title: Computation of the leading order contributions to the Lamb shift for the H atom using spectral regularization
authors: John Mashford
text: 
   
  
  
  Computation of the leading order contributions to the Lamb shift for the H atom using spectral regularization
      John Mashford 
  
  School of Mathematics and Statistics 
  
  University of Melbourne, Victoria 3010, Australia 
  
  E-mail: mashford@unimelb.edu.au
      December 30, 2023
  ==============================================================================================================================================
  
  
  
  
  
  
  
  The Uehling contribution to the Lamb shift can be computed exactly in terms of the Uehling potential function. However derivations of this function are complex involving avoiding divergences using intricate techniques from early quantum field theory (QFT) or else more modern approaches using charge and mass renormalization. In the present paper we derive the Uehling potential function in a fairly conceptually straightforward way not involving renormalization in which the vacuum polarization tensor is viewed as a Lorentz invariant 2-tensor valued measure on Minkowski space. Furthermore we compute a complex matrix valued potential function for the electron self-energy contribution to the Lamb shift. The resulting potential function is derived in a conceptually simple way not involving renormalization and can be used for higher order computations in QFT involving multiple loops.
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  The Lamb shift is a phenomenon which is closely tied up with the instigation and development of quantum field theory (QFT). At the time of its discovery in 1947 by Lamb and Retherford <cit.> it created considerable interest in the physics community and provided impetus for the development of the renormalization approach for dealing with the divergences present when considering radiative corrections. 
  
  Schwinger, Weisskopf and Oppenheimer suggested that the energy level shift might be the result of the interaction of the electron with the radiation field. The shift came out to be infinite in all theories at the time and was therefore ignored <cit.>. 
  
  Bethe <cit.> provided the first proposal for a solution to the problem. His calculation diverged logarithmically so he introduced a cutoff K for the light quanta which could be emitted and absorbed by the atom. He took K to be mc^2 (where m is the electron mass) and obtained a result of 1040 MHz which was within quite good agreement with the experimental value of ≈1000 MHz at the time <cit.>. 
  
  There followed rapidly a great deal of research by the pioneers of QFT such as Feynman <cit.>, Schwinger <cit.> and Dyson <cit.> resulting in the renormalization program whereby the divergences in QFT can be circumvented.
  
  The Lamb shift has continued to play a very important role in physics, both from the experimental point of view where its value has been determined to greater  and greater degrees of accuracy, and the theoretical point of view, where the theoretical prediction of its value has been refined to higher and higher orders of precision. The current level of agreement  between theory and experiment is as follows <cit.>.
  
      :   1057.847(9)
      :   1057.834 12 (23)(13)
  
  From the theoretical point of view research has developed on two fronts <cit.>. Firstly approaches to formulating the problems and then for dealing with the troublesome divergences have been developed. Secondly numerical methods for computing the results of these theoretical approaches have been developed. The current state of the art description of the theory of the Lamb shift is given by Yerokhin et al. in <cit.>.
  
  Research has also been carried out on hydrogenic ions and results are often presented as power series in Zα. Self-energy for low Z values is often obtained by extrapolating from high Z calculations <cit.>.
  
  The leading order contributions to the Lamb shift are the self-energy contribution (electron self-energy) and the Uehling contribution (photon self-energy or vacuum polarization). The theoretical problem associated with the Uehling contribution to the Lamb shift was solved in 1935 by Uehling <cit.> who derived the Uehling potential function. However his derivation using QFT as formulated at the time was, and all subsequent derivations using more modern techniques involving charge and mass  renormalization <cit.> have been, complex and to a certain extent obscure during the process of negotiating divergences. Work on the Uehling contribution has focussed on developing numerical techniques for applying the Uehling potential function in the case of the point nucleus or more elaborate nuclear models. Theoretical work on the self-energy has taken various directions but there does not seem to have been developed an analogue for the self-energy to the Uehling potential function.
  
  We have, in previous papers <cit.> presented an approach called spectral regularization for analyzing problematic objects in QFT such as the vacuum polarization tensor Π^μν or the self-energy function Σ in terms of their existence as measures on Minkowski space. In the present paper we  provide some theoretical results on Lorentz invariant 2-tensor valued measures in Sections 2 and 3 and then in Section 4 derive the Uehling potential function without requiring renormalization. Then in Sections 5 and 6 we present some theoretical results concerning invariant C^4×4 valued measures and then in section 7 we derive a complex matrix valued potential function,  which is analogous to the Uehling potential function, and can be used to compute the self-energy contribution to the Lamb shift.
  
  
  
  § LORENTZ INVARIANT 2-TENSOR VALUED MEASURES
  
  
  Let ℬ( R^4) denote the Borel algebra for R^4. A tensor valued measure μ:ℬ( R^4)→ C^4×4 will be said to be Lorentz invariant if
  
      μ^αβ(ΛΥ)=Λ^α_ρΛ^β_σμ^ρσ(Υ),∀Λ∈ O(1,3),Υ∈ℬ( R^4).
  
  
  
  
   §.§ Existence of a spectral function when the measure can be defined by a density with respect to Lebesgue measure
  
  
  Suppose that Π^μν is a Lorentz invariant 2-tensor valued measure which is associated with a density which we also denote as Π^μν. Then
  
      Π^μν(Υ)=∫_ΥΠ^μν(p) dp,∀Υ∈ℬ( R^4).
  
  Thus, for any Λ∈ O(1,3),
  
      Π^μν(ΛΥ)=∫_ΛΥΠ^μν(p) dp=∫_ΥΠ^μν(Λ p) dp,
  
  and
  
      Π^μν(ΛΥ)=Λ^μ_ρΛ^ν_σΠ^ρσ(Υ)=Λ^μ_ρΛ^ν_σ∫_ΥΠ^ρσ(p) dp,
  
  for all Υ∈ℬ( R^4). Therefore for each Λ∈ O(1,3),
  
      Π^μν(Λ p)=Λ^μ_ρΛ^ν_σΠ^ρσ(p),
  
  for almost all p∈ R^4. We will consider the (non-pathological) case where  Π can and has been adjusted so that Eq. <ref> holds for all Λ∈ O(1,3) and p∈ R^4.
  
  Suppose also that Π is causal and is supported on {p∈ R^4:p^2>0,p^0>0}. (The cone {p∈ R^4:p^2=0,p^0≥0} is a set of Lebesgue measure zero). Then Π is determined by its values on {(m, 0):m>0}. Define 
  
      λ^μν(m)=Π^μν((m, 0)), m>0.
  
  Then
  
      λ^μν(m)   =Π^μν((m,0))
         =Π^μν(R(m,0))
         =R^μ_ρR^ν_σΠ^ρσ((m,0))
         =R^μ_ρR^ν_σλ^ρσ(m),
  
  for all m>0,R∈, where Rotations ≅ O(3) is the rotation subgroup of O(1,3). Then (suppressing the argument m)
  
      λ^0ν=R^0_ρR^ν_σλ^ρσ=R^ν_σλ^0σ,∀ R∈.
  
  It follows that 
  
      λ^0ν=λ^00η^0ν.
  
  Similarly
  
      λ^μ0=λ^00η^μ0.
  
  Therefore
  
      λ = ([ λ^00    0;    0    A ]),
  
  for some A∈ C^3×3. Thus, since
  
      λ^μν=R^μ_ρR^ν_σλ^ρσ,∀ R∈,
  
  we must have
  
      λ^ij=B^i_kB^j_lλ^kl,∀ B∈ O(3)i,j,k,l∈{1,2,3}.
  
  Taking
  
      B=([  0  1  0; -1  0  0;  0  0  1 ]),
  
  shows that 
  
      λ^12=B^1_kB^2_lλ^kl=-λ^21.
  
  Taking
  
      B=([ 0 1 0; 1 0 0; 0 0 1 ]),
  
  shows that 
  
      λ^12=B^1_kB^2_lλ^kl=λ^21.
  
  Therefore
  
      λ^12=λ^21=0.
  
  By a similar argument
  
      λ^31=λ^13=λ^32=λ^23=0.
  
  Therefore for all m>0
  
      (λ^μν(m))=(λ_0(m),λ_1(m),λ_2(m),λ_3(m))λ_0(m),λ_1(m),λ_2(m),λ_3(m)∈ C.
  
  Now 
  
      λ^11=B^1_kB^1_lλ^kl.
  
  Taking B to be of the form of Eq. <ref> results in
  
      λ^11=B^1_kB^1_lλ^kl=λ^22.
  
  Similarly
  
      λ^11=λ^33.
  
  Thus
  
      λ(m)=(λ^μν(m))=(λ_0(m),λ_1(m),λ_1(m),λ_1(m)),
  
  for some locally integrable functions λ_0,λ_1:(0,∞)→ C.
  
  
  
   §.§ Canonical form of a causal Lorentz invariant 2-tensor valued measure
  
  
  Define
  
      Π_c^μν(Υ)=∫_m=0^∞∫_Υ(η^μνp^2σ_1(m)+p^μp^νσ_2(m)) Ω_m(dp) dm,
  
  for Υ∈ℬ( R^4), where σ_1,σ_2:(0,∞)→ C are locally integrable and for m>0 Ω_m is the standard Lorentz invariant measure on the hyperboloid H_m={p∈ R^4:p^2=m^2,p^0>0} defined by
  
      ∫ψ(p) Ω_m(dp)=∫_ R^3ψ(ω_m( p), p)d p/ω_m( p),
  
  for ψ a measurable function for which the integral exists, in which
  
      ω_m( p)=(m^2+ p^2)^1/2.
   
  Then
  
      Π_c^μν(Υ)   =∫_m=0^∞∫_ p∈ R^3χ_Υ(ω_m( p), p)(η^μνm^2σ_1(m)+(ω_m( p), p)^μ(ω_m( p), p)^νσ_2(m))
         d p/ω_m( p) dm,
  
  where χ_Γ denotes the characteristic function of Γ defined by
  
      χ_Γ(p)={[ 1p∈Γ;    0 ].
  
  Now make the coordinate transformation
  
      q = q(m, p)=(ω_m( p), p),m>0, p∈ R^3.
  
  The Jacobian of the transformation is 
  
      J(m, p)=mω_m( p)^-1.
  
  Therefore
  
      Π_c^μν(Υ)   =∫_q^2>0,q^0>0χ_Υ(q)(η^μνm^2σ_1(m)+q^μq^νσ_2(m))1/ω_m( p)1/mω_m( p)^-1 dq
         =∫_q^2>0,q^0>0χ_Υ(q)(η^μνmσ_1(m)+q^μq^νm^-1σ_2(m)) dq,
  
  where m=m(q)=(q^2)^1/2. Therefore the density associated with the canonical form is
  
      Π_c^μν(q)={[ η^μνζ(q)σ_1(ζ(q))+q^μq^νζ(q)^-1σ_2(ζ(q)),  q^2>0,q^0>0;                                                      0 ].
  
  where ζ(q)=(q^2)^1/2. Therefore the λ function associated with Π_c^μν is  
  
      λ_c^μν(m)   =Π_c^μν((m, 0))
         =η^μνmσ_1(m)+(m, 0)^μ(m, 0)^νm^-1σ_2(m)
         =η^μνmσ_1(m)+η^0μη^0νmσ_2(m).
  
  Since σ_1 and σ_2 are arbitrary locally integrable functions, comparing with Eq. <ref> we see that this is in the most general form of a density for a causal Lorentz invariant 2-tensor valued measure. Hence the canonical form is the most general form.
  
  
  
   §.§ The spectral calculus for causal Lorentz invariant 2-tensor valued measures
  
  
  Let Π^μν:ℬ( R^4)→ C be a causal Lorentz invariant 2-tensor valued measure which can be defined by a locally integrable density. Then from the previous section, Π^μν has the form
  
      Π^μν(Υ)=∫_m=0^∞∫_ R^4χ_Υ(p)(p^2η^μνσ_1(m)+p^μp^νσ_2(m)) Ω_m(dp) dm,
  
  for some locally integrable spectral functions σ_1,σ_2:(0,∞)→ C. Assume that the spectral functions are continuous. Now let 
  
      Υ(a,b,ϵ)=⋃_m∈(a,b)S(m,ϵ),
  
  where
  
      S(m,ϵ)={p∈ R^4:p^2=m^2,| p|<ϵ,p^0>0},
  
  be the hyperbolic cylinder defined in <cit.>. 
  Let 
  
      g^μν(a,b,ϵ)=Π^μν(Υ(a,b,ϵ)),
  
  for a,b∈(0,∞),a<b,ϵ>0. We have
  
      g^μν(a,b,ϵ)   =Π^μν(Υ(a,b,ϵ))
         =∫_m=0^∞∫_ p∈ R^3χ_Υ(a,b,ϵ)(ω_m( p), p)[η^μνm^2σ_1(m)+(ω_m( p), p)^μ(ω_m( p), p)^νσ_2(m)]
         d p/ω_m( p) dm
         =∫_m=a^b∫_B_ϵ( 0)[η^μνm^2σ_1(m)+(ω_m( p), p))^μ(ω_m( p), p)^νσ_2(m)]d p/ω_m( p) dm
         ≈∫_a^b[η^μνm^2σ_1(m)+η^μ0η^ν0m^2σ_2(m)]1/m dm (4/3πϵ^3).
  
  Therefore taking
  
      g^μν_a(b)   =lim_ϵ→0ϵ^-3g^μν(a,b,ϵ),
  
  we see that
  
      g_a^μν(b)   =∫_a^b[η^μνmσ_1(m)+η^μ0η^ν0mσ_2(m)] dm (4/3π).
  
  Therefore we can recover the spectral functions σ_1 and σ_2 as follows.
  
      g_a^ii'(b)=-bσ_1(b)(4/3π),∀ i=1,2,3.
  
  Therefore
  
      σ_1(b)=-3/4π1/bg_a^ii'(b).
  
  Also
  
      g_a^00'(b)=b(σ_1(b)+σ_2(b))(4/3π).
  
  Therefore
  
      σ_2(b)=3/4π1/bg_a^00'(b)-σ_1(b).
  
  Conversely, if a causal Lorentz invariant measure Π^μν is such that the functions g_a^μν defined by Eq. <ref> exist and are continuously differentiable then Π^μν has the form of Eq. <ref> and the spectral functions σ_1 and σ_2 can be recovered using Eqns. <ref>-<ref>.
  
  
  
  § THE VACUUM POLARIZATION TENSOR
  
  
  The vacuum polarization tensor is given by
  
      Π^μν(k)   =-(∫dp/(2π)^4i(-e)γ^μiS(p)i(-e)γ^νiS(p-k))
         =-e^2/(2π)^4∫(γ^μ1/ p-m+iϵγ^ν1/ p- k-m+iϵ) dp.
  
  (the leading minus sign is associated with the fermion loop). Therefore, using the ansatz  
  
      1/p^2-m^2+iϵ→ -iπΩ_m(p),
  
  and arguing as in <cit.> the measure associated with the vacuum polarization tensor is given by 
  
      Π^μν(Υ)=-e^2/16π^2∫χ_Υ(k+p)(γ^μ( p+m)γ^ν( k-m)) Ω_m(dk) Ω_m(dp),
  
  for Υ∈ℬ( R^4).
  theoremTheorem
  
  The vacuum polarization tensor valued measure is Lorentz invariant.
  
  Proof Rescale by -e^2/16π^2. Then
  
      Π_μν(Λ(κ)Υ)   =∫χ_ΛΥ(k+p)(γ_μ( p+m)γ_ν( k-m)) Ω_m(dk) Ω_m(dp) 
         =∫χ_Υ(Λ^-1k+Λ^-1p)(γ_μ( p+m)γ_ν( k-m)) Ω_m(dk) Ω_m(dp) 
         =∫χ_Υ(k+p)(γ_μ(κ pκ^-1+m)γ_ν(κ kκ^-1-m)) Ω_m(dk) Ω_m(dp) 
         =∫χ_Υ(k+p)(κκ^-1γ_μκ( p+m)κ^-1γ_νκ( k-m)κ^-1) Ω_m(dk) Ω_m(dp) 
         =∫χ_Υ(k+p)(Λ^-1ρ_μγ_ρ( p+m)Λ^-1σ_νγ_σ( k-m)) Ω_m(dk) Ω_m(dp) 
         =Λ^-1ρ_μΛ^-1σ_ν∫χ_Υ(k+p)(γ_ρ( p+m)γ_σ( k-m)) Ω_m(dk) Ω_m(dp) 
         =Λ^-1ρ_μΛ^-1σ_νΠ_ρσ(Υ),
  
  for all κ∈ K where 
  
      K={(
      [     a     0;     0 a^†-1 ]):a∈ GL(2, C),|(a)|=1}⊂ U(2,2),
  
  is the group defined in <cit.>, Λ=Λ(κ) is the Lorentz transformation corresponding to κ and we have used the intertwining property Σ_1(κ p)=κΣ_1(p)κ^-1=κ pκ^-1 of the map Σ_1=(p↦ p) <cit.> which implies that κγ_μκ^-1=Λ(κ)^ρ_μγ_ρ,∀κ∈ K.
  
  Therefore
  
      Π^αβ(ΛΥ)   =η^αμη^βνΠ_μν(ΛΥ)
         =η^αμη^βνΛ^-1ρ_μΛ^-1σ_νΠ_ρσ(Υ).
  
  Now
  
      Λ^-1ρ_μη^αμ=(Λ^-1η)^ρα,
  
  and
  
      Λ^TηΛ=η,
  
  from which it follows that
  
      Λ^-1η=ηΛ^T.
  
  Therefore
  
      Λ^-1ρ_μη^αμ=(ηΛ^T)^ρα=η^ρμΛ^T_μ^α=η^ρμΛ^α_μ.
  
  Similarly
  
      Λ^-1σ_νη^βν=η^σνΛ^β_ν.
  
  Therefore
  
      Π^αβ(ΛΥ)   =η^ρμΛ^α_μη^σνΛ^β_νΠ_ρσ(Υ)
         =Λ^α_μΛ^β_νΠ^μν(Υ).
  
  
  
  
  
   §.§ Determination of the form of the vacuum polarization tensor using spectral calculus
  
  
  It is straightforward to show that the vacuum polarization tensor valued measure is causal. 
  
  The vacuum polarization tensor has the form
  
      Π^μν(Υ)=∫_m^'=0^∞∫_ R^4χ_Υ(p)(p^2η^μν-p^μp^ν) Ω_m^'(dp) ρ(m^') dm^',
  
  for some continuous spectral function ρ:(0,∞)→ C.
  
  Proof
  
  One can readily compute using the gamma matrix trace identities that
  
      (γ^μ( p+m)γ^ν( k-m))=4(p^μk^ν-η^μνp.k+k^μp^ν-m^2η^μν).
  
  Thus
  
      g^μν(a,b,ϵ)   =Π^μν(Υ(a,b,ϵ))
         =-e^2/16π^2∫χ_Υ(a,b,ϵ)(k+p)(γ^μ( p+m)γ^ν( k-m)) Ω_m(dk) Ω_m(dp)
         =-e^2/4π^2∫χ_Υ(a,b,ϵ)(k+p)(p^μk^ν-η^μνp.k+k^μp^ν-m^2η^μν ) Ω_m(dk) Ω_m(dp)
         =-e^2/4π^2∫χ_(a,b)(ω_m( k)+ω_m( p))χ_B_ϵ( 0)( k+ p)((ω_m( p), p)^μ(ω_m( k), k)^ν-
         η^μν(ω_m( p)ω_m( k)- p. k)+(ω_m( k), k)^μ(ω_m( p), p)^ν-m^2η^μν) d k/ω_m( k)d p/ω_m( p)
         =-e^2/4π^2∫χ_(a,b)(ω_m( k)+ω_m( p))χ_B_ϵ( 0)- p( k)((ω_m( p), p)^μ(ω_m( k), k)^ν-
         η^μν(ω_m( p)ω_m( k)- p. k)+(ω_m( k), k)^μ(ω_m( p), p)^ν-m^2η^μν) d k/ω_m( k)d p/ω_m( p)
         ≈-e^2/4π^2∫χ_(a,b)(2ω_m( p))((ω_m( p), p)^μ(ω_m( p),- p)^ν-η^μν(ω_m( p)^2+ p^2)+
         (ω_m( p),- p)^μ(ω_m( p), p)^ν-m^2η^μν)1/ω_m( p)^2 d p (4/3πϵ^3)
  
  Now
  
      χ_(a,b)(2ω_m( p))=1   ⇔ a<2(m^2+ p^2)^1/2<b
         ⇔a^2/4<m^2+ p^2<b^2/4
         ⇔a^2/4-m^2< p^2<b^2/4-m^2
         ⇔ mZ(a)<| p|<mZ(b),
  
  where
  
      Z(b)=(b^2/4m^2-1)^1/2, b≥2m.
  
  Therefore
  
      g_a^μν(b)   =lim_ϵ→0ϵ^-3g^μν(a,b,ϵ)
         =-e^2/4π^2∫_r=mZ(a)^mZ(b)∫_θ=0^π∫_ϕ=0^2π((ω_m( p), p)^μ(ω_m( p),- p)^ν-η^μν(m^2+2r^2)+
         (ω_m( p),- p)^μ(ω_m( p), p)^ν-m^2η^μν)r^2/m^2+r^2sin(θ) dϕ dθ dr (4/3π).
  
  where p= p(r,θ,ϕ) (spherical polar coordinates). 
  Hence
  
      g_a^μμ'(b)   =-e^2/4π^2∫_θ=0^π∫_ϕ=0^2π((ω_m( p), p)^μ(ω_m( p),- p)^μ-η^μμ(m^2+2r^2)+
         (ω_m( p),- p)^μ(ω_m( p), p)^μ-m^2η^μμ)r^2/m^2+r^2sin(θ) dϕ dθ (4/3π)
         mZ^'(b),
  
  where r=mZ(b). Thus
  
      g_a^00'(b)   =-e^2/4π^2∫_θ=0^π∫_ϕ=0^2π(m^2+r^2-(m^2+2r^2)+m^2+r^2-m^2)
         r^2/m^2+r^2sin(θ) dϕ dθ (4/3π)(mZ^'(b))
         =0.
  
  Therefore by the spectral theorem and Eq. <ref>,
  Π^μν has the form of Eq. <ref> with σ_1 and σ_2 continuous functions and 
  
      σ_1(b)=-σ_2(b),∀ b>0.
  
  
  
  Thus, letting ρ=σ_1 and making the coordinate transformation defined by Eq. <ref>, we have that
  
      Π^μν(Υ)   =∫_m^'=0^∞∫_ R^3χ_Υ(ω_m^'( p), p)(η^μνm^'2-(ω_m^'( p), p)^μ(ω_m^'( p), p)^ν)ρ(m^')
         d p/ω_m^'( p) dm^'
         =∫_q^2>0,q^0>0χ_Υ(q)(q^2η^μν-q^μq^ν)(q^2)^-1/2ρ((q^2)^1/2) dq.
  
  Hence the density Π^μν for the measure Π^μν is given by
  
      Π^μν(q)=(q^2η^μν-q^μq^ν)π(q),
  
  where
  
      π(q)={[ (q^2)^-1/2ρ((q^2)^1/2)q^2>0,q^0>0;                                 0 ].
  
  (The fact that Π^μν has the form of Eq. <ref> is well known but has previously been established through manipulating infinite quantities during renormalization (see <cit.>, p. 478).) 
  
  Contracting Eq. <ref> with the Minkowski space metric tensor we obtain
  
      π(q)=1/3q^2Π(q)q^2≠0.
  
  where
  
      Π(q)=η_μνΠ^μν(q).
  
  
  
  
   §.§ Computation of the vacuum polarization function using spectral calculus
  
  
  It can be shown using the spectral calculus <cit.>, that the contraction of the spectral vacuum polarization tensor Π^μν is given by
  
      Π(q)={[ s^-1σ(s)q^2>0, q^0>0;                    0 ].
  
  where s=(q^2)^1/2 and σ is given by
  
      σ(s)={[ 2/πe^2m^3Z(s)(3+2Z^2(s))s≥2m;                            0 ].
  
  in which Z is given by Eq. <ref>.
  
  Therefore the spectral vacuum polarization function is given by
  
      π(q)=Π(q)/3q^2=1/3s^-3σ(s)=2/3πe^2m^3s^-3Z(s)(3+2Z^2(s)),s≥2m,
  
  for q^2>0,q^0>0 where s=(q^2)^1/2.
  
  It can also be shown by making the transformation Ω_m*Ω_m→Ω_im*Ω_im that, in the spacelike domain, π can be considered to be the function given by
  
      π(q)=1/3s^-3σ(s)=2/3πe^2m^3s^-3Z(s)(3+2Z^2(s)),s≥2m,
  
  where s=(-q^2)^1/2.
  
  
  
  § THE UEHLING CONTRIBUTION TO THE LAMB SHIFT 
  
  
  The Feynman amplitude associated with the tree diagram for electron proton scattering is given by
  
      ℳ_aα_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)=u_e(p_1^',α_1^')ie_1γ^ρu_e(p_1,α_1)iD_ρσ(q)u_p(p_2^',α_2^')ie_2γ^σu_p(p_2,α_2),
  
  where
  
      D_Fαβ(q)=-η_αβ/q^2+iϵ,
  
  is the photon propagator, q=p_1^'-p_1 is the momentum transfer, e_1=-e,e_2=e are the charges of the electron and proton respectively and u_e(p,α) and u_p(p,α) are Dirac spinors for the electron and the proton for α∈{0,1}. Therefore
  
      ℳ_aα_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)=iℳ_0α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)/q^2+iϵ,
  
  where
  
      ℳ_0α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)=u_e(p_1^',α_1^')e_1γ^ρu_e(p_1,α_1)η_ρσu_p(p_2^',α_2^')e_2γ^σu_p(p_2,α_2),
  
  The Feynman amplitude associated with the vacuum polarization diagram is given by
  
      ℳ_bα_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)   =u_e(p_1^',α_1^')ie_1γ^ρu_e(p_1,α_1)iD_Fρμ(q)iΠ^μν(q)iD_Fνσ(q)
         u_p(p_2^',α_2^')ie_2γ^σu_p(p_2,α_2),
  
   Therefore, since
  
      D_Fρμ(q)Π^μν(q)D_Fνσ(q)   =(1/q^2)^2η_ρμ(q^2η^μν-q^μq^ν)π(q)η_νσ, 
         =(η_ρσ/q^2-(1/q^2)^2q_ρq_σ)π(q),
  
  and, by a well known conservation property
  
      u(p_1^',α_1^')q_ργ^ρu(p_1,α_1)=0,
  
  we have that
  
      ℳ_bα_1^'α_1^'α_1α_2(p_1^',p_2^',p_1,p_2)=iℳ_0α_1^'α_1^'α_1α_2(p_1^',p_2^',p_1,p_2)π(q)/q^2+iϵ.
  
  Now consider the NRQED (non-relativistic quantum electrodynamics) approximation in which
  
      ℳ_0α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)   =e_1e_2δ_α_1^'α_1δ_α_2^',α_2=-e^2δ_α_1^'α_1δ_α_2^',α_2,
  
  and q^0 is negligible compared with | q|. In this case
  
      ℳ_bα_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)=-ie^2δ_α_1^'α_1δ_α_2^'α_2π((0, q))/(0, q)^2=ie^2δ_α_1^'α_1δ_α_2^'α_2π((0, q))/ q^2,
  
  and we can write
  
      ℳ_b( q)=ie^2π(0, q)/ q^2.
  
  Therefore using the Born approximation the potential function associated with the Uehling contribution to the Lamb shift is <cit.>
  
      Δ V( x)   =i(2π)^-3∫ℳ_b( q)e^i q. x d q
         =i(2π)^-3∫(ie^2π((0, q))/ q^2)e^i q. x d q
         =-(2π)^-3e^2∫π((0, q))/ q^2e^i q. x d q,
  
  if the integral exists. The argument of the inverse Fourier transform is orthogonally invariant and thus Δ V is orthogonally invariant. Therefore we can write
  
      Δ V(r)   =Δ V(0,0,r)
         =-(2π)^-3e^2∫_s=0^∞∫_θ=0^π∫_ϕ=0^2ππ(s)/s^2e^isrcos(θ)s^2sin(θ) dϕ dθ ds
         =-(2π)^-2e^2∫_s=0^∞π(s)/irs(e^isr-e^-isr) ds
         =i(2π)^-2e^21/r∫_s=0^∞π(s)/s(e^isr-e^-isr) ds.
  
  Hence
  
      Δ V(r)=i(2π)^-2e^21/r(∫_s=0^∞π(s)/se^isr ds-∫_s=0^∞π(s)/se^-isr ds).
  
  It is shown in <cit.> that these integrals are convergent for all r>0. We would like to analytically continue Δ V to the upper imaginary axis of the complex plane since we interested in spacelike points in Minkowski (configuration) space for which x^2<0, corresponding to pure imaginary r. Therefore we seek a complex analytic function Δ V_ associated with Δ V. 
  By a well known Paley-Weiner theorem the function
  
      r↦∫_s=0^∞π(s)/rse^irs ds,
  
  is analytic in the upper half plane while the function
  
      r↦∫_s=0^∞π(s)/rse^-irs,
  
  is analytic in the lower half plane (but not in the upper half plane). 
  
  Consider the transformation 𝒯 taking functions F of the form 
  
      F(r)=∫_s=0^∞f(s)(e^isr-e^-isr) ds,
  
  to the function 𝒯(F) defined by
  
      (𝒯(F))(r)=i∫_s=0^∞f(s)e^-isr ds.
  
  where f:(0,∞)→ C is such that both integrals defined by Eqns. <ref> and <ref> are convergent for all r>0.
  𝒯 takes non-divergent functions into non-divergent functions. The mathematical properties and justification of the transform 𝒯 may be considered elsewhere. The important thing from the point of view of physics is that the use of 𝒯 gives the correct answers. We find that it does for the cases of the computation of the Uehling potential function and also for the case of the electron self-energy contribution to the Lamb shift.
  
  In the case of the Ueling contribution we compute using 
  
      f(s)=π(s)/s=σ(s)/3s^4,
  
  and the Δ V function that we derive is
  
      Δ V(r)   =i(2π)^-2e^21/rF(r)
         → i(2π)^-2e^21/r(𝒯(F))(r)
         =i(2π)^-2e^21/ri∫_s=0^∞σ(s)/3s^4e^-sr ds
         =-(2π)^-2e^22/3πe^2m^31/r∫_s=2m^∞Z(s)(3+2Z^2(s))/s^4e^-sr ds
         =-(2π)^-2e^22/3πe^2m^31/r∫_x=1^∞(x^2-1)^1/2(2x^2+1)/(16m^4)x^4e^-2mxr(2m) dx
         =-α^2/3π1/r∫_x=1^∞(x^2-1)^1/2(2x^2+1)x^-4e^-2mxr dx,
  
  where α=e^2/4π is the fine structure constant. 
  
  The result that we have obtained is precisely the Uehling vacuum polarization potential function which has previously only been calculated through negotiating infinities and divergences using complex calculations involving methods such as charge and mass renormalization <cit.>. 
  
  From this potential function the Uehling contribution to the Lamb shift can be exactly calculated <cit.> according to
  
      Δ E=<ψ|Δ V|ψ>=4π∫_r=0^∞|ψ(r)|^2Δ V(r)r^2 dr,
  
  where ψ is the H atom 2s wave function, and theory agrees with experiment to a very high order of precision <cit.>.  
  
  
  
  § K INVARIANT C^4×4 MATRIX VALUED MEASURES
  
  
  As described in <cit.> the group K⊂ U(2,2)⊂ C^4×4 defined by Eq. <ref> acts in natural ways on R^4 and C^4. A C^4×4 matrix valued measure is a vector valued measure taking values in the vector space C^4×4. A C^4×4 valued measure μ:ℬ(  R^4)→ C^4×4 will be said to be K invariant if
  
      μ(κΥ)=κμ(Υ)κ^-1,∀κ∈ K,Υ∈ℬ( R^4).
  
  
  
  
   §.§ Existence of a spectral function when the measure can be defined by a density with respect to Lebesgue measure 
  
   
  Suppose that μ is such a measure which can be defined by a locally integrable density, which we also denote as μ. Then
  
      μ(Υ)=∫_Υμ(p) dp,∀Υ∈ℬ( R^4).
  
  By K invariance
  
      μ(Λ(κ)Υ)=κμ(Υ)κ^-1=κ∫_Υμ(p) dp κ^-1.
  
  But
  
      μ(Λ(κ)Υ)=∫_Λ(κ)Υμ(p) dp=∫_Υμ(Λ(κ)p) dp,
  
  where Λ(κ) is the element of O(1,3)^↑+ corresponding to κ∈ K. Both these equations are true for all κ∈ K,Υ∈ℬ( R^4). Therefore for all κ∈ K
  
      μ(Λ(κ)p)=κμ(p)κ^-1,
  
  (for almost all p∈ R^4). We will consider the case where μ can be, and has been, adjusted so that Eq. <ref> holds for all κ∈ K,p∈ R^4.
  
  Conversely, given a locally integrable matrix valued function which satisfies Eq. <ref> then the object μ:ℬ( R^4)→ C^4×4 defined by Eq. <ref> is a K invariant C^4×4 valued measure. 
  
  Suppose that μ is such a function and assume that μ is causal and is supported on {p∈ R^4:p^2>0, p^0>0}.  Define M:(0,∞)→ C^4×4 by
  
      M(m)=μ((m, 0)).
  
  Now
  
      μ(Λ(κ)(m, 0))=κ M(m)κ^-1.
  
  Thus μ is determined if M is given. We will call the function M the spectrum of μ.
  
  Since (m, 0) is invariant under elements of Rotations we have
  
      M(m)     =    ([ M_1 M_2; M_3 M_4 ]) 
          =    ([ a 0; 0 a ])([ M_1 M_2; M_3 M_4 ])([ a 0; 0 a ])^-1
          =    ([ aM_1a^-1 aM_2a^-1; aM_3a^-1 aM_4a^-1 ]),
  
  for all a∈ SU(2). Therefore each block M_i,i=1,2,3,4 of M commutes with each element of SU(2). Suppose that 
  
      M_i=([ b_i11 b_i12; b_i21 b_i22 ]).
  
  Now
  
      a=([  0  1; -1  0 ])∈ SU(2).
  
  Therefore
  
      ([  0  1; -1  0 ])
      ([ b_i11 b_i12; b_i21 b_i22 ])=
      ([ b_i11 b_i12; b_i21 b_i22 ])
      ([  0  1; -1  0 ]),
  
  from which it follows that b_i21=-b_i12 and b_i11=b_i22. Also
  
      a=([ 0 i; i 0 ])∈ SU(2).
  
  Therefore
  
      ([ 0 i; i 0 ])
      ([ b_i11 b_i12; b_i21 b_i22 ])=
      ([ b_i11 b_i12; b_i21 b_i22 ])
      ([ 0 i; i 0 ]),
  
  from which it follows that b_i21=b_i12 and b_i11=b_i22. Therefore
  
      M_i=λ_i=λ_iI_2,
  
  for some λ_i∈ C.
  
  Conversely, let λ_1,λ_2,λ_3,λ_4:(0,∞)→ C be locally integrable functions and define μ:{p∈ R^4:p^2>0,p^0>0}→ C^4×4 by
  
      μ(κ(m, 0))=κ M(m)κ^-1,∀κ∈ K,m>0,
  
  where
  
      M=(
      [ λ_1 λ_2; λ_3 λ_4 ])∈ C^4×4.
  
  It is straightforward to show that μ is well defined. 
  Let p↦κ(p) be any function such that κ(p)p=((p^2)^1/2, 0),∀ p∈ R^4p^2>0,p^0>0. Then for all κ∈ K,p∈ R^4 for which p^2>0,p^0>0 
  
      μ(κ p)   =μ(κκ(p)^-1κ(p) p)
         =μ(κκ(p)^-1(m, 0))
         =κκ(p)^-1M(m)(κκ(p)^-1)^-1
         =κμ(κ(p)^-1(m,0))κ^-1
         =κμ(κ(p)^-1κ(p) p)κ^-1
         =κμ(p)κ^-1.
  
  where m=(p^2)^1/2. Therefore μ is K invariant.
  
  Hence the function defined by Eq. <ref> is the most general form of the density for a K invariant C^4×4 valued measure which can be defined by a locally integrable density on Minkowski space.
  
  
  
   §.§ Canonical form of a K invariant C^4×4 valued measure
  
  
  If σ:(0,∞)→ C is a locally integrable function define μ_σ:ℬ( R^4)→ C^4×4 by
  
      μ_σ(Υ)=∫_m=0^∞∫_Υ( p+m) Ω_m(dp)σ(m) dm.
  
  Then
  
      μ_σ(Λ(κ)(Υ))     =    ∫_m=0^∞∫_Λ(κ)(Υ)( p+m) Ω_m(dp)σ(m) dm 
          =    ∫_m=0^∞∫_Υ(Σ_1(Λ(κ)p)+m) Ω_m(dp)σ(m) dm 
          =    ∫_m=0^∞∫_Υ(κ pκ^-1+m) Ω_m(dp)σ(m) dm 
          =    κ∫_m=0^∞∫_Υ( p+m) Ω_m(dp)σ(m) dm κ^-1
          =    κμ(Υ)κ^-1,
  
  for all Υ∈ℬ( R^4) where Σ_1 denotes the map p↦ p and we have used the intertwining property Σ_1(κ p)=κΣ_1(p)κ^-1 of Σ_1 <cit.>. Therefore μ_σ is K invariant. Now, making the coordinate transformation defined by Eq. <ref>, we have 
  
      μ_σ(Υ)     =    ∫_m=0^∞∫_Υ( p+m) Ω_m(dp)σ(m) dm 
          =    ∫_m=0^∞∫_ R^3χ_Υ((ω_m( p), p))(Σ_1((ω_m( p), p))+m)d p/ω_m( p)σ(m) dm 
          =    ∫_q^2>0,q^0>0χ_Υ(q)( q+ζ(q))σ(ζ(q))/ζ(q) dq,
  
  where ζ(q)=(q^2)^1/2. Therefore the density associated with μ_σ is given by
  
      μ_σ(q)={[ ( q+ζ(q))ζ(q)^-1σ(ζ(q))q^2>0,q^0>0;                                  0 ].
  
  Therefore the spectral function M=M_σ:(0,∞)→ C^4×4 associated with μ_σ is
  
      M_σ(m)     =    μ_σ((m, 0)) 
          =     (mγ^0+m)m^-1σ(m)
          =      (γ^0+1)σ(m)
          =    ([ 2σ(m)     0;     0     0 ]),
  
   where we use the Dirac representation for the gamma matrices.
  
  More generally if σ_1,σ_2:(0,∞)→ C are locally integrable functions then the matrix valued measure μ_σ_1,σ_2:ℬ( R^4)→ C^4×4 defined by
  
      μ_σ_1,σ_2(Υ)=∫_m=0^∞∫_Υ( p+m) Ω_m(dp)σ_1(m) dm+∫_m=0^∞∫_Υ( p-m) Ω_m(dp)σ_2(m) dm,
  
  is K invariant with spectral function M given by
  
      M_σ_1,σ_2(m)=([  2σ_1(m)        0;        0 -2σ_2(m) ]).
  
  
  
  
   §.§ The spectral calculus for K invariant C^4×4 valued measures
  
  
  Suppose that μ is a K invariant C^4×4 valued measure which can be defined by a locally integrable density of the form of μ_σ_1σ_2 for some locally integrable spectral functions σ_1,σ_2. Suppose that σ_1 and σ_2 are continuous on (0,∞). Let for a,b,ϵ∈(0,∞),a<b, g(a,b,ϵ) be defined by
  
      g(a,b,ϵ)=μ(Υ(a,b,ϵ)),
  
  where Υ(a,b,ϵ) is the hypercylinder defined by Eq. <ref>. 
  Then
  
      g(a,b,ϵ)     =    ∫_m=0^∞∫_Υ(a,b,ϵ)( p+m) Ω_m(dp)σ_1(m) dm+ 
            ∫_m=0^∞∫_Υ(a,b,ϵ)( p-m) Ω_m(dp)σ_2(m) dm 
          =    ∫_m=a^b∫_B_ϵ( 0)(Σ_1((ω_m( p), p))+m) d p/ω_m( pσ_1(m) dm+ 
            ∫_m=a^b∫_B_ϵ( 0)(Σ_1((ω_m( p), p))-m) d p/ω_m( p)σ_2(m) dm 
         ≈   4/3πϵ^3(∫_m=a^b(mγ^0+m) 1/mσ_1(m) dm+ 
            ∫_m=a^b(mγ^0-m) 1/mσ_2(m) dm) 
          =    4/3πϵ^3∫_a^b([  σ_1(m)       0;       0 -σ_2(m) ])dm,
  
  where Σ_1 denotes the map p↦ p. Therefore if we define g_a:(0,∞)→ C^4×4 by
  
      g_a(b)=lim_ϵ→0ϵ^-3g(a,b,ϵ),
  
  then
  
      σ_1(b)=3/4πg_a^00'(b),σ_2(b)=-3/4πg_a^33'(b), b>0.
  
  i.e.
  
      σ(b)=M(b)=3/4πg_a^'(b).
  
  
  Conversely if μ:ℬ( R^4)→ C^4×4 is a causal K invariant measure and if the function g_a defined by Eq. <ref> exists and is is continuously differentiable with g_a^αβ(b)=0α≠β  then μ has the form of Eq. <ref> and the spectral functions can be recovered using Eqns. <ref>.
   
  
  
  § THE SELF-ENERGY OF THE ELECTRON
  
  
  The Feynman integral associated with the self-energy of the electron is
  
      iΣ(p)=∫dk/(2π)^4iD_μν(k)i(-e)γ^μiS_F(p-k)i(-e)γ^ν,
  
  where
  
      D_μν(k)=-1/k^2+iϵ,
  
  is the photon propagator and
  
      S_F(p)=1/ p-m+iϵ,
  
  is the fermion propagator. This can be written as
  
      iΣ(p)=-e^2/(2π)^4∫1/k^2+iϵγ^μ p- k+m/(p-k)^2-m^2+iϵγ_μ dk.
  
  We make the following formal computation
  
      iΣ(Υ)     =    ∫_ΥiΣ(p) dp 
          =    ∫χ_Υ(p)(-e^2/(2π)^4)1/k^2+iϵγ^μ p- k+m/(p-k)^2-m^2+iϵγ_μ dk dp 
          =    ∫χ_Υ(p)(-e^2/(2π)^4)1/k^2+iϵγ^μ p- k+m/(p-k)^2-m^2+iϵγ_μ dp dk 
          =     -e^2/(2π)^4∫χ_Υ(p+k)1/k^2+iϵγ^μ p+m/p^2-m^2+iϵγ_μ dp dk 
          =    e^2/16π^2∫χ_Υ(p+k)γ^μ( p+m)γ_μ Ω_m^±(dp) Ω_0^±(dk),
  
  where we have used the ansatz <cit.>
  
      1/p^2-m^2+iϵ→-iπΩ_m^±(p),∀ m≥0.
  
  We take the case 
  
      iΣ(Υ)=e^2/16π^2∫χ_Υ(p+k)γ^μ( p+m)γ_μ Ω_m(dp) Ω_0^+(dk),
  
  in which case Σ has existence as a well defined mathematical object (a tempered C^4×4 valued measure).
  
  Now γ^μ pγ_μ=-2 p and γ^μmγ_μ=4m. Therefore
  
      iΣ(Υ)=e^2/16π^2∫χ_Υ(p+k)(4m-2 p) Ω_m(dp) Ω_0^+(dk).
  
  
  iΣ is K invariant.
  
  Proof
  
  Let κ∈ K and Υ∈ℬ( R^4). Then
  
      iΣ(κ(Υ))     =    e^2/16π^2∫χ_κ(Υ)(p+k)(4m-2 p) Ω_m(dp) Ω_0^+(dk) 
          =    e^2/16π^2∫χ_Υ(κ^-1(p+k))(4m-2 p) Ω_m(dp) Ω_0^+(dk) 
          =    e^2/16π^2∫χ_Υ(p+k)(4m-2κ pκ^-1) Ω_m(dp) Ω_0^+(dk) 
          =    e^2/16π^2κ∫χ_Υ(p+k)(4m-2 p) Ω_m(dp) Ω_0^+(dk)κ^-1
          =    κ iΣ(Υ)κ^-1
  
  
  
  It can be shown that iΣ is causal. We will now use the spectral calculus to compute the spectrum of iΣ.
  
      g(a,b,ϵ)   =   iΣ(Υ(a,b,ϵ))
          =    e^2/16π^2∫χ_Υ(a,b,ϵ)(p+k)(4m-2 p) Ω_m(dp) Ω_0^+(dk) 
          =    e^2/16π^2∫χ_(a,b)(ω_m( p)+ω_0( k))χ_B_ϵ( 0)( p+ k)(4m-2 p) 
            ω_m( p)^-1ω_0( k)^-1 d p d k
          =    e^2/16π^2∫χ_(a,b)(ω_m( p)+ω_0( k))χ_B_ϵ( 0)- p( k)(4m-2 p) 
            ω_m( p)^-1ω_0( k)^-1 d p d k
          =    e^2/16π^2∫χ_(a,b)(ω_m( p)+ω_0( k))χ_B_ϵ( 0)- p( k)(4m-2 p) 
            ω_m( p)^-1ω_0( k)^-1 d k d p
         ≈   4/3πϵ^3e^2/16π^2∫χ_(a,b)(ω_m( p)+ω_0( p))(4m-2 p) 
            ω_m( p)^-1ω_0( p)^-1 d p,
  
  with p=(ω_m( p), p).
  
  Now ω_m( p)+ω_0( p)=c⇒ c≥ m, and for all c≥ m
  
      ω_m( p)+ω_0( p)=c   ⇔(r^2+m^2)^1/2+r=c
         ⇔ r^2+m^2=(c-r)^2=c^2+r^2-2cr
         ⇔2cr=c^2-m^2
         ⇔ r=(2c)^-1(c^2-m^2),
  
  where r=| p|. Therefore
  
      g_a(b)     =    lim_ϵ→0ϵ^-3g(a,b,ϵ)
         =   4/3πe^2/16π^2∫χ_(a,b)(ω_m( p)+ω_0( p))(4m-2 p)ω_m( p)^-1ω_0( p)^-1 d p
          =    4/3πe^2/16π^2∫_r=Z_1(a)^Z_1(b)∫_θ=0^π∫_ϕ=0^2π(4m-2 p)ω_m(r)^-1ω_0(r)^-1r^2sin(θ) dϕ dθ dr,
  
  where p=p(r,θ,ϕ)=(ω_m(r),rsinθcos(ϕ),rsin(θ)sin(ϕ),rcos(θ)) and Z_1:[m,∞)→[0,∞) is defined by
  
      Z_1(b)=(2b)^-1(b^2-m^2).
  
  Now
  
      Z_1^'(b)=(2b^2)^-1(b^2+m^2),
  
  and
  
      ∫_θ=0^π∫_ϕ=0^2π psin(θ) dϕ dθ=4πω_m(r)γ^0.
  
  Hence
  
      g_a(b)     =    4/3πe^2/16π^2∫_r=Z_1(a)^Z_1(b)(4π)(4m-2ω_m(r)γ^0)ω_m(r)^-1r dr.
  
  Therefore applying the spectral calculus and using Leibniz' integral rule we obtain
  
      σ(b)    =3/4πg_a^'(b)
         =e^2/4π(4m-2ω_m(Z_1(b))γ^0)ω_m(Z_1(b))^-1Z_1(b)Z_1^'(b)b≥ m.
  
  If p∈ R^4 is timelike and κ∈ K is such that κ p=Λ(κ)p=((p^2)^1/2,0) then 
  
      iΣ(p)=iΣ(κ^-1κ p)=κ^-1iΣ(((p^2)^1/2,0))κ=κ^-1σ((p^2)^1/2)κ.
  
  Also, in the spacelike domain, if p∈ R^4 is spacelike and κ∈ K is such that κ p=(0,0,0,(-p^2)^1/2) then
  
      iΣ(p)=iΣ(κ^-1κ p)=κ^-1iΣ((0,0,0,(-p^2)^1/2))κ=κ^-1σ((-p^2)^1/2)κ.
  
  In particular
  
      iΣ((0,0,0,ζ))=σ(ζ),∀ζ>0.
  
  
  
  
  § THE ELECTRON SELF-ENERGY CONTRIBUTION TO THE LAMB SHIFT
  
  
  The Feynman amplitude for the electron self-energy contribution to the Lamb shift is given by
  
      ℳ=ℳ_0+ℳ_1,
  
  where
  
      ℳ_0α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)     =    u_e(p_2^',α_2^')iD_μν(p_2^'-p_2)ie_2γ^μu_e(p_2,α_2) 
            u_p(p_1^',α_1^')ie_1γ^νu_p(p_1,α_1), 
      ℳ_1α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)     =    u_e(p_2^',α_2^')iΣ(p_2^')iS(p_2^')iD_μν(p_2^'-p_2)ie_2γ^μu_e(p_2,α_2) 
            u_p(p_1^',α_1^')ie_1γ^νu_p(p_1,α_1),
  
  in which e_1=e and e_2=-e are the charges of the proton and the electron respectively,
  
      u_e(p,α)=m_e^-1( p+m_e)e_α,
  
  
      u_p(p,α)=m_p^-1( p+m_p)e_α,
  
  are Dirac spinors for the electron and the proton respectively with m_e= the mass of the electron, m_p= the mass of the proton, e_α is the αth standard basis element for C^4,
  
      D_μν(q)=-η_μν/q^2+iϵ,
  
  is the photon propagator,
  
      S(p)=1/ p-m_e+iϵ= p+m_e/p^2-m_e^2+iϵ,
  
  is the electron propagator and
  
      iΣ(p)“="-e^2/(2π)^4∫1/k^2+iϵγ^μ p- k+m_e/(p-k)^2-m_e^2+iϵγ_μ dk,
  
  is the the function associated with the Feynman integral for the self-energy of the electron.
  
  ℳ_0 corresponds to the tree level Feynman amplitude for electron-proton scattering and ℳ_1 is the perturbation due to the electron self-energy. Now
  
      ℳ_1α_1^'α_2^'α_1α_2(p_1^',p_2^',p_1,p_2)   =u_e(p_2^',α_2^')iΣ(p_2^')S(p_2^')D_μν(p_2^'-p_2)e_2γ^μu_e(p_2,α_2)
         u_p(p_1^',α_1^')e_1γ^νu_p(p_1,α_1)
         =m_e^-2m_p^-2e_2e_1( p_2^'+m_e)e_α_2^'iΣ(p_2^')S(p_2^')D_μν(p_2^'-p_2)γ^μ
         ( p_2+m_e)e_α_2( p_1^'+m_p)e_α_1^'γ^ν( p_1+m_p)e_α_1
         =m_e^-2m_p^-2e_2e_1e_α_2^'^†(γ^0 p_2^'γ^0+m_e)γ^0iΣ(p_2^')S(p_2^')D_μν(p_2^'-p_2)γ^μ
         ( p_2+m_e)e_α_2e_α_1^'^†(γ^0 p_1^'γ^0+m_p)γ^0γ^ν( p_1+m_p)e_α_1
         =m_e^-2m_p^-2e_2e_1e_α_2^'^†γ^0( p_2^'+m_e)iΣ(p_2^')S(p_2^')D_μν(p_2^'-p_2)γ^μ
         ( p_2+m_e)e_α_2e_α_1^'^†γ^0( p_1^'+m_p)γ^ν( p_1+m_p)e_α_1.
  
  Raising indices using the metric tensor g=γ^0 <cit.> we obtain
  
      ℳ_1^α_1^'α_2^'_α_1α_2(p_1^',p_2^',p_1,p_2)   =m_e^-2m_p^-2e_2e_1e_α_2^'^†( p_2^'+m_e)iΣ(p_2^')S(p_2^')D_μν(p_2^'-p_2)γ^μ
         ( p_2+m_e)e_α_2e_α_1^'^†( p_1^'+m_p)γ^ν( p_1+m_p)e_α_1.
  
  In <cit.> the notion of a covariant kernel was defined. We have the following.
  
  ℳ_1 is a covariant kernel.
  
  Proof
  Let κ∈ K. Then
  
      ℳ_1^α_1^'α_2^'_α_1α_2(κ p_1^',κ p_2^',κ p_1,κ p_2)   =m_e^-2m_p^-2e_2e_1e_α_2^'^†(κ p_2^'κ^-1+m_e)κ iΣ(p_2^')κ^-1κ S(p_2^')κ^-1
         D_μν(κ p_2^'-κ p_2)γ^μ(κ p_2κ^-1+m_e)e_α_2e_α_1^'^†(κ p_1^'κ^-1+m_p)γ^ν
         (κ p_1κ^-1+m_p)e_α_1
         =-m_e^-2m_p^-2e_2e_1e_α_2^'^†κ( p_2^'+m_e)iΣ(p_2^')S(p_2^')η^μνq^-2κ^-1γ_μ
         κ( p_2+m_e)κ^-1e_α_2e_α_1^†κ( p_1^'+m_p)κ^-1γ_νκ( p_1+m_p)κ^-1e_α_1
         =-m_e^-2m_p^-2e_2e_1e_α_2^'^†κ( p_2^'+m_e)iΣ(p_2^')S(p_2^')η^μνq^-2Λ^-1ρ_μγ_ρ
         ( p_2+m_e)κ^-1e_α_2e_α_1^†κ( p_1^'+m_p)Λ^-1σ_νγ_σ( p_1+m_p)κ^-1e_α_1
         =-m_e^-2m_p^-2e_2e_1e_α_2^'^†κ( p_2^'+m_e)iΣ(p_2^')S(p_2^')η^ρσ
         q^-2γ_ρ( p_2+m_e)κ^-1e_α_2e_α_1^†κ( p_1^'+m_p)γ_σ( p_1+m_p)κ^-1e_α_1
         =-m_e^-2m_p^-2e_2e_1
         [κ( p_2^'+m_e)iΣ(p_2^')S(p_2^')η^ρσq^-2γ_ρ( p_2+m_e)κ^-1]^α_2^'_α_2
         [κ( p_1^'+m_p)γ_σ( p_1+m_p)κ^-1]^α_1^'_α_1
         =-m_e^-2m_p^-2e_2e_1
         κ^α_2^'_β_2^'[( p_2^'+m_e)iΣ(p_2^')S(p_2^')η^ρσq^-2γ_ρ( p_2+m_e)]^β_2^'_β_2κ^-1β_2_α_2
         κ^α_1^'_β_1^'[( p_1^'+m_p)γ_σ( p_1+m_p)]^β_1^'_β_1κ^-1β_1_α_1
         =κ^α_2^'_β_2^'κ^-1β_2_α_2κ^α_1^'_β_1^'κ^-1β_1_α_1ℳ_1^β_1^'β_2^'_β_1β_2(p_1^',p_2^',p_1,p_2),
  
  where Λ=Λ(κ) is the Lorentz transformation associated with κ and q=p_2^'-p_2 is the momentum transfer for the scattered electron. Here we have used the intertwining property Σ_1(κ p)=κΣ_1(p)κ^-1,∀κ∈ K,p∈ R^4 where Σ_1 denotes the map p↦ p <cit.>.
  
  
  Our problem is considerably simplified if we make a non-relativistic approximation for the behaviour of the nucleus of the H atom since the proton is comparatively heavy and does not move much. In this approximation Dirac spinors u_p(p,α) satify
  
      u_p(p_1^',α_1^')γ^μu_p(p_1,α_1)=δ_α_1^'α_1η^μ0,
  
  see <cit.>. This implies that
  
      m_p^-2( p_1^'+m_p)γ^μ( p_1+m_p)=η^μ0γ^0.
  
  Therefore the Feynman amplitude ℳ_1 for the electron self-energy contribution to electron-proton scattering (in the NR approximation for the proton) is
  
      ℳ_1^α^'_α(p^',p)     = e^2m_e^-2e_α^'^†( p^'+m_e)iΣ(p^')S(p^')q^-2γ^0( p+m_e)e_α,
  
  where p is the incoming electron momentum, p^' is the outgoing electron momentum, q=p^'-p is the momentum transfer and we have suppressed the proton polarization indices which play no further part in the calculation. Thus the Feynman amplitude matrix valued function is given by
  
      ℳ_1=ℳ_1(p^',p)=e^2m_e^-2( p^'+m_e)iΣ(p^')S(p^')q^-2γ^0( p+m_e).
  
  ℳ_1 is not covariant with respect to all of the group K which is not surprising since we are using an NR approximation. However we have the following .
  
  ℳ_1 is covariant with respect to the rotation subgroup of K, that is, the group
  
      ={(
      [ a 0; 0 a ]):a∈ U(2)}⊂ K.
  
  
  Proof
  
  Let R∈ Rotations. Then R^-1=R^†. Thus
  
      ℳ_1(Rp^',Rp)   =e^2m_e^-2(R p^'R^-1+m_e)RiΣ(p^')R^-1RS(p^')R^-1(Rq)^-2γ^0(R pR^-1+m_e)
         =Re^2m_e^-2( p^'+m_e)iΣ(p^')S(p^')q^-2R^-1γ^0R( p+m_e)R^-1
         =Re^2m_e^-2( p^'+m_e)iΣ(p^')S(p^')q^-2γ^0( p+m_e)R^-1
         =Rℳ_1(p^',p)R^-1,
  
   since
  
      R^-1γ^0R=R^†γ^0R=γ^0.
  
  (R∈ K⊂ U(2,2) and the metric for U(2,2) is g=γ^0.)
  We have used the intertwining property of the Feynman slash and the fact that the Feynman fermion propagator S is K invariant <cit.>. 
  
  
  
  Now carry out a translation in momentum space to the center of mass frame with origin c=1/2(p+p^'). Then 
  
      p   → p-c=1/2(p-p^')=-1/2q,
      
      p^'   → p^'-c=1/2(p^'-p)=1/2q.
  
  In this frame
  
      ℳ_1(q)=ℳ_1(p^',p)=q^-2𝒩(1/2q),
  
  where
  
      𝒩(q)=e^2m_e^-2( q+m_e)iΣ(q)S(q)γ^0(- q+m_e).
  
  𝒩 is rotationally covariant in the sense that
  
      𝒩(Rq)=R𝒩(q)R^-1,∀ R∈.
  
  
  Now K acts on C^∞( R^4, C^4) according to <cit.>
  
      (κψ)(x)=κψ(κ^-1x)=κψ(Λ(κ)^-1x),
  
  where Λ(κ) is the Lorentz transformation corresponding to κ.
  lemmaLemma
   
  Let A∈ C^∞( R^4, C^4) transform as a 4-vector under Lorentz transformations. Then
  
      κ( Aψ)= A(κψ),∀κ∈ K,ψ∈ C^∞( R^4, C^4).
  
  
  ProofLet x∈ R^4. Then
  
      (κ( Aψ))(x)   =κ( Aψ)(Λ^-1x)
         =κ A(Λ^-1x)ψ(Λ^-1x)
         =κγ^μA_μ(Λ^-1x)ψ(Λ^-1x)
         =κγ_μA^μ(Λ^-1x)ψ(Λ^-1x)
         =κγ_μΛ^-1μ_νA^ν(x)ψ(Λ^-1x)
         =κΣ_1(Λ^-1A(x))ψ(Λ^-1x)
         =κκ^-1 A(x)κψ(Λ^-1x)
         = A(x)(κψ)(x)
         =( A(κψ))(x).
  
  Similarly
   
  
      κ(∂ψ)=∂(κψ),∀κ∈ K,ψ∈ C^∞( R^4, C^4).
  
  
   
  Let ψ∈ C^∞( R^4, C^4) be a unique solution to the Dirac equation with respect to a set ℬ of boundary conditions. Then
  
      ψ(κ x)=κψ(x),∀κ∈ K,x∈ R^4.
  
  
  Proof
  
  The Dirac equation is
  
      (iγ^μD_μ-m)ψ=0,
  
  where
  
      D_μ=∂_μ+ieA_μ.
  
  Let ψ∈ C^∞( R^4, C^4) be the unique solution with respect to ℬ. Then
  
      (i D-m)ψ=0.
  
  Therefore by Lemmas <ref> and <ref> 
  
      0=κ(i D-m)ψ=iκ(∂ψ)-eκ( Aψ)-κ mψ=i∂(κψ)-e A(κψ)-mκψ=(i D-m)(κψ).
  
  But ψ is uniquely determined by Eq. <ref> and ℬ. Therefore κψ=ψ and hence
  
      κψ(Λ(κ)^-1x)=ψ(x),∀ x∈ R^4,
  
  from which it follows that
  
      ψ(Λ(κ)x)=κψ(x),∀κ∈ K,x∈ R^4.
  
  
  (We have assumed that the transformed boundary conditions ℬ agree with ℬ.)
  
  
  Now suppose that we have a static rotationally invariant Dirac eigenfunction ψ for the 2s state of the H atom.
  In general suppose that we have a Feynman amplitude ℳ e.g. the tree level amplitude ℳ_0 or the amplitude ℳ_0+ℳ_1 where ℳ_1 is the electron self-energy amplitude. We propose that the energy associated with the bound state defined by ψ is given by the integral
  
      E=ω∫ψ^†( x)ℳ(0, q)ψ( x)e^i q. x d x d q,
  
  where we will determine ω by examination of the tree level diagram in the NR approximation. For this diagram 
  
      ℳ_0(q)=ie^2| q|^-2.
  
  However, as is well known, Feynman amplitudes are only defined up to multiplication by an element of U(1) since physical predictions are made on the basis of the modulus squared of the Feynman amplitude. We choose to take
  
      ℳ_0(q)=e^2| q|^-2,
  
  Then
  
      E_0   =ω∫ψ^†( x)ℳ_0(0, q)ψ( x)e^i q. x d q d x
         =ω e^2∫|ψ( x)|^2(∫(| q|^-2e^i q. x d q) d x.
  
  But as is well known
  
      ∫| q|^-2e^i q. x d q=2π^2/| x|.
  
  Therefore
  
      E_0=8π^3ω e^2∫_r=0^∞|ψ(r)|^2r dr.
  
  However we know that
  
      E_0=4π∫_r=0^∞|ψ(r)|^2V_0(r)r^2 dr=4π∫_r=0^∞|ψ(r)|^2(-e^2/4π r)r^2 dr=-e^2∫_r=0^∞|ψ(r)|^2r dr.
  
  Therefore 8π^3ω=-1 and thus 
  
      ω=-(2π)^-3.
  
  Now the perturbation of the H atom energy level due to the electron self energy is given by
  
      Δ E=ω∫ F( q) d q,
  
  where
  
      F( q)=∫ψ^†( x)ℳ_1(0, q)ψ( x)e^i q. x d x.
  
   
  F is rotationally invariant.
  
  Proof
  
      F(A q)   =∫ψ^†( x)ℳ_1(0,A q)ψ( x)e^iA q. x d x
         =∫ψ^†( x)Rℳ_1(0, q)R^†ψ( x)e^i q.A^† x d x
         =∫ψ^†(A^† x)ℳ_1(0, q)ψ(A^† x)e^i q.A^† x d x
         =∫ψ^†( x)ℳ_1(0, q)ψ( x)e^i q. x d x
         =F( q),
  
  for all A∈ O(3) where R∈⊂ K is such that
  
      Λ(R) = (
      [ 1 0; 0 A ]),
  
  and we have used the rotational covariance of ℳ_1 and Theorem <ref>.
  
  
  Therefore we have
  
      Δ E   =ω(4π)∫_s=0^∞F(0,0,s)s^2 ds
         =4πω∫_s=0^∞∫_ R^3ψ^†( x)ℳ_1(0,0,0,s)ψ( x)e^i(0,0,s). xs^2 d x ds
         =-4πω∫_s=0^∞∫_ R^3ψ^†( x)N(s)ψ( x)e^i(0,0,s). x d x ds
         =-(4π)(2π)ω∫_s=0^∞∫_r=0^∞∫_θ=0^πψ^†(r)N(s)ψ(r)e^isrcos(θ)r^2sin(θ) dθ dr ds
         =(4π)(2π)^-2∫_s=0^∞∫_r=0^∞ψ^†(r)N(s)ψ(r)1/irs(e^isr-e^-isr)r^2 dr ds,
  
  where
  
      N(s)=𝒩((0,0,0,s/2)).
  
  Thus writing
  
      Δ V(r)   =(2π)^-2∫_s=0^∞ N(s)1/isr(e^isr-e^-isr) ds
         =(2π)^-21/ir∫_s=0^∞N(s)/s(e^isr-e^-isr) ds,
  
  we have
  
      Δ E=4π∫_r=0^∞ψ^†(r)Δ V(r)ψ(r)r^2dr.
  
  (It can be shown using Fubini's theorem that the order of the integrations can be interchanged.)
  
  Note that Δ V:(0,∞)→ C^4×4 is a complex matrix valued potential function. We would like to analytically continue Δ V to the upper imaginary axis of the complex plane, since, for this bound state problem, we are considering spacelike points in Minkowski space for which x^2<0 corresponding to pure imaginary r 
  
  Applying the transform 𝒯 defined in Section <ref> with 
  
      F(r)=∫_s=0^∞f(s)(e^-sr-e^-isr) ds,
  
  and
  
      f(s)=N(s)/s,
  
  we obtain
  
      Δ V(r)   =(2π)^-21/irF(r)
         →(2π)^-21/ir(𝒯(F))(r)
         =(2π)^-21/iri∫_s=0^∞N(s)/se^-sr ds
         =(2π)^-21/r∫_s=m^∞N(s)/se^-sr ds.
  
  We call this the SE potential function and it is analogous to the Uehling potential function. The code for a C++ program to compute the SE contribution to the Lamb shift using the SE potential function is given in the Appendix. The output of the program is shown in Figure <ref>. The output converges to 1077 MHz whereas the accepted value for the SE contribution to the Lamb shift is 1085 MHz <cit.>. Thus the computed value differs by 0.7% from the accepted value. However this is not surprising since the program carries out more than 250,000,000 iterations which would be associated with considerable computational error. Accuracy could be increased by using the reduced mass of the electron-proton system rather than the electron mass and also by using the relativistic Dirac H atom 2s eigenfunction rather than the non-relativistic Schrödinger eigenfunction.
  
  
  A great deal of research was done many years ago e.g. <cit.>  on optimizing the numerical computation of the Uehling contribution to the Lamb shift on the basis of the Uehling potential function. One may envisage carrying out similar optimization for the numerical calculation of the electron self-energy contribution to the Lamb shift on the basis of the SE potential function.
  
  Moreover the existence of the simple form of the SE potential obtained without the need for renormalization may simplify many computations in QFT involving multiple loops which are often hampered and complicated by nested divergences e.g. <cit.>. 
  
  
  
  § ACKNOWLEDGEMENTS
  
  
  The author is grateful to Christopher Chantler for very helpful discussions and to Vladimir Yerokhin and Randolph Pohl for very helpful comments.
  
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  § APPENDIX: C++ CODE TO COMPUTE THE ELECTRON SELF-ENERGY CONTRIBUTION TO THE LAMB SHIFT
  
  
  
  
  