Computation of the leading order contributions to the Lamb shift for the H atom using spectral regularization
CComputation of the leading order contributions to the Lamb shift for the H atom using spectral regularization
John MashfordSchool of Mathematics and StatisticsUniversity of Melbourne, Victoria 3010, AustraliaE-mail: [email protected] 24, 2020
Abstract
The Uehling contribution to the Lamb shift can be computed exactly interms of the Uehling potential function. However derivations of this functionare complex involving avoiding divergences using intricate techniques from earlyquantum field theory (QFT) or else more modern approaches using charge andmass renormalization. In the present paper we derive the Uehling potentialfunction in a fairly conceptually straightforward way not involving renormal-ization in which the vacuum polarization tensor is viewed as a Lorentz invariant2-tensor valued measure on Minkowski space. Furthermore we compute a com-plex matrix valued potential function for the electron self-energy contributionto the Lamb shift. The resulting potential function is derived in a conceptu-ally simple way not involving renormalization and can be used for higher ordercomputations in QFT involving multiple loops.
Contents a r X i v : . [ phy s i c s . g e n - ph ] S e p Lorentz invariant 2-tensor valued measures 4 K invariant C × matrix valued measures 20 K invariant C × valued measure . . . . . . . . . 235.3 The spectral calculus for K invariant C × valued measures . . . . . 24 The Lamb shift is a phenomenon which is closely tied up with the instigation anddevelopment of quantum field theory (QFT). At the time of its discovery in 1947 byLamb and Retherford [1, 2] it created considerable interest in the physics communityand provided impetus for the development of the renormalization approach for dealingwith the divergences present when considering radiative corrections.Schwinger, Weisskopf and Oppenheimer suggested that the energy level shift mightbe the result of the interaction of the electron with the radiation field. The shift cameout to be infinite in all theories at the time and was therefore ignored [3].Bethe [3] provided the first proposal for a solution to the problem. His calculationdiverged logarithmically so he introduced a cutoff K for the light quanta which couldbe emitted and absorbed by the atom. He took K to be mc (where m is the electron2ass) and obtained a result of 1040 MHz which was within quite good agreementwith the experimental value of ≈ . . et al. in [7].Research has also been carried out on hydrogenic ions and results are often pre-sented as power series in Zα . Self-energy for low Z values is often obtained byextrapolating from high Z calculations [9].The leading order contributions to the Lamb shift are the self-energy contribution(electron self-energy) and the Uehling contribution (photon self-energy or vacuumpolarization). The theoretical problem associated with the Uehling contribution tothe Lamb shift was solved in 1935 by Uehling [14] who derived the Uehling potentialfunction. However his derivation using QFT as formulated at the time was, andall subsequent derivations using more modern techniques involving charge and massrenormalization [15] have been, complex and to a certain extent obscure during theprocess of negotiating divergences. Work on the Uehling contribution has focussedon developing numerical techniques for applying the Uehling potential function inthe case of the point nucleus or more elaborate nuclear models. Theoretical workon the self-energy has taken various directions but there does not seem to have beendeveloped an analogue for the self-energy to the Uehling potential function.3e have, in previous papers [16, 17] presented an approach called spectral regu-larization for analyzing problematic objects in QFT such as the vacuum polarizationtensor Π µν or the self-energy function Σ in terms of their existence as measures onMinkowski space. In the present paper we provide some theoretical results on Lorentzinvariant 2-tensor valued measures in Sections 2 and 3 and then in Section 4 derivethe Uehling potential function without requiring renormalization. Then in Sections5 and 6 we present some theoretical results concerning invariant C × valued mea-sures 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 computethe self-energy contribution to the Lamb shift. Let B ( R ) denote the Borel algebra for R . A tensor valued measure µ : B ( R ) → C × will be said to be Lorentz invariant if µ αβ (ΛΥ) = Λ αρ Λ βσ µ ρσ (Υ) , ∀ Λ ∈ O (1 , , Υ ∈ B ( R ) . (3) Suppose that Π µν is a Lorentz invariant 2-tensor valued measure which is associatedwith a density which we also denote as Π µν . ThenΠ µν (Υ) = (cid:90) Υ Π µν ( p ) dp, ∀ Υ ∈ B ( R ) . (4)Thus, for any Λ ∈ O (1 , µν (ΛΥ) = (cid:90) ΛΥ Π µν ( p ) dp = (cid:90) Υ Π µν (Λ p ) dp, and Π µν (ΛΥ) = Λ µρ Λ ν σ Π ρσ (Υ) = Λ µρ Λ ν σ (cid:90) Υ Π ρσ ( p ) dp, for all Υ ∈ B ( R ). Therefore for each Λ ∈ O (1 , µν (Λ p ) = Λ µρ Λ ν σ Π ρσ ( p ) , (5)4or almost all p ∈ R . We will consider the (non-pathological) case where Π can andhas been adjusted so that Eq. 5 holds for all Λ ∈ O (1 ,
3) and p ∈ R .Suppose also that Π is causal and is supported on { p ∈ R : p > , p > } .(The cone { p ∈ R : p = 0 , p ≥ } is a set of Lebesgue measure zero). Then Π isdetermined by its values on { ( m, (cid:42) m > } . Define λ µν ( m ) = Π µν (( m, (cid:42) , for m > . (6)Then λ µν ( m ) = Π µν (( m, (cid:42) µν ( R ( m, (cid:42) R µρ R ν σ Π ρσ (( m, (cid:42) R µρ R ν σ λ ρσ ( m ) , for all m > , R ∈ Rotations, where Rotations ∼ = O (3) is the rotation subgroup of O (1 , m ) λ ν = R ρ R ν σ λ ρσ = R ν σ λ σ , ∀ R ∈ Rotations . It follows that λ ν = λ η ν . Similarly λ µ = λ η µ . Therefore λ = (cid:32) λ A (cid:33) , for some A ∈ C × . Thus, since λ µν = R µρ R ν σ λ ρσ , ∀ R ∈ Rotations , we must have λ ij = B ik B jl λ kl , ∀ B ∈ O (3) where i, j, k, l ∈ { , , } . B = − , shows that λ = B k B l λ kl = − λ . Taking B = , (7)shows that λ = B k B l λ kl = λ . Therefore λ = λ = 0 . By a similar argument λ = λ = λ = λ = 0 . Therefore for all m > λ µν ( m )) = diag( λ ( m ) , λ ( m ) , λ ( m ) , λ ( m )) for some λ ( m ) , λ ( m ) , λ ( m ) , λ ( m ) ∈ C . Now λ = B k B l λ kl . Taking B to be of the form of Eq. 7 results in λ = B k B l λ kl = λ . Similarly λ = λ . Thus λ ( m ) = ( λ µν ( m )) = diag( λ ( m ) , λ ( m ) , λ ( m ) , λ ( m )) , (8)for some locally integrable functions λ , λ : (0 , ∞ ) → C .6 .2 Canonical form of a causal Lorentz invariant 2-tensorvalued measure Define Π µνc (Υ) = (cid:90) ∞ m =0 (cid:90) Υ ( η µν p σ ( m ) + p µ p ν σ ( m )) Ω m ( dp ) dm, (9)for Υ ∈ B ( R ), where σ , σ : (0 , ∞ ) → C are locally integrable and for m > m is the standard Lorentz invariant measure on the hyperboloid H m = { p ∈ R : p = m , p > } defined by (cid:90) ψ ( p ) Ω m ( dp ) = (cid:90) R ψ ( ω m ( (cid:42) p ) , (cid:42) p ) d (cid:42) pω m ( (cid:42) p ) , (10)for ψ a measurable function for which the integral exists, in which ω m ( (cid:42) p ) = ( m + (cid:42) p ) . (11)ThenΠ µνc (Υ) = (cid:90) ∞ m =0 (cid:90) (cid:42) p ∈ R χ Υ ( ω m ( (cid:42) p ) , (cid:42) p )( η µν m σ ( m ) + ( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν σ ( m )) d (cid:42) pω m ( (cid:42) p ) dm, where χ Γ denotes the characteristic function of Γ defined by χ Γ ( p ) = (cid:40) p ∈ Γ0 otherwise. (12)Now make the coordinate transformation q = q ( m, (cid:42) p ) = ( ω m ( (cid:42) p ) , (cid:42) p ) , m > , (cid:42) p ∈ R . (13)The Jacobian of the transformation is J ( m, (cid:42) p ) = mω m ( (cid:42) p ) − . µνc (Υ) = (cid:90) q > ,q > χ Υ ( q )( η µν m σ ( m ) + q µ q ν σ ( m )) 1 ω m ( (cid:42) p ) 1 mω m ( (cid:42) p ) − dq = (cid:90) q > ,q > χ Υ ( q )( η µν mσ ( m ) + q µ q ν m − σ ( m )) dq, where m = m ( q ) = ( q ) . Therefore the density associated with the canonical formis Π µνc ( q ) = (cid:40) η µν ζ ( q ) σ ( ζ ( q )) + q µ q ν ζ ( q ) − σ ( ζ ( q )) , for q > , q >
00 otherwise, (14)where ζ ( q ) = ( q ) . Therefore the λ function associated with Π µνc is λ µνc ( m ) = Π µνc (( m, (cid:42) η µν mσ ( m ) + ( m, (cid:42) µ ( m, (cid:42) ν m − σ ( m )= η µν mσ ( m ) + η µ η ν mσ ( m ) . Since σ and σ are arbitrary locally integrable functions, comparing with Eq. 8 wesee that this is in the most general form of a density for a causal Lorentz invariant2-tensor valued measure. Hence the canonical form is the most general form. Let Π µν : B ( R ) → C be a causal Lorentz invariant 2-tensor valued measure whichcan be defined by a locally integrable density. Then from the previous section, Π µν has the formΠ µν (Υ) = (cid:90) ∞ m =0 (cid:90) R χ Υ ( p )( p η µν σ ( m ) + p µ p ν σ ( m )) Ω m ( dp ) dm, (15)for some locally integrable spectral functions σ , σ : (0 , ∞ ) → C . Assume that thespectral functions are continuous. Now letΥ( a, b, (cid:15) ) = (cid:91) m ∈ ( a,b ) S ( m, (cid:15) ) , (16)8here S ( m, (cid:15) ) = { p ∈ R : p = m , | (cid:42) p | < (cid:15), p > } , (17)be the hyperbolic cylinder defined in [16]. Let g µν ( a, b, (cid:15) ) = Π µν (Υ( a, b, (cid:15) )) , (18)for a, b ∈ (0 , ∞ ) , a < b, (cid:15) >
0. We have g µν ( a, b, (cid:15) ) = Π µν (Υ( a, b, (cid:15) ))= (cid:90) ∞ m =0 (cid:90) (cid:42) p ∈ R χ Υ( a,b,(cid:15) ) ( ω m ( (cid:42) p ) , (cid:42) p )[ η µν m σ ( m ) + ( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν σ ( m )] d (cid:42) pω m ( (cid:42) p ) dm = (cid:90) bm = a (cid:90) B (cid:15) ( (cid:42) ) [ η µν m σ ( m ) + ( ω m ( (cid:42) p ) , (cid:42) p )) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν σ ( m )] d (cid:42) pω m ( (cid:42) p ) dm ≈ (cid:90) ba [ η µν m σ ( m ) + η µ η ν m σ ( m )] 1 m dm ( 43 π(cid:15) ) . Therefore taking g µνa ( b ) = lim (cid:15) → (cid:15) − g µν ( a, b, (cid:15) ) , (19)we see that g µνa ( b ) = (cid:90) ba [ η µν mσ ( m ) + η µ η ν mσ ( m )] dm ( 43 π ) . (20)Therefore we can recover the spectral functions σ and σ as follows. g ii (cid:48) a ( b ) = − bσ ( b )( 43 π ) , ∀ i = 1 , , . Therefore σ ( b ) = − π b g ii (cid:48) a ( b ) . (21)Also g (cid:48) a ( b ) = b ( σ ( b ) + σ ( b ))( 43 π ) . (22)9herefore σ ( b ) = 34 π b g (cid:48) a ( b ) − σ ( b ) . (23)Conversely, if a causal Lorentz invariant measure Π µν is such that the functions g µνa defined by Eq. 19 exist and are continuously differentiable then Π µν has the form ofEq. 15 and the spectral functions σ and σ can be recovered using Eqns. 21-23. The vacuum polarization tensor is given byΠ µν ( k ) = − Tr( (cid:90) dp (2 π ) i ( − e ) γ µ iS ( p ) i ( − e ) γ ν iS ( p − k ))= − e (2 π ) (cid:90) Tr( γ µ p/ − m + i(cid:15) γ ν p/ − k/ − m + i(cid:15) ) dp. (the leading minus sign is associated with the fermion loop). Therefore, using the ansatz p − m + i(cid:15) → − iπ Ω m ( p ) , and arguing as in [16] the measure associated with the vacuum polarization tensor isgiven byΠ µν (Υ) = − e π (cid:90) χ Υ ( k + p )Tr( γ µ ( p/ + m ) γ ν ( k/ − m )) Ω m ( dk ) Ω m ( dp ) , (24)for Υ ∈ B ( R ). Theorem 1.
The vacuum polarization tensor valued measure is Lorentz invariant. roof Rescale by − e π . ThenΠ µν (Λ( κ )Υ) = (cid:90) χ ΛΥ ( k + p )Tr( γ µ ( p/ + m ) γ ν ( k/ − m )) Ω m ( dk ) Ω m ( dp )= (cid:90) χ Υ (Λ − k + Λ − p )Tr( γ µ ( p/ + m ) γ ν ( k/ − m )) Ω m ( dk ) Ω m ( dp )= (cid:90) χ Υ ( k + p )Tr( γ µ ( κp/κ − + m ) γ ν ( κk/κ − − m )) Ω m ( dk ) Ω m ( dp )= (cid:90) χ Υ ( k + p )Tr( κκ − γ µ κ ( p/ + m ) κ − γ ν κ ( k/ − m ) κ − ) Ω m ( dk ) Ω m ( dp )= (cid:90) χ Υ ( k + p )Tr(Λ − ρµ γ ρ ( p/ + m )Λ − σν γ σ ( k/ − m )) Ω m ( dk ) Ω m ( dp )= Λ − ρµ Λ − σν (cid:90) χ Υ ( k + p )Tr( γ ρ ( p/ + m ) γ σ ( k/ − m )) Ω m ( dk ) Ω m ( dp )= Λ − ρµ Λ − σν Π ρσ (Υ) , for all κ ∈ K where K = { (cid:32) a a †− (cid:33) : a ∈ GL (2 , C ) , | det( a ) | = 1 } ⊂ U (2 , , (25)is the group defined in [18], Λ = Λ( κ ) is the Lorentz transformation corresponding to κ and we have used the intertwining property Σ ( κp ) = κ Σ ( p ) κ − = κp/κ − of themap Σ = ( p (cid:55)→ p/ ) [18] which implies that κγ µ κ − = Λ( κ ) ρµ γ ρ , ∀ κ ∈ K .Therefore Π αβ (ΛΥ) = η αµ η βν Π µν (ΛΥ)= η αµ η βν Λ − ρµ Λ − σν Π ρσ (Υ) . Now Λ − ρµ η αµ = (Λ − η ) ρα , and Λ T η Λ = η, from which it follows that Λ − η = η Λ T . − ρµ η αµ = ( η Λ T ) ρα = η ρµ Λ Tµ α = η ρµ Λ αµ . Similarly Λ − σν η βν = η σν Λ βν . Therefore Π αβ (ΛΥ) = η ρµ Λ αµ η σν Λ βν Π ρσ (Υ)= Λ αµ Λ βν Π µν (Υ) . (cid:50) It is straightforward to show that the vacuum polarization tensor valued measure iscausal.
Theorem 2.
The vacuum polarization tensor has the form Π µν (Υ) = (cid:90) ∞ m (cid:48) =0 (cid:90) R χ Υ ( p )( p η µν − p µ p ν ) Ω m (cid:48) ( dp ) ρ ( m (cid:48) ) dm (cid:48) , for some continuous spectral function ρ : (0 , ∞ ) → C . Proof
One can readily compute using the gamma matrix trace identities thatTr( γ µ ( p/ + m ) γ ν ( k/ − m )) = 4( p µ k ν − η µν p.k + k µ p ν − m η µν ) . g µν ( a, b, (cid:15) ) = Π µν (Υ( a, b, (cid:15) ))= − e π (cid:90) χ Υ( a,b,(cid:15) ) ( k + p )Tr( γ µ ( p/ + m ) γ ν ( k/ − m )) Ω m ( dk ) Ω m ( dp )= − e π (cid:90) χ Υ( a,b,(cid:15) ) ( k + p )( p µ k ν − η µν p.k + k µ p ν − m η µν ) Ω m ( dk ) Ω m ( dp )= − e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) k ) + ω m ( (cid:42) p )) χ B (cid:15) ( (cid:42) ) ( (cid:42) k + (cid:42) p )(( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) k ) , (cid:42) k ) ν − η µν ( ω m ( (cid:42) p ) ω m ( (cid:42) k ) − (cid:42) p . (cid:42) k ) + ( ω m ( (cid:42) k ) , (cid:42) k ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν − m η µν ) d (cid:42) kω m ( (cid:42) k ) d (cid:42) pω m ( (cid:42) p )= − e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) k ) + ω m ( (cid:42) p )) χ B (cid:15) ( (cid:42) ) − (cid:42) p ( (cid:42) k )(( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) k ) , (cid:42) k ) ν − η µν ( ω m ( (cid:42) p ) ω m ( (cid:42) k ) − (cid:42) p . (cid:42) k ) + ( ω m ( (cid:42) k ) , (cid:42) k ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν − m η µν ) d (cid:42) kω m ( (cid:42) k ) d (cid:42) pω m ( (cid:42) p ) ≈ − e π (cid:90) χ ( a,b ) (2 ω m ( (cid:42) p ))(( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) p ) , − (cid:42) p ) ν − η µν ( ω m ( (cid:42) p ) + (cid:42) p )+( ω m ( (cid:42) p ) , − (cid:42) p ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν − m η µν ) 1 ω m ( (cid:42) p ) d (cid:42) p ( 43 π(cid:15) )Now χ ( a,b ) (2 ω m ( (cid:42) p )) = 1 ⇔ a < m + (cid:42) p ) < b ⇔ a < m + (cid:42) p < b ⇔ a − m < (cid:42) p < b − m ⇔ mZ ( a ) < | (cid:42) p | < mZ ( b ) , where Z ( b ) = ( b m − , b ≥ m. (26)13herefore g µνa ( b ) = lim (cid:15) → (cid:15) − g µν ( a, b, (cid:15) )= − e π (cid:90) mZ ( b ) r = mZ ( a ) (cid:90) πθ =0 (cid:90) πφ =0 (( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) p ) , − (cid:42) p ) ν − η µν ( m + 2 r )+( ω m ( (cid:42) p ) , − (cid:42) p ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) ν − m η µν ) r m + r sin( θ ) dφ dθ dr ( 43 π ) . where (cid:42) p = (cid:42) p ( r, θ, φ ) (spherical polar coordinates). Hence g µµ (cid:48) a ( b ) = − e π (cid:90) πθ =0 (cid:90) πφ =0 (( ω m ( (cid:42) p ) , (cid:42) p ) µ ( ω m ( (cid:42) p ) , − (cid:42) p ) µ − η µµ ( m + 2 r )+( ω m ( (cid:42) p ) , − (cid:42) p ) µ ( ω m ( (cid:42) p ) , (cid:42) p ) µ − m η µµ ) r m + r sin( θ ) dφ dθ ( 43 π ) mZ (cid:48) ( b ) , where r = mZ ( b ). Thus g (cid:48) a ( b ) = − e π (cid:90) πθ =0 (cid:90) πφ =0 ( m + r − ( m + 2 r ) + m + r − m ) r m + r sin( θ ) dφ dθ ( 43 π )( mZ (cid:48) ( b ))= 0 . Therefore by the spectral theorem and Eq. 22, Π µν has the form of Eq. 15 with σ and σ continuous functions and σ ( b ) = − σ ( b ) , ∀ b > . (27) (cid:50) Thus, letting ρ = σ and making the coordinate transformation defined by Eq. 13,14e have thatΠ µν (Υ) = (cid:90) ∞ m (cid:48) =0 (cid:90) R χ Υ ( ω m (cid:48) ( (cid:42) p ) , (cid:42) p )( η µν m (cid:48) − ( ω m (cid:48) ( (cid:42) p ) , (cid:42) p ) µ ( ω m (cid:48) ( (cid:42) p ) , (cid:42) p ) ν ) ρ ( m (cid:48) ) d (cid:42) pω m (cid:48) ( (cid:42) p ) dm (cid:48) = (cid:90) q > ,q > χ Υ ( q )( q η µν − q µ q ν )( q ) − ρ (( q ) ) dq. Hence the density Π µν for the measure Π µν is given byΠ µν ( q ) = ( q η µν − q µ q ν ) π ( q ) , (28)where π ( q ) = (cid:40) ( q ) − ρ (( q ) ) for q > , q >
00 otherwise. (29)(The fact that Π µν has the form of Eq. 28 is well known but has previously beenestablished through manipulating infinite quantities during renormalization (see [19],p. 478).)Contracting Eq. 28 with the Minkowski space metric tensor we obtain π ( q ) = 13 q Π( q ) for q (cid:54) = 0 . (30)where Π( q ) = η µν Π µν ( q ) . (31) It can be shown using the spectral calculus [16, 17], that the contraction of the spectralvacuum polarization tensor Π µν is given byΠ( q ) = (cid:40) s − σ ( s ) if q > , q >
00 otherwise, (32)15here s = ( q ) and σ is given by σ ( s ) = (cid:40) π e m Z ( s )(3 + 2 Z ( s )) if s ≥ m Z is given by Eq. 26.Therefore the spectral vacuum polarization function is given by π ( q ) = Π( q )3 q = 13 s − σ ( s ) = 23 π e m s − Z ( s )(3 + 2 Z ( s )) , if s ≥ m, (34)for q > , q > s = ( q ) .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 ) = 13 s − σ ( s ) = 23 π e m s − Z ( s )(3 + 2 Z ( s )) , if s ≥ m, (35)where s = ( − q ) . The Feynman amplitude associated with the tree diagram for electron proton scat-tering is given by M aα (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) ie γ ρ u e ( p , α ) iD ρσ ( q ) u p ( p (cid:48) , α (cid:48) ) ie γ σ u p ( p , α ) , where D F αβ ( q ) = − η αβ q + i(cid:15) , is the photon propagator, q = p (cid:48) − p is the momentum transfer, e = − e, e = e are the charges of the electron and proton respectively and u e ( p, α ) and u p ( p, α ) areDirac spinors for the electron and the proton for α ∈ { , } . Therefore M aα (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = i M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) q + i(cid:15) , (36)16here M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) e γ ρ u e ( p , α ) η ρσ u p ( p (cid:48) , α (cid:48) ) e γ σ u p ( p , α ) , The Feynman amplitude associated with the vacuum polarization diagram is givenby M bα (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) ie γ ρ u e ( p , α ) iD F ρµ ( q ) i Π µν ( q ) iD F νσ ( q ) u p ( p (cid:48) , α (cid:48) ) ie γ σ u p ( p , α ) , Therefore, since D F ρµ ( q )Π µν ( q ) D F νσ ( q ) = ( 1 q ) η ρµ ( q η µν − q µ q ν ) π ( q ) η νσ , = ( η ρσ q − ( 1 q ) q ρ q σ ) π ( q ) , and, by a well known conservation property u ( p (cid:48) , α (cid:48) ) q ρ γ ρ u ( p , α ) = 0 , we have that M bα (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = i M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) π ( q ) q + i(cid:15) . Now consider the NRQED (non-relativistic quantum electrodynamics) approximationin which M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = e e δ α (cid:48) α δ α (cid:48) ,α = − e δ α (cid:48) α δ α (cid:48) ,α , and q is negligible compared with | (cid:42) q | . In this case M bα (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = − ie δ α (cid:48) α δ α (cid:48) α π ((0 , (cid:42) q ))(0 , (cid:42) q ) = ie δ α (cid:48) α δ α (cid:48) α π ((0 , (cid:42) q )) (cid:42) q , and we can write M b ( (cid:42) q ) = ie π (0 , (cid:42) q ) (cid:42) q . (37)17herefore using the Born approximation the potential function associated with theUehling contribution to the Lamb shift is [16]∆ V ( (cid:42) x ) = i (2 π ) − (cid:90) M b ( (cid:42) q ) e i(cid:42) q .(cid:42) x d (cid:42) q = i (2 π ) − (cid:90) ( ie π ((0 , (cid:42) q )) (cid:42) q ) e i(cid:42) q .(cid:42) x d (cid:42) q = − (2 π ) − e (cid:90) π ((0 , (cid:42) q )) (cid:42) q e i(cid:42) q .(cid:42) x d (cid:42) q , if the integral exists. The argument of the inverse Fourier transform is orthogonallyinvariant and thus ∆ V is orthogonally invariant. Therefore we can write∆ V ( r ) = ∆ V (0 , , r )= − (2 π ) − e (cid:90) ∞ s =0 (cid:90) πθ =0 (cid:90) πφ =0 π ( s ) s e isr cos( θ ) s sin( θ ) dφ dθ ds = − (2 π ) − e (cid:90) ∞ s =0 π ( s ) irs ( e isr − e − isr ) ds = i (2 π ) − e r (cid:90) ∞ s =0 π ( s ) s ( e isr − e − isr ) ds. Hence ∆ V ( r ) = i (2 π ) − e r ( (cid:90) ∞ s =0 π ( s ) s e isr ds − (cid:90) ∞ s =0 π ( s ) s e − isr ds ) . (38)It is shown in [16] that these integrals are convergent for all r >
0. We would like toanalytically continue ∆ V to the upper imaginary axis of the complex plane since weinterested in spacelike points in Minkowski (configuration) space for which x < r . Therefore we seek a complex analytic function∆ V analytic associated with ∆ V . By a well known Paley-Weiner theorem the function r (cid:55)→ (cid:90) ∞ s =0 π ( s ) rs e irs ds, is analytic in the upper half plane while the function r (cid:55)→ (cid:90) ∞ s =0 π ( s ) rs e − irs ,
18s 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 ) = (cid:90) ∞ s =0 f ( s )( e isr − e − isr ) ds, (39)to the function T ( F ) defined by( T ( F ))( r ) = i (cid:90) ∞ s =0 f ( s ) e − isr ds. (40)where f : (0 , ∞ ) → C is such that both integrals defined by Eqns. 39 and 40 areconvergent for all r > T takes non-divergent functions into non-divergent functions.The mathematical properties and justification of the transform T may be consideredelsewhere. 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 computationof the Uehling potential function and also for the case of the electron self-energycontribution to the Lamb shift.In the case of the Ueling contribution we compute using f ( s ) = π ( s ) s = σ ( s )3 s , (41)and the ∆ V function that we derive is∆ V ( r ) = i (2 π ) − e r F ( r ) → i (2 π ) − e r ( T ( F ))( r )= i (2 π ) − e r i (cid:90) ∞ s =0 σ ( s )3 s e − sr ds = − (2 π ) − e π e m r (cid:90) ∞ s =2 m Z ( s )(3 + 2 Z ( s )) s e − sr ds = − (2 π ) − e π e m r (cid:90) ∞ x =1 ( x − (2 x + 1)(16 m ) x e − mxr (2 m ) dx = − α π r (cid:90) ∞ x =1 ( x − (2 x + 1) x − e − mxr dx, where α = e π is the fine structure constant.The result that we have obtained is precisely the Uehling vacuum polarization19otential function which has previously only been calculated through negotiatinginfinities and divergences using complex calculations involving methods such as chargeand mass renormalization [15].From this potential function the Uehling contribution to the Lamb shift can beexactly calculated [20] according to∆ E = < ψ | ∆ V | ψ > = 4 π (cid:90) ∞ r =0 | ψ ( r ) | ∆ V ( r ) r dr, (42)where ψ is the H atom 2s wave function, and theory agrees with experiment to a veryhigh order of precision [7]. K invariant C × matrix valued measures As described in [18] the group K ⊂ U (2 , ⊂ C × defined by Eq. 25 acts in naturalways on R and C . A C × matrix valued measure is a vector valued measure takingvalues in the vector space C × . A C × valued measure µ : B ( R ) → C × will besaid to be K invariant if µ ( κ Υ) = κµ (Υ) κ − , ∀ κ ∈ K, Υ ∈ B ( R ) . (43) 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) = (cid:90) Υ µ ( p ) dp, ∀ Υ ∈ B ( R ) . (44)By K invariance µ (Λ( κ )Υ) = κµ (Υ) κ − = κ (cid:90) Υ µ ( p ) dp κ − . (45)But µ (Λ( κ )Υ) = (cid:90) Λ( κ )Υ µ ( p ) dp = (cid:90) Υ µ (Λ( κ ) p ) dp, (46)20here Λ( κ ) is the element of O (1 , ↑ + corresponding to κ ∈ K . Both these equationsare true for all κ ∈ K, Υ ∈ B ( R ). Therefore for all κ ∈ Kµ (Λ( κ ) p ) = κµ ( p ) κ − , (47)(for almost all p ∈ R ). We will consider the case where µ can be, and has been,adjusted so that Eq. 47 holds for all κ ∈ K, p ∈ R .Conversely, given a locally integrable matrix valued function which satisfies Eq. 47then the object µ : B ( R ) → C × defined by Eq. 44 is a K invariant C × valuedmeasure.Suppose that µ is such a function and assume that µ is causal and is supportedon { p ∈ R : p > , p > } . Define M : (0 , ∞ ) → C × by M ( m ) = µ (( m, (cid:42) . (48)Now µ (Λ( κ )( m, (cid:42) κM ( m ) κ − . (49)Thus µ is determined if M is given. We will call the function M the spectrum of µ .Since ( m, (cid:42) M ( m ) = (cid:32) M M M M (cid:33) = (cid:32) a a (cid:33) (cid:32) M M M M (cid:33) (cid:32) a a (cid:33) − = (cid:32) aM a − aM a − aM a − aM a − (cid:33) , for all a ∈ SU (2). Therefore each block M i , i = 1 , , , M commutes with eachelement of SU (2). Suppose that M i = (cid:32) b i b i b i b i (cid:33) . (50)Now a = (cid:32) − (cid:33) ∈ SU (2) . (51)21herefore (cid:32) − (cid:33) (cid:32) b i b i b i b i (cid:33) = (cid:32) b i b i b i b i (cid:33) (cid:32) − (cid:33) , (52)from which it follows that b i = − b i and b i = b i . Also a = (cid:32) ii (cid:33) ∈ SU (2) . (53)Therefore (cid:32) ii (cid:33) (cid:32) b i b i b i b i (cid:33) = (cid:32) b i b i b i b i (cid:33) (cid:32) ii (cid:33) , (54)from which it follows that b i = b i and b i = b i . Therefore M i = λ i = λ i I , (55)for some λ i ∈ C .Conversely, let λ , λ , λ , λ : (0 , ∞ ) → C be locally integrable functions anddefine µ : { p ∈ R : p > , p > } → C × by µ ( κ ( m, (cid:42) κM ( m ) κ − , ∀ κ ∈ K, m > , (56)where M = (cid:32) λ λ λ λ (cid:33) ∈ C × . (57)It is straightforward to show that µ is well defined. Let p (cid:55)→ κ ( p ) be any function suchthat κ ( p ) p = (( p ) , (cid:42) , ∀ p ∈ R with p > , p >
0. Then for all κ ∈ K, p ∈ R forwhich p > , p > µ ( κp ) = µ ( κκ ( p ) − κ ( p ) p )= µ ( κκ ( p ) − ( m, (cid:42) κκ ( p ) − M ( m )( κκ ( p ) − ) − = κµ ( κ ( p ) − ( m, (cid:42) κ − = κµ ( κ ( p ) − κ ( p ) p ) κ − = κµ ( p ) κ − . m = ( p ) . Therefore µ is K invariant.Hence the function defined by Eq. 56 is the most general form of the density for a K invariant C × valued measure which can be defined by a locally integrable densityon Minkowski space. K invariant C × valued measure If σ : (0 , ∞ ) → C is a locally integrable function define µ σ : B ( R ) → C × by µ σ (Υ) = (cid:90) ∞ m =0 (cid:90) Υ ( p/ + m ) Ω m ( dp ) σ ( m ) dm. (58)Then µ σ (Λ( κ )(Υ)) = (cid:90) ∞ m =0 (cid:90) Λ( κ )(Υ) ( p/ + m ) Ω m ( dp ) σ ( m ) dm = (cid:90) ∞ m =0 (cid:90) Υ (Σ (Λ( κ ) p ) + m ) Ω m ( dp ) σ ( m ) dm = (cid:90) ∞ m =0 (cid:90) Υ ( κp/κ − + m ) Ω m ( dp ) σ ( m ) dm = κ (cid:90) ∞ m =0 (cid:90) Υ ( p/ + m ) Ω m ( dp ) σ ( m ) dm κ − = κµ (Υ) κ − , for all Υ ∈ B ( R ) where Σ denotes the map p (cid:55)→ p/ and we have used the intertwiningproperty Σ ( κp ) = κ Σ ( p ) κ − of Σ [18]. Therefore µ σ is K invariant. Now, makingthe coordinate transformation defined by Eq. 13, we have µ σ (Υ) = (cid:90) ∞ m =0 (cid:90) Υ ( p/ + m ) Ω m ( dp ) σ ( m ) dm = (cid:90) ∞ m =0 (cid:90) R χ Υ (( ω m ( (cid:42) p ) , (cid:42) p ))(Σ (( ω m ( (cid:42) p ) , (cid:42) p )) + m ) d (cid:42) pω m ( (cid:42) p ) σ ( m ) dm = (cid:90) q > ,q > χ Υ ( q )( q/ + ζ ( q )) σ ( ζ ( q )) ζ ( q ) dq, where ζ ( q ) = ( q ) . Therefore the density associated with µ σ is given by µ σ ( q ) = (cid:40) ( q/ + ζ ( q )) ζ ( q ) − σ ( ζ ( q )) if q > , q >
00 otherwise. (59)23herefore the spectral function M = M σ : (0 , ∞ ) → C × associated with µ σ is M σ ( m ) = µ σ (( m, (cid:42) mγ + m ) m − σ ( m )= ( γ + 1) σ ( m )= (cid:32) σ ( m ) 00 0 (cid:33) , where we use the Dirac representation for the gamma matrices.More generally if σ , σ : (0 , ∞ ) → C are locally integrable functions then thematrix valued measure µ σ ,σ : B ( R ) → C × defined by µ σ ,σ (Υ) = (cid:90) ∞ m =0 (cid:90) Υ ( p/ + m ) Ω m ( dp ) σ ( m ) dm + (cid:90) ∞ m =0 (cid:90) Υ ( p/ − m ) Ω m ( dp ) σ ( m ) dm, (60)is K invariant with spectral function M given by M σ ,σ ( m ) = (cid:32) σ ( m ) 00 − σ ( m ) (cid:33) . (61) K invariant C × valued mea-sures Suppose that µ is a K invariant C × valued measure which can be defined by a locallyintegrable density of the form of µ σ σ for some locally integrable spectral functions σ , σ . Suppose that σ and σ are continuous on (0 , ∞ ). Let for a, b, (cid:15) ∈ (0 , ∞ ) , a < b , g ( a, b, (cid:15) ) be defined by g ( a, b, (cid:15) ) = µ (Υ( a, b, (cid:15) )) , (62)24here Υ( a, b, (cid:15) ) is the hypercylinder defined by Eq. 16. Then g ( a, b, (cid:15) ) = (cid:90) ∞ m =0 (cid:90) Υ( a,b,(cid:15) ) ( p/ + m ) Ω m ( dp ) σ ( m ) dm + (cid:90) ∞ m =0 (cid:90) Υ( a,b,(cid:15) ) ( p/ − m ) Ω m ( dp ) σ ( m ) dm = (cid:90) bm = a (cid:90) B (cid:15) ( (cid:42) ) (Σ (( ω m ( (cid:42) p ) , (cid:42) p )) + m ) d (cid:42) pω m ( (cid:42) p σ ( m ) dm + (cid:90) bm = a (cid:90) B (cid:15) ( (cid:42) ) (Σ (( ω m ( (cid:42) p ) , (cid:42) p )) − m ) d (cid:42) pω m ( (cid:42) p ) σ ( m ) dm ≈ π(cid:15) ( (cid:90) bm = a ( mγ + m ) 1 m σ ( m ) dm + (cid:90) bm = a ( mγ − m ) 1 m σ ( m ) dm )= 43 π(cid:15) (cid:90) ba (cid:32) σ ( m ) 00 − σ ( m ) (cid:33) dm, (63)where Σ denotes the map p (cid:55)→ p/ . Therefore if we define g a : (0 , ∞ ) → C × by g a ( b ) = lim (cid:15) → (cid:15) − g ( a, b, (cid:15) ) , (64)then σ ( b ) = 34 π g (cid:48) a ( b ) , σ ( b ) = − π g (cid:48) a ( b ) , for b > . (65)i.e. σ ( b ) = M ( b ) = 34 π g (cid:48) a ( b ) . (66)Conversely if µ : B ( R ) → C × is a causal K invariant measure and if the function g a defined by Eq. 64 exists and is is continuously differentiable with g αβa ( b ) = 0 for α (cid:54) = β then µ has the form of Eq. 60 and the spectral functions can be recovered usingEqns. 65. 25 The self-energy of the electron
The Feynman integral associated with the self-energy of the electron is i Σ( p ) = (cid:90) dk (2 π ) iD µν ( k ) i ( − e ) γ µ iS F ( p − k ) i ( − e ) γ ν , (67)where D µν ( k ) = − k + i(cid:15) , (68)is the photon propagator and S F ( p ) = 1 p/ − m + i(cid:15) , (69)is the fermion propagator. This can be written as i Σ( p ) = − e (2 π ) (cid:90) k + i(cid:15) γ µ p/ − k/ + m ( p − k ) − m + i(cid:15) γ µ dk. (70)We make the following formal computation i Σ(Υ) = (cid:90) Υ i Σ( p ) dp = (cid:90) χ Υ ( p )( − e (2 π ) ) 1 k + i(cid:15) γ µ p/ − k/ + m ( p − k ) − m + i(cid:15) γ µ dk dp = (cid:90) χ Υ ( p )( − e (2 π ) ) 1 k + i(cid:15) γ µ p/ − k/ + m ( p − k ) − m + i(cid:15) γ µ dp dk = − e (2 π ) (cid:90) χ Υ ( p + k ) 1 k + i(cid:15) γ µ p/ + mp − m + i(cid:15) γ µ dp dk = e π (cid:90) χ Υ ( p + k ) γ µ ( p/ + m ) γ µ Ω ± m ( dp ) Ω ± ( dk ) , where we have used the ansatz [21]1 p − m + i(cid:15) → − iπ Ω ± m ( p ) , ∀ m ≥ . (71)We take the case i Σ(Υ) = e π (cid:90) χ Υ ( p + k ) γ µ ( p/ + m ) γ µ Ω m ( dp ) Ω +0 ( dk ) , (72)26n which case Σ has existence as a well defined mathematical object (a tempered C × valued measure).Now γ µ p/γ µ = − p/ and γ µ mγ µ = 4 m . Therefore i Σ(Υ) = e π (cid:90) χ Υ ( p + k )(4 m − p/ ) Ω m ( dp ) Ω +0 ( dk ) . (73) Theorem 3. i Σ is K invariant. Proof
Let κ ∈ K and Υ ∈ B ( R ). Then i Σ( κ (Υ)) = e π (cid:90) χ κ (Υ) ( p + k )(4 m − p/ ) Ω m ( dp ) Ω +0 ( dk )= e π (cid:90) χ Υ ( κ − ( p + k ))(4 m − p/ ) Ω m ( dp ) Ω +0 ( dk )= e π (cid:90) χ Υ ( p + k )(4 m − κp/κ − ) Ω m ( dp ) Ω +0 ( dk )= e π κ (cid:90) χ Υ ( p + k )(4 m − p/ ) Ω m ( dp ) Ω +0 ( dk ) κ − = κi Σ(Υ) κ − (cid:50) It can be shown that i Σ is causal. We will now use the spectral calculus to compute27he spectrum of i Σ. g ( a, b, (cid:15) ) = i Σ(Υ( a, b, (cid:15) ))= e π (cid:90) χ Υ( a,b,(cid:15) ) ( p + k )(4 m − p/ ) Ω m ( dp ) Ω +0 ( dk )= e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) p ) + ω ( (cid:42) k )) χ B (cid:15) ( (cid:42) ) ( (cid:42) p + (cid:42) k )(4 m − p/ ) ω m ( (cid:42) p ) − ω ( (cid:42) k ) − d (cid:42) p d (cid:42) k = e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) p ) + ω ( (cid:42) k )) χ B (cid:15) ( (cid:42) ) − (cid:42) p ( (cid:42) k )(4 m − p/ ) ω m ( (cid:42) p ) − ω ( (cid:42) k ) − d (cid:42) p d (cid:42) k = e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) p ) + ω ( (cid:42) k )) χ B (cid:15) ( (cid:42) ) − (cid:42) p ( (cid:42) k )(4 m − p/ ) ω m ( (cid:42) p ) − ω ( (cid:42) k ) − d (cid:42) k d (cid:42) p ≈ π(cid:15) e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) p ) + ω ( (cid:42) p ))(4 m − p/ ) ω m ( (cid:42) p ) − ω ( (cid:42) p ) − d (cid:42) p , with p = ( ω m ( (cid:42) p ) , (cid:42) p ).Now ω m ( (cid:42) p ) + ω ( (cid:42) p ) = c ⇒ c ≥ m , and for all c ≥ mω m ( (cid:42) p ) + ω ( (cid:42) p ) = c ⇔ ( r + m ) + r = c ⇔ r + m = ( c − r ) = c + r − cr ⇔ cr = c − m ⇔ r = (2 c ) − ( c − m ) , where r = | (cid:42) p | . Therefore g a ( b ) = lim (cid:15) → (cid:15) − g ( a, b, (cid:15) )= 43 π e π (cid:90) χ ( a,b ) ( ω m ( (cid:42) p ) + ω ( (cid:42) p ))(4 m − p/ ) ω m ( (cid:42) p ) − ω ( (cid:42) p ) − d (cid:42) p = 43 π e π (cid:90) Z ( b ) r = Z ( a ) (cid:90) πθ =0 (cid:90) πφ =0 (4 m − p/ ) ω m ( r ) − ω ( r ) − r sin( θ ) dφ dθ dr, (74)28here p = p ( r, θ, φ ) = ( ω m ( r ) , r sin θ cos( φ ) , r sin( θ ) sin( φ ) , r cos( θ )) and Z : [ m, ∞ ) → [0 , ∞ ) is defined by Z ( b ) = (2 b ) − ( b − m ) . (75)Now Z (cid:48) ( b ) = (2 b ) − ( b + m ) , (76)and (cid:90) πθ =0 (cid:90) πφ =0 p/ sin( θ ) dφ dθ = 4 πω m ( r ) γ . Hence g a ( b ) = 43 π e π (cid:90) Z ( b ) r = Z ( a ) (4 π )(4 m − ω m ( r ) γ ) ω m ( r ) − r dr. Therefore applying the spectral calculus and using Leibniz’ integral rule we obtain σ ( b ) = 34 π g (cid:48) a ( b )= e π (4 m − ω m ( Z ( b )) γ ) ω m ( Z ( b )) − Z ( b ) Z (cid:48) ( b ) for b ≥ m. If p ∈ R is timelike and κ ∈ K is such that κp = Λ( κ ) p = (( p ) , (cid:42) i Σ( p ) = i Σ( κ − κp ) = κ − i Σ((( p ) , (cid:42) κ = κ − σ (( p ) ) κ. (77)Also, in the spacelike domain, if p ∈ R is spacelike and κ ∈ K is such that κp =(0 , , , ( − p ) ) then i Σ( p ) = i Σ( κ − κp ) = κ − i Σ((0 , , , ( − p ) )) κ = κ − σ (( − p ) ) κ. (78)In particular i Σ((0 , , , ζ )) = σ ( ζ ) , ∀ ζ > . (79)29 The electron self-energy contribution to the Lambshift
The Feynman amplitude for the electron self-energy contribution to the Lamb shiftis given by M = M + M , (80)where M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) iD µν ( p (cid:48) − p ) ie γ µ u e ( p , α ) u p ( p (cid:48) , α (cid:48) ) ie γ ν u p ( p , α ) , M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) i Σ( p (cid:48) ) iS ( p (cid:48) ) iD µν ( p (cid:48) − p ) ie γ µ u e ( p , α ) u p ( p (cid:48) , α (cid:48) ) ie γ ν u p ( p , α ) , in which e = e and e = − e are the charges of the proton and the electron respec-tively, u e ( p, α ) = m − e ( p/ + m e ) e α , (81) u p ( p, α ) = m − p ( p/ + m p ) e α , (82)are Dirac spinors for the electron and the proton respectively with m e = the mass ofthe electron, m p = the mass of the proton, e α is the α th standard basis element for C , D µν ( q ) = − η µν q + i(cid:15) , (83)is the photon propagator, S ( p ) = 1 p/ − m e + i(cid:15) = p/ + m e p − m e + i(cid:15) , (84)is the electron propagator and i Σ( p )“ = ” − e (2 π ) (cid:90) k + i(cid:15) γ µ p/ − k/ + m e ( p − k ) − m e + i(cid:15) γ µ dk, (85)is the the function associated with the Feynman integral for the self-energy of theelectron. M corresponds to the tree level Feynman amplitude for electron-proton scatter-30ng and M is the perturbation due to the electron self-energy. Now M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = u e ( p (cid:48) , α (cid:48) ) i Σ( p (cid:48) ) S ( p (cid:48) ) D µν ( p (cid:48) − p ) e γ µ u e ( p , α ) u p ( p (cid:48) , α (cid:48) ) e γ ν u p ( p , α )= m − e m − p e e ( p/ (cid:48) + m e ) e α (cid:48) i Σ( p (cid:48) ) S ( p (cid:48) ) D µν ( p (cid:48) − p ) γ µ ( p/ + m e ) e α ( p/ (cid:48) + m p ) e α (cid:48) γ ν ( p/ + m p ) e α = m − e m − p e e e † α (cid:48) ( γ p/ (cid:48) γ + m e ) γ i Σ( p (cid:48) ) S ( p (cid:48) ) D µν ( p (cid:48) − p ) γ µ ( p/ + m e ) e α e † α (cid:48) ( γ p/ (cid:48) γ + m p ) γ γ ν ( p/ + m p ) e α = m − e m − p e e e † α (cid:48) γ ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) D µν ( p (cid:48) − p ) γ µ ( p/ + m e ) e α e † α (cid:48) γ ( p/ (cid:48) + m p ) γ ν ( p/ + m p ) e α . Raising indices using the metric tensor g = γ [21] we obtain M α (cid:48) α (cid:48) α α ( p (cid:48) , p (cid:48) , p , p ) = m − e m − p e e e † α (cid:48) ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) D µν ( p (cid:48) − p ) γ µ ( p/ + m e ) e α e † α (cid:48) ( p/ (cid:48) + m p ) γ ν ( p/ + m p ) e α . In [21] the notion of a covariant kernel was defined. We have the following.
Theorem 4. M is a covariant kernel. roof Let κ ∈ K . Then M α (cid:48) α (cid:48) α α ( κp (cid:48) , κp (cid:48) , κp , κp ) = m − e m − p e e e † α (cid:48) ( κp/ (cid:48) κ − + m e ) κi Σ( p (cid:48) ) κ − κS ( p (cid:48) ) κ − D µν ( κp (cid:48) − κp ) γ µ ( κp/ κ − + m e ) e α e † α (cid:48) ( κp/ (cid:48) κ − + m p ) γ ν ( κp/ κ − + m p ) e α = − m − e m − p e e e † α (cid:48) κ ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) η µν q − κ − γ µ κ ( p/ + m e ) κ − e α e † α κ ( p/ (cid:48) + m p ) κ − γ ν κ ( p/ + m p ) κ − e α = − m − e m − p e e e † α (cid:48) κ ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) η µν q − Λ − ρµ γ ρ ( p/ + m e ) κ − e α e † α κ ( p/ (cid:48) + m p )Λ − σν γ σ ( p/ + m p ) κ − e α = − m − e m − p e e e † α (cid:48) κ ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) η ρσ q − γ ρ ( p/ + m e ) κ − e α e † α κ ( p/ (cid:48) + m p ) γ σ ( p/ + m p ) κ − e α = − m − e m − p e e [ κ ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) η ρσ q − γ ρ ( p/ + m e ) κ − ] α (cid:48) α [ κ ( p/ (cid:48) + m p ) γ σ ( p/ + m p ) κ − ] α (cid:48) α = − m − e m − p e e κ α (cid:48) β (cid:48) [( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) η ρσ q − γ ρ ( p/ + m e )] β (cid:48) β κ − β α κ α (cid:48) β (cid:48) [( p/ (cid:48) + m p ) γ σ ( p/ + m p )] β (cid:48) β κ − β α = κ α (cid:48) β (cid:48) κ − β α κ α (cid:48) β (cid:48) κ − β α M β (cid:48) β (cid:48) β β ( p (cid:48) , p (cid:48) , p , p ) , where Λ = Λ( κ ) is the Lorentz transformation associated with κ and q = p (cid:48) − p is themomentum transfer for the scattered electron. Here we have used the intertwiningproperty Σ ( κp ) = κ Σ ( p ) κ − , ∀ κ ∈ K, p ∈ R where Σ denotes the map p (cid:55)→ p/ [18]. (cid:50) Our problem is considerably simplified if we make a non-relativistic approximationfor the behaviour of the nucleus of the H atom since the proton is comparatively heavyand does not move much. In this approximation Dirac spinors u p ( p, α ) satify u p ( p (cid:48) , α (cid:48) ) γ µ u p ( p , α ) = δ α (cid:48) α η µ , (86)see [16]. This implies that m − p ( p/ (cid:48) + m p ) γ µ ( p/ + m p ) = η µ γ . (87)32herefore the Feynman amplitude M for the electron self-energy contribution toelectron-proton scattering (in the NR approximation for the proton) is M α (cid:48) α ( p (cid:48) , p ) = e m − e e † α (cid:48) ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) q − γ ( p/ + m e ) e α , where p is the incoming electron momentum, p (cid:48) is the outgoing electron momentum, q = p (cid:48) − p is the momentum transfer and we have suppressed the proton polarizationindices which play no further part in the calculation. Thus the Feynman amplitudematrix valued function is given by M = M ( p (cid:48) , p ) = e m − e ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) q − γ ( p/ + m e ) . (88) M is not covariant with respect to all of the group K which is not surprising sincewe are using an NR approximation. However we have the following . Theorem 5. M is covariant with respect to the rotation subgroup of K , that is, thegroup Rotations = { (cid:32) a a (cid:33) : a ∈ U (2) } ⊂ K. (89) Proof
Let R ∈ Rotations. Then R − = R † . Thus M ( Rp (cid:48) , Rp ) = e m − e ( Rp/ (cid:48) R − + m e ) Ri Σ( p (cid:48) ) R − RS ( p (cid:48) ) R − ( Rq ) − γ ( Rp/R − + m e )= Re m − e ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) q − R − γ R ( p/ + m e ) R − = Re m − e ( p/ (cid:48) + m e ) i Σ( p (cid:48) ) S ( p (cid:48) ) q − γ ( p/ + m e ) R − = R M ( p (cid:48) , p ) R − , since R − γ R = R † γ R = γ . ( R ∈ K ⊂ U (2 ,
2) and the metric for U (2 ,
2) is g = γ .) We have used the intertwiningproperty of the Feynman slash and the fact that the Feynman fermion propagator S is K invariant [21]. (cid:50) Now carry out a translation in momentum space to the center of mass frame with33rigin c = ( p + p (cid:48) ). Then p → p − c = 12 ( p − p (cid:48) ) = − q,p (cid:48) → p (cid:48) − c = 12 ( p (cid:48) − p ) = 12 q. In this frame M ( q ) = M ( p (cid:48) , p ) = q − N ( 12 q ) , (90)where N ( q ) = e m − e ( q/ + m e ) i Σ( q ) S ( q ) γ ( − q/ + m e ) . (91) N is rotationally covariant in the sense that N ( Rq ) = R N ( q ) R − , ∀ R ∈ Rotations . (92)Now K acts on C ∞ ( R , C ) according to [21]( κψ )( x ) = κψ ( κ − x ) = κψ (Λ( κ ) − x ) , (93)where Λ( κ ) is the Lorentz transformation corresponding to κ . Lemma 1.
Let A ∈ C ∞ ( R , C ) transform as a 4-vector under Lorentz transforma-tions. Then κ ( A/ψ ) = A/ ( κψ ) , ∀ κ ∈ K, ψ ∈ C ∞ ( R , C ) . (94) Proof x ∈ R . Then ( κ ( A/ψ ))( x ) = κ ( A/ψ )(Λ − x )= κA/ (Λ − x ) ψ (Λ − x )= κγ µ A µ (Λ − x ) ψ (Λ − x )= κγ µ A µ (Λ − x ) ψ (Λ − x )= κγ µ Λ − µν A ν ( x ) ψ (Λ − x )= κ Σ (Λ − A ( x )) ψ (Λ − x )= κκ − A/ ( x ) κψ (Λ − x )= A/ ( x )( κψ )( x )= ( A/ ( κψ ))( x ) . (cid:50) Similarly
Lemma 2. κ ( ∂/ψ ) = ∂/ ( κψ ) , ∀ κ ∈ K, ψ ∈ C ∞ ( R , C ) . (95) Theorem 6.
Let ψ ∈ C ∞ ( R , C ) be a unique solution to the Dirac equation withrespect to a set B of boundary conditions. Then ψ ( κx ) = κψ ( x ) , ∀ κ ∈ K, x ∈ R . (96) Proof
The Dirac equation is ( iγ µ D µ − m ) ψ = 0 , (97)where D µ = ∂ µ + ieA µ . (98)Let ψ ∈ C ∞ ( R , C ) be the unique solution with respect to B . Then( iD/ − m ) ψ = 0 . (99)Therefore by Lemmas 1 and 20 = κ ( iD/ − m ) ψ = iκ ( ∂/ψ ) − eκ ( A/ψ ) − κmψ = i∂/ ( κψ ) − eA/ ( κψ ) − mκψ = ( iD/ − m )( κψ ) . ψ is uniquely determined by Eq. 99 and B . Therefore κψ = ψ and hence κψ (Λ( κ ) − x ) = ψ ( x ) , ∀ x ∈ R , from which it follows that ψ (Λ( κ ) x ) = κψ ( x ) , ∀ κ ∈ K, x ∈ R . (100)(We have assumed that the transformed boundary conditions ∼ B agree with B .) (cid:50) Now suppose that we have a static rotationally invariant Dirac eigenfunction ψ forthe 2s state of the H atom. In general suppose that we have a Feynman amplitude M e.g. the tree level amplitude M or the amplitude M + M where M is the electronself-energy amplitude. We propose that the energy associated with the bound statedefined by ψ is given by the integral E = ω (cid:90) ψ † ( (cid:42) x ) M (0 , (cid:42) q ) ψ ( (cid:42) x ) e i(cid:42) q .(cid:42) x d (cid:42) x d (cid:42) q , (101)where we will determine ω by examination of the tree level diagram in the NR ap-proximation. For this diagram M ( q ) = ie | (cid:42) q | − . (102)However, as is well known, Feynman amplitudes are only defined up to multiplicationby an element of U (1) since physical predictions are made on the basis of the modulussquared of the Feynman amplitude. We choose to take M ( q ) = e | (cid:42) q | − , (103)Then E = ω (cid:90) ψ † ( (cid:42) x ) M (0 , (cid:42) q ) ψ ( (cid:42) x ) e i(cid:42) q .(cid:42) x d (cid:42) q d (cid:42) x = ωe (cid:90) | ψ ( (cid:42) x ) | ( (cid:90) ( | (cid:42) q | − e i(cid:42) q .(cid:42) x d (cid:42) q ) d (cid:42) x. But as is well known (cid:90) | (cid:42) q | − e i(cid:42) q .(cid:42) x d (cid:42) q = 2 π | (cid:42) x | . (104)36herefore E = 8 π ωe (cid:90) ∞ r =0 | ψ ( r ) | r dr. (105)However we know that E = 4 π (cid:90) ∞ r =0 | ψ ( r ) | V ( r ) r dr = 4 π (cid:90) ∞ r =0 | ψ ( r ) | ( − e πr ) r dr = − e (cid:90) ∞ r =0 | ψ ( r ) | r dr. (106)Therefore 8 π ω = − ω = − (2 π ) − . (107)Now the perturbation of the H atom energy level due to the electron self energy isgiven by ∆ E = ω (cid:90) F ( (cid:42) q ) d (cid:42) q , (108)where F ( (cid:42) q ) = (cid:90) ψ † ( (cid:42) x ) M (0 , (cid:42) q ) ψ ( (cid:42) x ) e i(cid:42) q .(cid:42) x d (cid:42) x. (109) Theorem 7. F is rotationally invariant. Proof F ( A (cid:42) q ) = (cid:90) ψ † ( (cid:42) x ) M (0 , A (cid:42) q ) ψ ( (cid:42) x ) e iA(cid:42) q .(cid:42) x d (cid:42) x = (cid:90) ψ † ( (cid:42) x ) R M (0 , (cid:42) q ) R † ψ ( (cid:42) x ) e i(cid:42) q .A † (cid:42) x d (cid:42) x = (cid:90) ψ † ( A † (cid:42) x ) M (0 , (cid:42) q ) ψ ( A † (cid:42) x ) e i(cid:42) q .A † (cid:42) x d (cid:42) x = (cid:90) ψ † ( (cid:42) x ) M (0 , (cid:42) q ) ψ ( (cid:42) x ) e i(cid:42) q .(cid:42) x d (cid:42) x = F ( (cid:42) q ) , for all A ∈ O (3) where R ∈ Rotations ⊂ K is such thatΛ( R ) = (cid:32) A (cid:33) , and we have used the rotational covariance of M and Theorem 6. (cid:50) E = ω (4 π ) (cid:90) ∞ s =0 F (0 , , s ) s ds = 4 πω (cid:90) ∞ s =0 (cid:90) R ψ † ( (cid:42) x ) M (0 , , , s ) ψ ( (cid:42) x ) e i (0 , ,s ) .(cid:42) x s d (cid:42) x ds = − πω (cid:90) ∞ s =0 (cid:90) R ψ † ( (cid:42) x ) N ( s ) ψ ( (cid:42) x ) e i (0 , ,s ) .(cid:42) x d (cid:42) x ds = − (4 π )(2 π ) ω (cid:90) ∞ s =0 (cid:90) ∞ r =0 (cid:90) πθ =0 ψ † ( r ) N ( s ) ψ ( r ) e isr cos( θ ) r sin( θ ) dθ dr ds = (4 π )(2 π ) − (cid:90) ∞ s =0 (cid:90) ∞ r =0 ψ † ( r ) N ( s ) ψ ( r ) 1 irs ( e isr − e − isr ) r dr ds, where N ( s ) = N ((0 , , , s . (110)Thus writing ∆ V ( r ) = (2 π ) − (cid:90) ∞ s =0 N ( s ) 1 isr ( e isr − e − isr ) ds = (2 π ) − ir (cid:90) ∞ s =0 N ( s ) s ( e isr − e − isr ) ds, we have ∆ E = 4 π (cid:90) ∞ r =0 ψ † ( r )∆ V ( r ) ψ ( r ) r dr. (111)(It can be shown using Fubini’s theorem that the order of the integrations can beinterchanged.)Note that ∆ V : (0 , ∞ ) → C × is a complex matrix valued potential function. Wewould like to analytically continue ∆ V to the upper imaginary axis of the complexplane, since, for this bound state problem, we are considering spacelike points inMinkowski space for which x < r Applying the transform T defined in Section 4 with F ( r ) = (cid:90) ∞ s =0 f ( s )( e − sr − e − isr ) ds, (112)and f ( s ) = N ( s ) s , (113)38e obtain ∆ V ( r ) = (2 π ) − ir F ( r ) → (2 π ) − ir ( T ( F ))( r )= (2 π ) − ir i (cid:90) ∞ s =0 N ( s ) s e − sr ds = (2 π ) − r (cid:90) ∞ s = m N ( s ) s e − sr ds. We call this the SE potential function and it is analogous to the Uehling potentialfunction. The code for a C++ program to compute the SE contribution to theLamb shift using the SE potential function is given in the Appendix. The outputof the program is shown in Figure 1. The output converges to 1077 MHz whereasthe accepted value for the SE contribution to the Lamb shift is 1085 MHz [7]. Thusthe computed value differs by 0.7% from the accepted value. However this is notsurprising since the program carries out more than 250,000,000 iterations which wouldbe associated with considerable computational error. Accuracy could be increased byusing the reduced mass of the electron-proton system rather than the electron massand also by using the relativistic Dirac H atom 2s eigenfunction rather than thenon-relativistic Schr¨odinger eigenfunction.A great deal of research was done many years ago e.g. [20, 22] on optimizing thenumerical computation of the Uehling contribution to the Lamb shift on the basis ofthe Uehling potential function. One may envisage carrying out similar optimizationfor the numerical calculation of the electron self-energy contribution to the Lambshift on the basis of the SE potential function.Moreover the existence of the simple form of the SE potential obtained without theneed for renormalization may simplify many computations in QFT involving multipleloops which are often hampered and complicated by nested divergences e.g. [23, 24].
Acknowledgements
The author is grateful to Christopher Chantler for very helpful discussions and toVladimir Yerokhin and Randolph Pohl for very helpful comments.39igure 1: Convergence to SE contribution to the Lamb shift for the H atom, MHz vs.iteration
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Appendix: C++ code to compute the electron self-energy contribution to the Lamb shift // self_energy.cpp : This file contains the ’main’ function.// Program execution begins and ends there.// omplex()// constructor{ real = 0.0;imaginary = 0.0;}};// omplex* Delta_E;complex** cx_vec, ** cx_vec_1;const double m_electron = 9.10938356e-31; // electron mass in Kg mksconst double c = 2.99792458e8; // speed of light m/s mksconst double e = 1.6021766208e-19; // electron charge in Coulombs mksconst double h = 6.626070040e-34; // Planck constant mksconst double h_bar = h / (2.0 * pi);const double epsilon_0 = 8.854187817e-12; // permittivity of free space mksconst double e1 = e / sqrt(epsilon_0); // electron charge in rationalized units//const double e1 = e/sqrt(4*pi*epsilon_0); // electron charge in Gaussian unitsconst double alpha = e1 * e1 / (4 * pi * h_bar * c); // fine structure constantdouble m = m_electron * c * c / e; // electron mass in natural unitsdouble a_0_mks = 4.0 * pi * h_bar * h_bar / (m * e1 * e1);// Bohr radius of the Hydrogen atom in metersdouble a_0 = 1.0 / (m * alpha);// a_0 in natural units eV^{-1}double factor = e / h; // factor for converting from eV to Hzdouble Lambda_r = 5.0 * a_0;double delta_r = (Lambda_r) / 10000;int N_s = 100000;double Lambda_s = 50.0 * m;double delta_s = (Lambda_s) / N_s;double delta = delta_r * delta_s;int main(){ std::ofstream outFile("out.txt");create_arrays();make_gamma_matrices();make_unit();std::cout << "electron mass = " << m << " eV\n";std::cout << "electron mass in Kg = " << m_electron << " Kg\n"; td::cout << "Inverse fine structure constant 1/alpha = " << 1 / alpha << "\n";std::cout << "Bohr radius of hydrogen atom = "<< a_0_mks << " m\n";std::cout << "Bohr radius in natural units = " << a_0<< " eV^{-1}\n\n";Delta_E->real = 0.0;Delta_E->imaginary = 0.0;int i, j;double answer = 0.0;for (i = 1;i<250; i++){ double eV_r, eV_i, MHz_r, MHz_i;eV_r = -Delta_E->real * 4.0 * pi * alpha * 4.0 * pi * delta /(2.0 * pi * 2.0 * pi);eV_i = -Delta_E->imaginary * 4.0 * pi * alpha * 4.0 * pi * delta /(2.0 * pi * 2.0 * pi);MHz_r = eV_r * factor / (1.0e6);MHz_i = eV_i * factor / (1.0e6);std::cout << "for i = " << i << " answer = ("<< eV_r << "," << eV_i << ") eV = ("<< MHz_r << "," << MHz_i << ") MHz\n";outFile << MHz_r << "\n";double r = i * delta_r;for (j = 1; j < N_s; j++){ double s = m + j * delta_s;p[0] = 0.0;p[1] = 0.0;p[2] = 0.0;p[3] = s / 2.0;zeta = s / 2.0;compute_N(p);compute_psi(r, cx_vec);vec_prod(N, cx_vec, cx_vec_1); ec_hermitian_prod(cx_vec, cx_vec_1, z4);double v_r = exp(-r * s) * r * z4->real / s;double v_i = exp(-r * s) * r * z4->imaginary / s;Delta_E->real += v_r;Delta_E->imaginary += v_i;}}return 0;}void compute_N(double* q_vec){ int i, j, k;for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ mat_1[i][j]->real = (q_vec[0] * gamma[0][i][j]->real +q_vec[1] * gamma[1][i][j]->real +q_vec[2] * gamma[2][i][j]->real +q_vec[3] * gamma[3][i][j]->real) / (m);mat_1[i][j]->imaginary = (q_vec[0] * gamma[0][i][j]->imaginary +q_vec[1] * gamma[1][i][j]->imaginary +q_vec[2] * gamma[2][i][j]->imaginary +q_vec[3] * gamma[3][i][j]->imaginary) / (m);mat_2[i][j]->real = -mat_1[i][j]->real;mat_2[i][j]->imaginary = -mat_1[i][j]->imaginary;}sum(mat_1, unit, mat_3);Sigma(q_vec, mat_4);S(q_vec, mat_5);prod(mat_4, mat_5, mat_6);prod(mat_3, mat_6, mat_4);sum(mat_2, unit, mat_5);prod(gamma[0], mat_5, mat_6);prod(mat_4, mat_6, N); void S(double* p_vec, complex*** answer)// density for the electron propagator S{ int i, j;double v = zeta * zeta - m * m;if (v != 0.0){ for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ answer[i][j]->real = (p_vec[0] * gamma[0][i][j]->real +p_vec[1] * gamma[1][i][j]->real +p_vec[2] * gamma[2][i][j]->real +p_vec[3] * gamma[3][i][j]->real +m * unit[i][j]->real) / v;answer[i][j]->imaginary = (p_vec[0] * gamma[0][i][j]->imaginary +p_vec[1] * gamma[1][i][j]->imaginary +p_vec[2] * gamma[2][i][j]->imaginary +p_vec[3] * gamma[3][i][j]->imaginary +m * unit[i][j]->imaginary) / v;}}}void Sigma(double* p_vec, complex*** answer)// density for the electron self-energy kernel Sigma{ int i, j;double b = zeta;double Z_1 = (b * b - m * m) / (2 * b);double Z_1_prime = (b * b + m * m) / (2 * b * b); ouble o_Z = sqrt(Z_1 * Z_1 + m * m);for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ answer[i][j]->real = (4.0 * m * unit[i][j]->real - 2.0 * o_Z *gamma[0][i][j]->real) * Z_1 * Z_1_prime / (4.0 * pi * o_Z);answer[i][j]->real *= 4.0 * pi * alpha;answer[i][j]->imaginary = (-2.0 * o_Z * gamma[0][i][j]->imaginary) *Z_1 * Z_1_prime / (4.0 * pi * o_Z);answer[i][j]->imaginary *= 4.0 * pi * alpha;}}void matrix_dagger_4(complex*** m1, complex*** m2){ int i, j;for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ m2[i][j]->real = m1[j][i]->real;m2[i][j]->imaginary = -m1[j][i]->imaginary;}}void make_gamma_matrices(){ gamma = new complex * **[4];int i, j, k;for (k = 0; k < 4; k++){ gamma[k] = new complex * *[4];for (i = 0; i < 4; i++){ gamma[k][i] = new complex * [4]; or (j = 0; j < 4; j++)gamma[k][i][j] = new complex;}for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ gamma[k][i][j]->real = 0.0;gamma[k][i][j]->imaginary = 0.0;}}// Diracgamma[0][0][0]->real = 1.0;gamma[0][1][1]->real = 1.0;gamma[0][2][2]->real = -1.0;gamma[0][3][3]->real = -1.0;// make_gamma[1]gamma[1][0][3]->real = 1.0;gamma[1][1][2]->real = 1.0;gamma[1][2][1]->real = -1.0;gamma[1][3][0]->real = -1.0;// make_gamma[2]gamma[2][0][3]->imaginary = -1.0;gamma[2][1][2]->imaginary = 1.0;gamma[2][2][1]->imaginary = 1.0;gamma[2][3][0]->imaginary = -1.0;// make_gamma[3]gamma[3][0][2]->real = 1.0;gamma[3][1][3]->real = -1.0;gamma[3][2][0]->real = -1.0;gamma[3][3][1]->real = 1.0;}void make_unit(){ nt i, j;unit = new complex * *[4];for (i = 0; i < 4; i++){ unit[i] = new complex * [4];for (j = 0; j < 4; j++)unit[i][j] = new complex;}for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ if (i == j){ unit[i][j]->real = 1.0;unit[i][j]->imaginary = 0.0;}else{ unit[i][j]->real = 0.0;unit[i][j]->imaginary = 0.0;}}}void create_arrays(){ int i, j;// create 4x4 mat matricesmat_1 = new complex * *[4];for (i = 0; i < 4; i++) mat_1[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_1[i][j] = new complex; at_2 = new complex * *[4];for (i = 0; i < 4; i++) mat_2[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_2[i][j] = new complex;mat_3 = new complex * *[4];for (i = 0; i < 4; i++) mat_3[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_3[i][j] = new complex;mat_4 = new complex * *[4];for (i = 0; i < 4; i++) mat_4[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_4[i][j] = new complex;mat_5 = new complex * *[4];for (i = 0; i < 4; i++) mat_5[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_5[i][j] = new complex;mat_6 = new complex * *[4];for (i = 0; i < 4; i++) mat_6[i] = new complex * [4];for (i = 0; i < 4; i++)for (j = 0; j < 4; j++)mat_6[i][j] = new complex;// create NN = new complex * *[4];for (i = 0; i < 4; i++) N[i] = new complex * [4];for (i = 0; i < 4; i++) or (j = 0; j < 4; j++)N[i][j] = new complex;z4 = new complex;Delta_E = new complex;p = new double[4];// create complex vectorscx_vec = new complex * [4];for (i = 0; i < 4; i++)cx_vec[i] = new complex;cx_vec_1 = new complex * [4];for (i = 0; i < 4; i++){ cx_vec_1[i] = new complex;}}void sum(complex*** M_1, complex*** M_2, complex*** answer){ // form the sum of two complex matricesint i, j;for (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ answer[i][j]->real = M_1[i][j]->real + M_2[i][j]->real;answer[i][j]->imaginary = M_1[i][j]->imaginary + M_2[i][j]->imaginary;}}void prod(complex*** M_1, complex*** M_2, complex*** answer){ // form the product of two complex matricesint i, j, k; or (i = 0; i < 4; i++)for (j = 0; j < 4; j++){ answer[i][j]->real = 0.0;answer[i][j]->imaginary = 0.0;for (k = 0; k < 4; k++){ answer[i][j]->real += M_1[i][k]->real * M_2[k][j]->real -M_1[i][k]->imaginary * M_2[k][j]->imaginary;answer[i][j]->imaginary += M_1[i][k]->imaginary * M_2[k][j]->real +M_1[i][k]->real * M_2[k][j]->imaginary;}}}void vec_prod(complex*** m1, complex** p_vec, complex** answer){ int i, j;for (i = 0; i < 4; i++){ answer[i]->real = 0.0;answer[i]->imaginary = 0.0;for (j = 0; j < 4; j++){ answer[i]->real += m1[i][j]->real * p_vec[j]->real -m1[i][j]->imaginary *p_vec[j]->imaginary;answer[i]->imaginary += m1[i][j]->imaginary * p_vec[j]->real +m1[i][j]->real *p_vec[j]->imaginary;}}}void vec_hermitian_prod(complex** cx_vec_1, complex** cx_vec_2, complex* z){ z->real = 0.0; ->imaginary = 0.0;int i;for (i = 0; i < 4; i++){ z->real += cx_vec_1[i]->real * cx_vec_2[i]->real +cx_vec_1[i]->imaginary * cx_vec_2[i]->imaginary;z->imaginary += cx_vec_1[i]->real * cx_vec_2[i]->imaginary -cx_vec_1[i]->imaginary * cx_vec_2[i]->real;}}double psi(double r){ // Hydrogen atom wave function for 2s orbitaldouble answer;double v = r / (2.0 * a_0);answer = (2.0 - r / a_0) * exp(-v);answer /= (4.0 * sqrt(2.0 * pi) * a_0 * sqrt(a_0));return(answer);}void compute_psi(double r, complex** psi_Dirac){// make Dirac wave function with NR Schroedinger wave function as first componentpsi_Dirac[0]->real = psi(r);psi_Dirac[0]->imaginary = 0.0;psi_Dirac[1]->real = 0.0;psi_Dirac[1]->imaginary = 0.0;psi_Dirac[2]->real = 0.0;psi_Dirac[2]->imaginary = 0.0;psi_Dirac[3]->real = 0.0;psi_Dirac[3]->imaginary = 0.0;}->imaginary = 0.0;int i;for (i = 0; i < 4; i++){ z->real += cx_vec_1[i]->real * cx_vec_2[i]->real +cx_vec_1[i]->imaginary * cx_vec_2[i]->imaginary;z->imaginary += cx_vec_1[i]->real * cx_vec_2[i]->imaginary -cx_vec_1[i]->imaginary * cx_vec_2[i]->real;}}double psi(double r){ // Hydrogen atom wave function for 2s orbitaldouble answer;double v = r / (2.0 * a_0);answer = (2.0 - r / a_0) * exp(-v);answer /= (4.0 * sqrt(2.0 * pi) * a_0 * sqrt(a_0));return(answer);}void compute_psi(double r, complex** psi_Dirac){// make Dirac wave function with NR Schroedinger wave function as first componentpsi_Dirac[0]->real = psi(r);psi_Dirac[0]->imaginary = 0.0;psi_Dirac[1]->real = 0.0;psi_Dirac[1]->imaginary = 0.0;psi_Dirac[2]->real = 0.0;psi_Dirac[2]->imaginary = 0.0;psi_Dirac[3]->real = 0.0;psi_Dirac[3]->imaginary = 0.0;}