Critique of Feynman Propagator, the $\E \cdot x$ gauge
Navin Khaneja
quant-ph hep-th

         


4em



Critique of Feynman Propagator, the · x gauge.
     Navin Khaneja To whom correspondence may be addressed. Email:navinkhaneja@gmail.com IIT Bombay, Powai - 400076, India.
    December 30, 2023
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3cm


 Abstract 

Consider Møller scattering. Electrons with momentum p and -p scatter by exchange of photon say in z direction to p+q and -(p+q).
The scattering amplitude is well known, given as Feynman propagator = (e ħ c)^2/ϵ_0 Vu̅(p+q) γ^μ u(p)  u̅(-(p+q)) γ_μ u(-p)/q^2, where V is the volume of the scattering electrons, e elementary charge and ϵ_0
permitivity of vacuum. But this is not completely correct.
Since we exchange photon momentum in z direction, we have two photon polarization x,y and hence the true scattering amplitude should be

    _1 = (e ħ c)^2/ϵ_0 Vu̅(p+q) γ^x u(p)  u̅(-(p+q)) γ_x u(-p)  + u̅(p+q) γ^y u(p)  u̅(-(p+q)) γ_y u(-p)  /q^2.
 But when electrons are non-relativistic, _1 ∼ 0. This is disturbing, how will we ever get the coulomb potential, where ∼(e ħ c)^2/ϵ_0 V q^2. Where is the problem ? The problem is with the gauge in Dirac equation.

For a plane wave along z direction, with electric field E_x sin (kz - ω t), the Lorentz gauge is  
    (A_0, A_x, A_y, A_z) = E_x/ωcos(kz-ω t)(0, 1, 0, 0)
. But this gauge is not suited for calculating optical transitions, because we don't recover the Rabi frequency q E_x d (d electric dipole moment). What we find is something orders of magnitude smaller. Nor is it suitable for calculating electron electron scattering because we don't recover Coulomb potential. What we find is something orders of magnitude smaller. Instead, we work with · x gauge  
    (A_0, A_x, A_y, A_z) = -E_x/2 ( x sin(kz-ω t), -cos(kz-ω t)/ω, 0, x/csin(kz-ω t) )
 (c light velocity) to find everything correct. What we get is new propagator.



§ INTRODUCTION




The heart of interactions in high energy physics is the beautiful electron electron scattering of Møller. The coulomb interaction between electrons. Fig. <ref> shows two electrons with momentum p and -p scatter by exchange of photon say in z direction to p+q and -(p+q).
The scattering amplitude is well known, given as Feynman propagator = (e ħ c)^2/ϵ_0 Vu̅(p+q) γ^μ u(p)  u̅(-(p+q)) γ_μ u(-p)/q^2, <cit.>, where V is the volume of the scattering electrons, e elementary charge and ϵ_0
permitivity of vacuum. But this is not completely correct.
Since we exchange photon momentum in z direction, we have two photon polarization x,y and hence the true scattering amplitude should be

    _1 = (e ħ c)^2/ϵ_0 Vu̅(p+q) γ^x u(p)  u̅(-(p+q)) γ_x u(-p)  + u̅(p+q) γ^y u(p)  u̅(-(p+q)) γ_y u(-p)  /q^2.
 But when electrons are non-relativistic, _1 ∼ 0. This is disturbing, how will we ever get the coulomb potential, where ∼(e ħ c)^2/ϵ_0 V q^2. Where is the problem ? The problem is with the Dirac equation, it is not all correct in presence of electromagnetic field.

For a plane wave along z direction, with electric field E_x sin (kz - ω t), the Lorentz gauge is  (A_0, A_x, A_y, A_z) = E_x/ωcos(kz-ω t)(0, 1, 0, 0). But this gauge is not suited for calculating optical transitions, because we don't recover the Rabi frequency q E_x d (d electric dipole moment). What we find is something orders of magnitude smaller. Nor is it suitable for calculating electron electron scattering because we don't recover Coulomb potential. What we find is something orders of magnitude smaller. Instead, we work with · x gauge  
    (A_0, A_x, A_y, A_z) = -E_x/2 ( x sin(kz-ω t), -cos(kz-ω t)/ω, 0, x/csin(kz-ω t) )
 (c light velocity) to find everything correct. What we get is a new Feynman propagator.


The paper is organized as follows. We first derive the amplitude of optical transition using Dirac equation with Lorentz gauge. We show it is way too small. We introduce the new gauge, we call the E · x gauge. We show we can correctly calculate the optical Rabi frequency using this gauge. We then use this gauge to calculate Møller scattering amplitude and show we recover the Coulomb potential.  The result is we get a new Feynman propagator.





§ DIRAC EQUATION AND LORENTZ GAUGE




§ INTRODUCTION


Take a classical electron, with coordinates (x, y, z) = (x_1, x_2, x_3) . Its Lagrangian in the electromagnetic field is


    L = m/2∑_i ẋ_̇i̇^2  + q ∑_i A_i ẋ_̇i̇ - q A_0,
 where q and m are electron charge and mass.  and V are vector and scalar potentials. 

The Euler Lagrange equations are the familiar Lorentz force law m v̇ = q (E + v × B), where v is the velocity vector, E_i = - ∂ A_i/∂ t - ∂ V/∂ x_i, B_i = ∂ A_k/∂ x_j - ∂ A_j/∂ x_k, the electric and magnetic fields.

The momentum p_i = ∂ L/∂ x_i and the Hamiltonian of the system H = p_i ∂/∂ẋ_i - L is


    H = ∑_j=x, y, z(p_j - q A_j)^2/2m + q A_0.
  



 §.§ Dirac and Schrödinger Equation


The Electron  Schrödinger Equation <cit.> is


    i  ∂ψ/∂ t =    ( ∑_j=x, y, z(-i ħ∂/∂ x_j - q A_j)^2/2m + q A_0  ) ψ,
 where ψ is electron wave-function. This equation is not very tractable, because it is nonlinear in , lets write a linear equation, which is the Dirac equation <cit.>, which takes the form


    i  ∂ϕ/∂ t =    ( ∑_j=x, y, z  c (-i  ħ∂/∂ x_j - q A_j) α_j  + β mc^2 + q A_0  ) ϕ.
 where α_j = σ_z ⊗σ_j and β = σ_x ⊗ are Dirac matrices, where σ_j are  the Pauli matrices, σ_z =  ( [  1  0;  0 -1 ] ). ϕ is electron spinor, for a electron wave with momentum k, takes the form ϕ =  [ [ cosθ/2; sinθ/2 ] ] ⊗↑ , where ↑ is spin up,  cosθ = ħ k/mc = υ/c, where υ = ħ k/m, is electron wave group velocity. Electron Orbitals are of size ∼ A^∘, their k ∼ 10^10 m , then υ∼ 10^6 m/s and cosθ∼10^6/3 × 10^8∼ 10^-3. Electron is non-relativistic, cosθ = υ/c∼ 0, θ∼π/2,
ϕ = 1/√(2) [ [ 1; 1 ] ] ⊗↑.

To fix ideas, take incoming EM wave,  along z direction, with electric field E_x sin (kz - ω t), the Lorentz gauge is  (A_0, A_x, A_y, A_z) = E_x/ωcos(kz-ω t)(0, 1, 0, 0). Electron wave with momentum q absorbs the photon with momentum k, and transits to momentum q + k. The transition is driven by Dirac matrix α_x, with transition amplitude


    =   [ [ cosθ/2; sinθ/2 ] ] ⊗↑ (σ_z ⊗σ_x_α_x)   [ [ cosθ/2; sinθ/2 ] ] ⊗↓ = q c A_x υ/c = q E_x  υ/ω


If we have electron orbital ϕ_0 then k' = M/M+mk of photon momentum goes to electron-nuclear relative coordinate, while k” = k momentum to CM (center of mass), where M is nucleus mass. The process drives the transition


    ϕ_0 ↑ ⟶ exp(i k'z)  ϕ_0 ↓,
 with amplitude =  q E_x  υ/ω. 

When orbital ϕ_1 is different from ϕ_0 we go to 
    ϕ_0 ↑ ⟶ exp(i k'z)  ϕ_0 ↓,
 with amplitude =  q c A_x  υ/c whose overlap with ϕ_1 is


    _1 = q  c A_x  υ/c i k' ⟨ϕ_1 |z| ϕ_0 ⟩_d_z  = i q E_x d_z  υ/c,
 where c k' ∼ω. 

But this is not suited for study of optical transitions, because we don't recover the Rabi frequency q E_x d. What we find is orders of magnitude smaller (down by υ/c). Instead we work with gauge


    (A_0, A_x, A_y, A_z) = -E_x/2 ( x sin(kz-ω t), -cos(kz-ω t)/ω, 0, x/csin(kz-ω t) ).



Now we have process driven by x term. For the  process, the amplitude of ϕ_0 →ϕ_0 is just 0, as ⟨ϕ_0 |x| ϕ_0 ⟩ = 0 and the  amplitude of ϕ_0 →ϕ_1 is simply  


    _1' = q  E_x  ⟨ϕ_1 |x| ϕ_0 ⟩_d_x  = q E_x d_x,


Dipole elements d_z, d_x are approx, Bohr radius ∼ A^∘. Due to the factor υ/c∼ 10^-3, _1 ≪_1'. Therefore transition between different atomic orbitals are largely driven by the x term. 



§ SCATTERING WITH · X TERM, THE NEGATIVE SIGN OF AMPLITUDE

Consider Moller scattering with x term


When electron changes momentum by q, in Lorentz gauge, photon of momentum -q is emitted. In · x gauge, the emitted photon can be more general with momentum -q + k, where k = n Δ (Δ ł = 2 π, l length of electron packet) makes angle θ with q, then the amplitude  of scattering a momentum exchange q is


    _0 = C u_4^†(p_4) u_3(p_2)  u_2^†(p_3) u_1(p_1)



    C = -1/4∑_n (2/π n)^2 1/( (q + n Δcos(θ))^2 + n^2 Δ^2 sin^2 θ )∫_0^πcos^2 θ d θ∫_0^2 πcos^2 ϕ dϕ,
 where ∫_l x sin(n 2 π/l x)= 2/n π. Then in limit Δ→ 0, we get C = -1/|q|^2.

We of course have (from A_z) the term


    _q = C u_4^†(p_4) γ_q u_3(p_2)  u_2^†(p_3) γ^q u_1(p_1)


The total amplitude including contribution from A_x, A_y in gauge


    = 1/4 |q|^2 { u_4^†(p_4) γ_μ u_3(p_2)  u_2^†(p_3) γ^μ u_1(p_1) -  ( 3 _0 + 5 _q     ) }.


The first term is the usual Feynman propagator, scaled by 1/4, but second term is new and gives big contribution to Coulomb potential.

That's it, we have a new propagator. In its full glory it reads


    =  (e ħ c)^2/4 ϵ_0 V q^2{ u_4^†(p_4) γ_μ u_3(p_2)  u_2^†(p_3) γ^μ u_1(p_1) -  ( 3 _0 + 5 _q     ) }.



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