid:104) ξ n j ( k j ) ξ ˆ n j ( k j ) (cid:105) dk dk dk = δ n ˆ n δ n ˆ n δ n ˆ n =: δ n ˆ n (2c) − i∆ ξ n j ( k j ) = k j ξ n j ( k j ) (2d) The electromagnetic four-potential wave field is denoted by A µ ( n , t ). It isassumed to be operator-valued and satisfies the partial difference-differentialequations [12,13]: δ ab ∆ a ∆ b A σ ( n , t ) − ( ∂ t ) A σ ( n , t ) = 0 (3)We assume that the four-potential function A σ ( n , t ) also satisfies the Lorenz-gauge condition, ∆ b A b ( n , t ) + ∂ t A ( n , t ) = 0 (4)For a classical electromagnetic field, the equations (3) and (4) with thefollowing definitions F ab ( n , t ) := ∆ a A b ( n , t ) − ∆ b A a ( n , t ), and F a ( n , t ) := ∆ a A ( n , t ) − ∂ t A a ( n , t ) yield exactly the electromagnetic field equations inthe discrete phase space-continuous time arena. However, after the secondquantization, the operator version of (4) poses a problem. One possible solutionis to replace (4) with a weaker expectation value equation: (cid:104) Ψ ( p ) | ∆ b A b ( n , t ) + ∂ t A ( n , t ) | Ψ ( p ) (cid:105) = 0 (5)Here, Ψ ( p ) indicates a physically admissible Hilbert space vector.The momentum-energy four-vector of an external photon belongs to a four-dimensional Minkowskian vector space. Therefore, we can denote this entityby ( k , k , k , k ) := ( k , k ) , (6)where η µν k µ k ν = 0 , ( k ) = δ ab k a k b , (7)and ν = ν ( k ) := (cid:112) δ ab k a k b = + √ k · k > . (8)The symbol ν = ν ( k ) physically stands for the frequency of the electromagneticwave propagation. Anadijiban Das et al.
We introduce four M-orthonormal or tetrad vectors by e µ ( λ ) ( k ) := ˜ e µ ( λ ) ( k , ν ( k )) , (9a) η ( λσ ) e µ ( λ ) ( k ) e ν ( σ ) ( k ) := η µν , (9b) η µν e ( λ ) µ ( k ) e ( σ ) ν ( k ) := η ( λσ ) . (9c)We choose the spatial direction of the photon momentum propagator vector k along the third axis. A compatible selection of polarization vectors e ( λ ) µ ( k )are furnished by [15]( k , k , k , k ) = (0 , , k , ν ( k )) , (10a) e ( λ ) µ ( k ) = δ ( λ ) µ , (10b) k µ e ( λ ) µ ( k ) = 0 for λ = 1 , , (10c) k µ e (3) µ ( k ) = k , k µ e (4) µ ( k ) = − ν ( k ) (10d)The quantization of the electromagnetic four-potential involves Hermitian op-erators A σ ( n , t ) satisfying the partial differential-difference equations (3). Aclass of exact solutions of (3) is furnished by the integrals [12,13]: A µ ( n , t ) = A † µ ( n , t ) = (cid:90) R d k [2 ν ( k )] − / a µ ( k ) (cid:89) j =1 ξ n j ( k j ) e − iνt + a † µ ( k ) (cid:89) j =1 ξ n j ( k j ) e iνt (11a) A ( − ) µ ( n , t ) = (cid:90) R d k [2 ν ( k )] − / a µ ( k ) (cid:89) j =1 ξ n j ( k j ) e − iνt = (cid:88) λ =1 (cid:90) R d k [2 ν ( k )] − / a ( λ ) ( k ) e ( λ ) µ ( k ) (cid:89) j =1 ξ n j ( k j ) e − iνt (11b) A (+) µ ( n , t ) = (cid:90) R d k [2 ν ( k )] − / a † µ ( k ) (cid:89) j =1 ξ n j ( k j ) e + iνt = (cid:88) λ =1 (cid:90) R d k [2 ν ( k )] − / a † ( λ ) ( k ) e ( λ ) µ ( k ) (cid:89) j =1 ξ n j ( k j ) e + iνt (11c) A µ ( n , t ) = A ( − ) µ ( n , t ) + A (+) µ ( n , t ) (11d)The operators A ( − ) µ ( n , t ) represent external photons terminating at ( n , t ),whereas A (+) µ ( n , t ) represent external photons emanating from ( n , t ). itle Suppressed Due to Excessive Length 5 The canonical quantization rules for operators a µ ( k ) and a † µ ( k ) are as-sumed to be the commutators:[ a µ ( k ) , a ν (ˆ k )] = [ a † µ ( k ) , a † ν (ˆ k )] = 0 (12a)[ a µ ( k ) , a † ν (ˆ k )] = − [ a † ν (ˆ k ) , a µ ( k )] = η µν δ ( k − ˆ k ) I (12b)Using (9) and (10), we obtain from (12)[ a ( λ ) ( k ) , a (ˆ λ ) (ˆ k )] = [ a † ( λ ) ( k ) , a † (ˆ λ ) (ˆ k )] = 0 (13a)[ a ( λ ) ( k ) , a † (ˆ λ ) (ˆ k )] = − [ a † (ˆ λ ) (ˆ k ) , a ( λ ) ( k )] = η ( λ ˆ λ ) δ ( k − ˆ k ) I (13b)[ a ( λ ) ( k ) , a † (ˆ λ ) (ˆ k )] = δ ( λ ˆ λ ) δ ( k − ˆ k ) I for λ, ˆ λ ∈ { , } . (13c)In the sequel, we shall denote polarization indices only as λ, ˆ λ ∈ { , } . Weuse expressions (11) and commutation relations (12) to obtain[ A (+) µ ( n , t ) , A (+) ν (ˆ n , ˆ t )] = [ A ( − ) µ ( n , t ) , A ( − ) ν (ˆ n , ˆ t )] = 0 (14a)[ A ( − ) µ ( n , t ) , A (+) ν (ˆ n , ˆ t )] = − iη µν D (+) ( n , t ; ˆ n , ˆ t ) I (14b)[ A (+) µ ( n , t ) , A ( − ) ν (ˆ n , ˆ t )] = − iη µν D ( − ) ( n , t ; ˆ n , ˆ t ) I (14c)[ A µ ( n , t ) , A ν (ˆ n , ˆ t )] = − iη µν D ( n , t ; ˆ n , ˆ t ) I (14d)[ A µ ( n , t ) , A ν (ˆ n , ˆ t )] | t =ˆ t = 0 for n (cid:54) = ˆ n (14e)Here, D ( ± ) ( n , t ; ˆ n , ˆ t ) and D ( n , t ; ˆ n , ˆ t ) are non-singular Green’s functions (orphoton propagators) to be discussed in the Appendix. The equation (14e)indicates physically micro-causality in regards to measurements of two photonssituated at two distinct discrete points of phase space at the same time. Let us introduce one possible representation of 4 × γ = [ γ AB ] := , γ = [ γ AB ] := − i i − i i γ = [ γ AB ] := −
11 0 0 00 − , γ = [ γ AB ] := − i − i i
00 0 0 i (15)Here, the bispinor indices A, B take values from { , , , } . The importantproperties of Dirac matrices are listed below: γ a † = γ a , γ † = − γ (16a) γ µ γ ν + γ ν γ µ = 2 η µν I (4 × (16b) Anadijiban Das et al.
The classical Dirac bispinor wave field ψ ( n , t ) = [ ψ A ( n , t )] is a 4 × × ψ ( n , t ) := iψ † ( n , t ) γ (17)The discrete phase space continuous time Dirac wave equations are fur-nished by γ a ∆ a ψ ( n , t ) + γ ∂ t ψ ( n , t ) + mψ ( n , t ) = 0 (4 × (18a)[ ∆ a ˜ ψ ( n , t )] γ a + [ ∂ t ˜ ψ ( n , t )] γ − m ˜ ψ ( n , t ) = 0 (1 × (18b)Here, the positive constant m > ψ ( n , t ) = ζ ( p , p ) (cid:89) j =1 ξ n j ( p j ) e ip t (19)By substituting (19) into (18a), we derive that η µν p µ p ν + m = 0 , (20a) p = ± E ( p ) , E ( p ) = + (cid:112) δ ab p a p b + m > u (1) ( p ) = [( m + E ) / E ] / − i ( m + E ) − p − i ( m + E ) − ( p + ip ) , (21a) u (2) ( p ) = [( m + E ) / E ] / − i ( m + E ) − ( p − ip ) i ( m + E ) − p , (21b) v (1) ( p ) = [( m + E ) / E ] / i ( m + E ) − p − i ( m + E ) − ( p + ip )10 , (21c) v (2) ( p ) = [( m + E ) / E ] / i ( m + E ) − ( p − ip ) − i ( m + E ) − p . (21d)The above solutions (21) satisfy˜ u ( r ) ( p ) · u ( s ) ( p ) := δ AB ˜ u A ( r ) ( p ) u B ( s ) ( p ) = − ˜ v ( r ) ( p ) · v ( s ) ( p ) = δ ( rs ) (22a)˜ u ( r ) ( p ) · v ( s ) ( p ) = ˜ v ( r ) ( p ) · u ( s ) ( p ) = 0 (22b) itle Suppressed Due to Excessive Length 7 The indices r, s ∈ { , } physically represent spin orientations of an electron ora positron. The functions u ( r ) ( p ) are physically associated with electron wavefields, whereas functions v ( r ) ( p ) are associated with the positron wave fields.We have to consider in the later sections an almost static approximation toequations (21). In these approximations, we can derive that E ( p ) = m + (cid:0) || p || / m (cid:1) + O (cid:0) || p || (cid:1) , (23a) u (1) = + (1 / (cid:0) || p || / m (cid:1) − i ( p / m )[1 − (1 /
2) ( || p || / m ) ] − i (cid:18) p + ip m (cid:19) [1 − (1 /
2) ( || p || / m ) ] + O (cid:0) || p || (cid:1) (23b) u (2) = + / (cid:0) || p || / m (cid:1) − i (cid:18) p − ip m (cid:19) [1 − (1 /
2) ( || p || / m ) ]+ i ( p / m )[1 − (1 /
2) ( || p || / m ) ] + O (cid:0) || p || (cid:1) (23c)Now, using Dirac matrices of equations (15) and equations (23), we deducethe almost static approximations˜ u ( r ) ( p (cid:48) ) γ a u ( s ) ( p ) = 0 + O (cid:0) || p (cid:48) || (cid:1) + O (cid:0) || p || (cid:1) (24a)˜ u ( r ) ( p (cid:48) ) γ u ( s ) ( p ) = − i [ δ ( rs ) + O (cid:0) || p (cid:48) || (cid:1) + O (cid:0) || p || (cid:1) ] (24b) Anadijiban Das et al.
A class of exact solutions of the Dirac Equations (18) is furnished by [12,13], ψ − ( n , t ) = (cid:90) R d p [ m/E ( p )] / (cid:88) r =1 α ( r ) ( p ) u ( r ) ( p ) (cid:89) j =1 ξ n j ( p j ) e − iEt (25a) ψ + ( n , t ) = (cid:90) R d p [ m/E ( p )] / (cid:88) r =1 β † ( r ) ( p ) v ( r ) ( p ) (cid:89) j =1 ξ n j ( p j ) e iEt (25b) ψ ( n , t ) = ψ − ( n , t ) + ψ + ( n , t ) (25c)˜ ψ + ( n , t ) = (cid:90) R d p [ m/E ( p )] / (cid:88) r =1 α † ( r ) ( p )˜ u ( r ) ( p ) (cid:89) j =1 ξ n j ( p j ) e iEt (25d)˜ ψ − ( n , t ) = (cid:90) R d p [ m/E ( p )] / (cid:88) r =1 β ( r ) ( p )˜ v ( r ) ( p ) (cid:89) j =1 ξ n j ( p j ) e − iEt (25e)˜ ψ ( n , t ) = ˜ ψ + ( n , t ) + ˜ ψ − ( n , t ) (25f)Now, we shall introduce the canonical or second quantization of the freeelectron-positron wave fields. We adopt a two-dimensional complex vectorspace called a Pre-Hilbert space. We postulate that the electron-positron field ψ ( n , t ) of equation (25c) is a Pre-Hilbert space vector and that the Fourier co-efficients α ( r ) , β ( r ) of equations (25a) and (25e) are linear operators acting onthe Pre-Hilbert space vectors. Morever, we assume that these linear operatorssatisfy anti-commutation rules [12,13],[ A, B ] + : = AB + BA = [ B, A ] + (26a)[ α ( r ) ( p ) , α ( s ) (ˆ p )] + = [ β ( r ) ( p ) , β ( s ) (ˆ p )] + =[ α † ( r ) ( p ) , α † ( s ) (ˆ p )] + = [ β † ( r ) ( p ) , β † ( s ) (ˆ p )] + = 0 (26b)[ α ( r ) ( p ) , β ( s ) (ˆ p )] + = [ α † ( r ) ( p ) , β † ( s ) (ˆ p )] + =[ α ( r ) ( p ) , β † ( s ) (ˆ p )] + = [ α † ( r ) ( p ) , β ( s ) (ˆ p )] + = 0 (26c)[ α ( r ) ( p ) , α † ( s ) (ˆ p )] = [ β ( r ) ( p ) , β † ( s ) (ˆ p )] = δ ( rs ) δ ( p − ˆ p ) I (26d)There are three physical motivations for choosing Pre-Hilbert space op-erators α ( r ) ( p ) , α † ( s ) (ˆ p ) , β ( r ) ( p ) , β † ( s ) (ˆ p ) satisfying anti-commutation relations(26) in contrast to Hilbert space operators a µ ( k ) , a † ν (ˆ k ) satisfying commuta-tion relations (12): (I) A Photon field obeys Bose-Einstein statistics, whereasthe electron-positron field obeys Fermi-Dirac statistics. (II) The total energy itle Suppressed Due to Excessive Length 9 of the electron field or positron field is a positive definite operator. (III) Thetotal electric charge of the electron field is negative, whereas the total electriccharge of the positron field is positive.Now, we shall work out the anti-commutation relations between variouselectron-positron wave fields. We use equations (25) and (26) to deduce[ ψ ( − ) ( n , t ) , ˜ ψ ( − ) (ˆ n , ˆ t )] + = [ ψ (+) ( n , t ) , ˜ ψ (+) (ˆ n , ˆ t )] + = 0 (27a)[ ψ ( n , t ) , ψ (ˆ n , ˆ t )] + = [ ˜ ψ ( n , t ) , ˜ ψ (ˆ n , ˆ t )] + = 0 (27b)[ ψ ( − ) ( n , t ) , ˜ ψ (+) (ˆ n , ˆ t )] + = (cid:90) R d p [ m/E ( p )] (cid:88) r =1 u ( r ) ( p )˜ u ( r ) ( p ) (cid:89) j =1 ξ n j ( p j ) ξ ˆ n j ( p j ) e − iE ( t − ˆ t ) =: iS (+) ( n , t ; ˆn , ˆ t ; m ) I (27c)[ ψ (+) ( n , t ) , ˜ ψ ( − ) (ˆ n , ˆ t )] + = iS ( − ) ( n , t ; ˆn , ˆ t ; m ) I (27d)[ ψ ( n , t ) , ˜ ψ (ˆ n , ˆ t )] + = iS ( n , t ; ˆn , ˆ t ; m ) I (27e) S ( n , t ; ˆn , ˆ t ; m ) := S (+) ( n , t ; ˆn , ˆ t ; m ) + S ( − ) ( n , t ; ˆn , ˆ t ; m ) (27f)The various non-singular Green’s functions S ( a ) ( n , t ; ˆn , ˆ t ; m ) will be elaboratedin the Appendix. S -matrix, and Møller Scattering The relativistic Lagrangian of the interacting photon and Dirac fields is takento be [14] L ( int ) ( n , t ) := − ieN [ ˜ ψ (ˆ n , ˆ t ) γ µ ψ ( n , t ) A µ ( n , t )] . (28)Here, | e | = (cid:112) π/ <
1, the electric charge of an electron is −| e | < | e | >
0, and N [ · · · ] stands fornormal ordering.The scattering matrix in the discrete phase space and continuous time,denoted by S -matrix, is defined by the operator-valued infinite series, S = I + (cid:80) ∞ j =1 S j := I + ∞ (cid:80) j =1 ( e ) j j ! ∞ (3) (cid:80) n (1) =1 · · · ∞ (3) (cid:80) n ( j ) =1 (cid:82) R dt (1) · · · (cid:82) R dt ( j ) T (cid:110) N [ ˜ ψ ( n (1) , t (1) ) γ µ ψ ( n (1) , t (1) ) A µ (1) ( n (1) , t (1) )] · · · N [ ˜ ψ ( n ( j ) , t ( j ) ) γ µ j ψ ( n (1) , t ( j ) ) A µ ( j ) ( n ( j ) , t ( j ) )] (cid:111) (29)Here, T denotes Wick’s time ordering operation. We distinguish the scat-tering matrix by the notation S -matrix from the usual notation of S-matrixin continuous space-time because the physics in the discrete phase space andcontinuous time is different from the physics in the space-time continuum. Table 1
Feynman Rules Description Factor in S -matrixPhoton propagator − iη µν D ( F +) ( n , t ; ˆn , ˆ t )Electron-positron propagator iS ( F +) ( n , t ; ˆn , ˆ t ; m )Electron-photon vertex γ µ Incoming or outgoing external photon lines: A ( − ) µ ( n , t ) or A (+) µ ( n , t )Incoming or outgoing external electron lines: ψ ( − ) ( n , t ) or ˜ ψ (+) ( n , t )Incoming or outgoing external positron lines: ˜ ψ ( − ) ( n , t ) or ψ (+) ( n , t ) We provide Feynman rules to evaluate succinctly each term of the S -matrix series in (29) in Tables 1 and 2.The actual construction of the S j ) element from Tables 1 and 2 is incom-plete until we determine the appropriate numerical factor c j ) to be multipliedto the j -th term, It turns out that the correct factor is c j ) = ( − l δ p (2 π ) j − ( P i + E i ) ( e ) j (30)Here, j is the number of vertices, ( − l is associated with each loop, and P i + E i is the sum of internal photon and electron propagators.Moreover, the vertex factor in 4-momentum space is defined by δ ( p , − q , − k ) := ∞ (3) (cid:88) n =0 (cid:89) j =1 ξ n j ( p j ) ξ n j ( q j ) ξ n j ( k j ) (31a) δ ( p , q , k ) := ∞ (3) (cid:88) n =0 (cid:89) j =1 ξ n j ( p j ) ξ n j ( q j ) ξ n j ( k j ) (31b)The function δ ( p , − q , − k ) above is different from the standard delta func-tion factor (2 π ) δ ( p , − q , − k ) found in the usual theory in space-time contin-uum. Thus, in the discrete phase space theory, exact conservation of the total3-momentum in a vertex is violated. In the quantum field theory of interac-tions over any discrete space, exact conservation of momentum in a vertex islost [6,17].If we denote the initial and final state vectors by | i (cid:105) and | f (cid:105) of a physicalprocess, then from (29) we can define (cid:104) f | S − I | i (cid:105) =: i (2 π ) δ ( E ( f ) − E ( i ) ) (cid:104) f | M | i (cid:105) (32)Here, E ( i ) stands for the total initial energy, whereas E ( f ) stands for the totalfinal energy. The equation (32) implies the exact conservation of the totalenergy in the interaction. The transition probability from the initial state | i (cid:105) to the final state | f (cid:105) per unit time is provided by w ( fi ) =: i (2 π ) δ ( E ( f ) − E ( i ) ) |(cid:104) f | M | i (cid:105)| (33) itle Suppressed Due to Excessive Length 11 Table 2
Feynman Rules Continued2 Anadijiban Das et al.
Fig. 1
The second order Feynman diagram for Møller scattering (”Fermi’s golden rule”).As an illustration of computing the second order element (cid:104) f | S − I | i (cid:105) in(29), we consider the case of the electron-electron scattering or Møller scatter-ing. The corresponding component of the Feynman diagram in 4-momentumspace is exhibited in Figure 1.Using the 4-momentum version of Table 1, 2, and equation (30), we obtainthe required second order S -matrix element as, (cid:104) p (cid:48) , p (cid:48) | S | p , p (cid:105) = [ c ] (cid:34) m (cid:112) E (cid:48) E (cid:48) E E (cid:35)(cid:90) R (cid:110) [˜ u ( p (cid:48) ) γ µ u ( p )] δ ( p , − p (cid:48) , − k ) δ ( p − p (cid:48) − k ) (cid:20) η µν k · k − ( k ) (cid:21) [˜ u ( p (cid:48) ) γ ν u ( p )] δ ( p , − p (cid:48) , + k ) δ ( p − p (cid:48) + k ) (cid:111) d k dk (34a) (cid:104) p (cid:48) , p (cid:48) | S | p , p (cid:105) = (cid:34) − i (2 π ) e m (cid:112) E (cid:48) E (cid:48) E E (cid:35) δ ( E (cid:48) + E (cid:48) − E − E ) (cid:90) R (cid:26) [˜ u ( p (cid:48) ) γ µ u ( p )] δ ( p , − p (cid:48) , − k ) (cid:20) η µν k · k − ( E (cid:48) − E ) (cid:21) [˜ u ( p (cid:48) ) γ ν u ( p )] δ ( p , − p (cid:48) , + k ) (cid:111) d k (34b)To compare and contrast the above equations (34) with the usual secondorder S-matrix elements for Møller scattering over space-time continuum, we itle Suppressed Due to Excessive Length 13 furnish from [18] (cid:104) p (cid:48) , p (cid:48) | S (2) | p , p (cid:105) = [ c (2) ] (cid:34) m (cid:112) E (cid:48) E (cid:48) E E (cid:35)(cid:90) R (cid:110) [˜ u ( p (cid:48) ) γ µ u ( p )] δ ( p , − p (cid:48) , − k ) δ ( p − p (cid:48) − k ) (cid:20) η µν k · k − ( k ) (cid:21) [˜ u ( p (cid:48) ) γ ν u ( p )] δ ( p , − p (cid:48) , + k ) δ ( p − p (cid:48) + k ) (cid:111) d k dk (35a) (cid:104) p (cid:48) , p (cid:48) | S (2) | p , p (cid:105) = − ( ie m ) / (4 π ) (cid:112) E (cid:48) E (cid:48) E E δ ( E (cid:48) + E (cid:48) − E − E ) (cid:90) R (cid:26) [˜ u ( p (cid:48) ) γ µ u ( p )] δ ( p , − p (cid:48) , − k ) η µν k · k − ( E (cid:48) − E ) [˜ u ( p (cid:48) ) γ ν u ( p )] δ ( p , − p (cid:48) , + k ) (cid:111) d k (35b)The last equation (35) can be further reduced to the following algebraic form[18], (cid:104) p (cid:48) , p (cid:48) | S (2) | p , p (cid:105) = − ( ie m ) / (4 π ) (cid:112) E (cid:48) E (cid:48) E E δ ( p (cid:48) + p (cid:48) − p − p )[˜ u ( p (cid:48) ) γ µ u ( p )] η µν η αβ ( p (cid:48) α − p α )( p (cid:48) β − p β ) [˜ u ( p (cid:48) ) γ ν u ( p )] (36a)Such a drastic reduction for equation (34) in the discrete case is not possibleat the current time. Let us consider the usual Møller scattering formula (35b)in the continuous case. Using the relation δ ( p − p (cid:48) − k ) = (2 π ) − (cid:82) R e i p · x d x ,(35b) becomes (cid:104) p (cid:48) , p (cid:48) | S (2) | p , p (cid:105) = − ( ie m ) / (4 π ) (cid:112) E (cid:48) E (cid:48) E E δ ( E (cid:48) + E (cid:48) − E − E ) (cid:82) R (cid:26) [˜ u ( p (cid:48) ) γ µ u ( p )] (cid:82) R e i ( p − p (cid:48) − k ) · x d x (2 π ) (cid:20) η µν k · k − ( E (cid:48) − E ) (cid:21) [˜ u ( p (cid:48) ) γ ν u ( p )] (cid:82) R e i ( p − p (cid:48) + k ) · x d x (2 π ) (cid:27) d k (37)Now, we assume slow motions of the two external electrons and consequentequations (23) and (24). We also assume conservation of electron spin implying r (cid:48) = r and r (cid:48) = r . Then equation (37) reduces to (cid:104) p (cid:48) , p (cid:48) | S (2) | p , p (cid:105) = − ie (2 π ) δ (cid:18) || p (cid:48) || m + || p (cid:48) || m − || p || m − || p || m (cid:19)(cid:82) R (cid:82) R e i ( p − p (cid:48) ) · x e i ( p − p (cid:48) ) · x (cid:82) R e − i ( x − x ) · k (2 π ) [ k · k ] d k d x d x +( Higher order terms ) (38)
We can read off from the above equations the Green’s function G ( x − x ) ofthe usual potential equation as, G ( x − x ) = (cid:90) R e − i ( x − x ) · k (2 π ) [ k · k ] d k (39a) G ( x − x ) = 1(4 π ) || x − x || (39b) δ ab ∂ ∂x a ∂x b G ( x − x ) = − δ ( x − x ) (39c)lim x → x G ( x − x ) → ∞ (39d)We obtain the usual singular Coulomb potential between two electrons as [19], V ( x , x ) = e G ( x − x ) = e (4 π ) || x − x || (40)Suppose that one of the charged particles has unit electric charge and issituated at the origin x = (0 , , − x = ( x , , W ( x , ,
0) := V ( x , ) = − π ) | x | < x →∞ W ( x , ,
0) = 0 (41b)Now we shall examine the matrix element (cid:104) p (cid:48) , p (cid:48) | S | p , p (cid:105) for Møllerscattering in discrete phase space. Using equations (31) and (34), we obtain (cid:104) p (cid:48) , p (cid:48) | S | p , p (cid:105) − ( ie πm ) (cid:112) E (cid:48) E (cid:48) E E δ ( E (cid:48) + E (cid:48) − E − E ) (cid:82) R (cid:110) [˜ u ( p (cid:48) ) γ µ u ( p )] (cid:80) ∞ (3) n =0 (cid:104)(cid:81) a =1 ξ n a ( p a ) ξ n a ( p (cid:48) a ) ξ n a ( k a ) (cid:105) η µν k · k − ( E (cid:48) − E ) [˜ u ( p (cid:48) ) γ ν u ( p )] (cid:80) ∞ (3) ˆn =0 (cid:104)(cid:81) b =1 ξ ˆ n b ( p b ) ξ ˆ n b ( p (cid:48) b ) ξ ˆ n b ( k a ) (cid:105)(cid:111) d k (42)Now we assume slow momenta for two external electrons and consequentequations (23) and (24). Then (42) reduces to (cid:104) p (cid:48) , p (cid:48) | S | p , p (cid:105) = − ie δ (cid:18) || p (cid:48) || m + || p (cid:48) || m − || p || m − || p || m (cid:19)(cid:80) ∞ (3) n =0 (cid:80) ∞ (3) ˆn =0 (cid:104)(cid:81) a =1 ξ n a ( p a ) ξ n a ( p (cid:48) a ) (cid:105) (cid:104)(cid:81) b =1 ξ ˆ n b ( p b ) ξ ˆ n b ( p (cid:48) b ) (cid:105)(cid:82) R (cid:104)(cid:81) j =1 ξ n j ( k j ) ξ ˆ n j ( k j )( k · k ) − (cid:105) d k +( Higher order terms ) (43) itle Suppressed Due to Excessive Length 15
Comparing (43) to (38), we decduce that the relevant Green’s function mustbe [20] G ( n ; ˆn ) = (cid:90) R (cid:89) j =1 ξ n j ( k j ) ξ ˆ n j ( k j )( k · k ) − d k (44a) δ ab ∆ a ∆ b G ( n , ˆn ) = − (cid:89) j =1 δ n j ˆ n j (44b) G (0; 0) = 2 (44c)Comparing (44) with (39), we obtain the new non-singular Coulomb potentialas in [20], V ( n , ˆn ) = e G ( n , ˆn ) (45)Suppose that one of the charged particles has unit electric charge and is sit-uated at the discrete origin ˆn = (0 , , − n = ( n , , W (2 n , ,
0) := − G (2 n , ,
0; 0 , , − (cid:34) n +1 ( n !)(2 n + 1)! (cid:112) (2 n )! (cid:35) < , (46a) W (2 n + 1 , ,
0) = 0 (46b) (cid:20) W (2 n + 2 , , W (2 n , , (cid:21) = (cid:20) (2 n ) + 6 n + 2(2 n ) + 12 n + 9 (cid:21) / < n →∞ W (2 n , ,
0) = 0 , (46d) W (0 , ,
0) = − (cid:8) W (2 n , , (cid:9) ∞ isa monotone increasing sequence of negative numbers. Comparing and con-trasting Coulomb potentials W ( x , ,
0) of ordinary space and W ( n , , Let us physically analyze the usual Coulomb potential W ( x , ,
0) in three di-mensional physical space R represented by the top graph of Figure 2. On Diracparticle of electric charge +1 is residing at the origin (0 , , − x , ,
0) falling off in the infinitehole of the singular potential well characterized by W ( x , ,
0) = − π ) | x | . Fig. 2
A graph of the usual singular potential Coulomb potential well W ( x , ,
0) versusthe new non-singular Coulomb potential well W ( n , ,
0) in discrete phase space.
On the other hand, the bottom graph of Figure 2, the Dirac particle of electriccharge +1 is residing at (0 , ,
0) of discrete phase space. The other Dirac par-ticle of electric charge − n , ,
0) falls off along the non-singular potentialwell at a finite depth characterized by the non-singular Coulomb potential W ( n , , Zitterbewegung a long time ago. itle Suppressed Due to Excessive Length 17
Fig. 3
Various contours C ( a ) in the complex k plane. Appendix: Discrete phase space, continuous time, and non-singularGreen’s functions for free relativistic field equations
The relativistic partial difference-differential equation for a real scalar field (orKlein-Gordon field) is provided by δ jl ∆ j ∆ l φ ( n , t ) − ( ∂ t ) φ ( n , t ) − µ φ ( n , t ) = 0 (47)Here, µ > ∆ ( a ) ( n , t ; ˆn , ˆ t ; µ ) = (2 π ) − (cid:82) R (cid:110)(cid:104)(cid:81) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) (cid:105)(cid:82) C ( a ) (cid:104)(cid:0) η αβ k α k β + µ (cid:1) − exp ( ik ( t − ˆ t )] dk (cid:105)(cid:111) d k (48)(Note that in the signature +2 convention, k = − k .) The Green’s functionsabove involve nine contours in the complex k -plane [18]. These contours areexhibited explicitly in Figure 3 with w = w ( k ) := + (cid:112) k · k + µ > δ jl ∆ j ∆ l ∆ ( a ) ( n , t ; ˆn , ˆ t ) − ( ∂ t ) φ ( n , t ) − µ ∆ ( a ) ( n , t ; ˆn , ˆ t ) = 0 for contours C ( a ) = C, C (+) , C ( − ) , C (1) = − δ n ˆ n δ n ˆ n δ n ˆ n δ ( t − ˆ t ) for contours C ( a ) = C ( R ) , C ( A ) , C ( P ) , C ( F +) , C ( F − ) The Green’s functions D ( a ) ( n , t ; ˆ n , ˆ t ) in equations (14) for a photon fieldare defined by [12,13], D ( a ) ( n , t ; ˆ n , ˆ t ) := ∆ ( a ) ( n , t ; ˆ n , ˆ t : 0) = 12 π (cid:82) R (cid:110)(cid:104)(cid:81) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) (cid:105)(cid:82) C ( a ) (cid:104)(cid:0) η αβ k α k β (cid:1) − exp ( ik ( t − ˆ t )] dk (cid:105)(cid:111) d k (49)Here, in Figure 3, we have to replace w = w ( k ) by ν = ν ( k ) := √ k · k . Bycarrying out some closed contour integrals over the complex k -plane in (49),one arrives at the following three-dimensional representations, D (+) ( n , t ; ˆ n , ˆ t ) = i (cid:90) R (cid:89) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) e − iν ( t − ˆ t ) ν ( k ) d k (50a) D ( − ) ( n , t ; ˆ n , ˆ t ) = − i (cid:90) R (cid:89) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) e + iν ( t − ˆ t ) ν ( k ) d k (50b) D ( n , t ; ˆ n , ˆ t ) = D (+) ( n , t ; ˆ n , ˆ t ) + D ( − ) ( n , t ; ˆ n , ˆ t )= (cid:90) R (cid:89) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) sin( ν ( t − ˆ t )) ν d k (50c) D ( n , t ; ˆ n , ˆ t ) | t =ˆ t = 0 (50d) D (1) ( n , t ; ˆ n , ˆ t ) = − i [ D (+) ( n , t ; ˆ n , ˆ t ) − D ( − ) ( n , t ; ˆ n , ˆ t )]= (cid:90) R (cid:89) j =1 ξ n j ( k j ) ξ ˆ n j ( k j ) cos( ν ( t − ˆ t )) ν d k (50e)Now, Green’s functions for Dirac field equations (18a) are provided by [12,13], S ( a ) ( n , t ; ˆ n , ˆ t ; m ) = ( γ b ∆ b + γ ∂ t − mI ) ∆ ( a ) ( n , t ; ˆ n , ˆ t ; m ) (51a) S ( a ) AB ( n , t ; ˆ n , ˆ t ; m ) = ( γ bAB ∆ b + γ AB ∂ t − mδ AB ) ∆ ( a ) ( n , t ; ˆ n , ˆ t ; m ) (51b) S ( a ) ( n , t ; ˆ n , ˆ t ) = (2 π ) − (cid:90) R (cid:89) j =1 ξ n j ( p j ) ξ ˆ n j ( p j ) (cid:90) C ( a ) ( iγ µ p µ − mI ) e − ip ( t − ˆ t ) η αβ p α p β + m dp (cid:41) d p (51c) itle Suppressed Due to Excessive Length 19 Here,
A, B ∈ { , , , } are bispinor indices. The 4 × γ b ∆ b + γ ∂ t + mI ) S ( a ) ( n , t ; ˆ n , ˆ t ; m ) = [0] × for contours C, C (+) , C ( − ) , C (1) = − δ n ˆ n δ n ˆ n δ n ˆ n δ ( t − ˆ t )[1] × for contours C ( R ) , C ( A ) , C ( P ) , C ( F +) , C ( F − ) References
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