variational derivation of relativistic fermion–antifermion wave equations in qed

Variational derivation of relativistic fermion–antifermion wave equations in QEDAndrei G. Terekidi and Jurij W. Darewych Citation: Journal of Mathematical Physics 45, 1474 (2004); doi: 10.1063/1.1649794 View online: http://dx.doi.org/10.1063/1.1649794 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/45/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Relativistic quaternionic wave equation. II J. Math. Phys. 48, 052303 (2007); 10.1063/1.2735441 Relativistic quaternionic wave equation J. Math. Phys. 47, 122301 (2006); 10.1063/1.2397555 Variational principles for eigenvalues of the Klein–Gordon equation J. Math. Phys. 47, 103506 (2006); 10.1063/1.2345108 Variational two-fermion wave equations in quantum electrodynamics: Muoniumlike systems J. Math. Phys. 46, 032302 (2005); 10.1063/1.1845602 Variational Two Fermion Wave Equations in QED AIP Conf. Proc. 646, 105 (2002); 10.1063/1.1524559

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Variational derivation of relativistic fermion–antifermionwave equations in QED

Andrei G. Terekidi and Jurij W. DarewychDepartment of Physics and Astronomy, York University,Toronto, Ontario, M3J 1P3, Canada

~Received 16 July 2003; accepted 17 December 2003!

We present a variational method for deriving relativistic two-fermion wave equa-tions in a Hamiltonian formulation of QED. A reformulation of QED is performed,in which covariant Green functions are used to solve for the electromagnetic fieldin terms of the fermion fields. The resulting modified Hamiltonian contains thephoton propagator directly. The reformulation permits one to use a simple Fock-space variational trial state to derive relativistic fermion–antifermion wave equa-tions from the corresponding quantum field theory. We verify that the energy ei-genvalues obtained from the wave equation agree with known results forpositronium. © 2004 American Institute of Physics.@DOI: 10.1063/1.1649794#

I. INTRODUCTION

The description of relativistic bound and quasibound~i.e., unstable! few body systems con-tinues to be an active area of research. The traditional method of treating relativistic bound statesin quantum field theory~QFT! is by means of the Bethe–Salpeter~BS! equation. However, thisapproach has a number of difficulties, including the appearance of relative-time coordinates andnegative-energy solutions. In practice, the interaction kernels~potentials! in the BS equation areobtained from covariant perturbation theory, which may be of questionable validity for stronglycoupled systems. In addition, the BS formalism is difficult to implement for systems of more thantwo particles.

An alternative approach might be the variational method, which is nonperturbative in prin-ciple. The variational method has not been widely used in quantum field theory, in contrast tononrelativistic systems describable by the Schro¨dinger theory, in part because of the difficulty ofconstructing realistic yet tractable trial states.

It has been pointed out in previous publications1,2 that various models in QFT, including QED,can be reformulated, using mediating-field Green functions, into a form particularly convenient forvariational calculations. This approach was applied recently to the study of relativistic two-bodystates in the scalar Yukawa~Wick–Cutkosky! theory.3–5 In the present paper we shall implementthis approach to the realistic QED theory, where comparison with experimentally verified resultsare possible. In particular, we shall use the reformulated QED Hamiltonian to derive a relativisticfermion–antifermion wave equation and discuss its solution.

The reformulation of QED is presented in Sec. II, while the Hamiltonian and equal timequantization are given in Sec. III. In Sec. IV we use the variational principle with simple Fock-space trial states to derive the relativistic fermion–antifermion equations, and present their ‘‘par-tial wave’’ decomposition for all possibleJPC states. The relativistic radial equations are presentedin Sec. V, while their nonrelativistic and semirelativistic limits are given in Sec. VI. In Sec. VII theenergy eigenvalues are shown to yield the correct fine and hyperfine structure for all states.Concluding remarks are given in Sec. VIII.

II. REFORMULATION OF FIELD EQUATIONS AND LAGRANGIAN

The Lagrangian of QED is (\5c51)

L5c~x!~ igm]m2m2egmAm~x!!c~x!2 14 ~]aAb~x!2]bAa~x!!~]aAb~x!2]bAa~x!!. ~1!

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http://dx.doi.org/10.1063/1.1649794

The corresponding Euler–Lagrange equations of motion are the coupled Dirac–Maxwell equa-tions,

~ igm]m2m!c~x!5egmAm~x!c~x!, ~2!

and

]m]mAn~x!2]n]mAm~x!5 j n~x!, ~3!

where

j n~x!5ec~x!gnc~x!. ~4!

Equations~2! and~3! can be decoupled in part by using the well-known formal solution6,7 of theMaxwell equation~3!, namely,

Am~x!5Am0 ~x!1E d4x8Dmn~x2x8! j n~x8!, ~5!

whereDmn(x2x8) is a Green function~or photon propagator in QFT terminology!, defined by

]a]aDmn~x2x8!2]m]aDan~x2x8!5gmnd4~x2x8!, ~6!

andAm0 (x) is a solution of the homogeneous~or ‘‘free field’’ ! equation~3! with j m(x)50.

We recall, in passing, that Eq.~6! does not define the covariant Green functionDmn(x2x8)uniquely. For one thing, one can always add a solution of the homogeneous equation@Eq. ~6! withgmn→0]. This allows for a certain freedom in the choice ofDmn , as is discussed in standard texts~e.g., Refs. 6 and 7!. In practice, the solution of Eq.~6!, like that of Eq.~3!, requires a choice ofgauge. However, we do not need to specify one at this stage.

Substitution of the formal solution~5! into Eq. ~2! yields the ‘‘partly reduced’’ equations,

~ igm]m2m!c~x!5egmS Am0 ~x!1E d4x8Dmn~x2x8! j n~x8! Dc~x!, ~7!

which is a nonlinear Dirac equation. To our knowledge no exact~analytic or numeric! solution ofEq. ~7! for classical fields have been reported in the literature. However, approximate solutionshave been discussed by various authors, particularly Barut and co-workers~see Refs. 8, 9, andcitations therein!. In any case, our interest here is in the quantized field theory.

The partially reduced equation~7! is derivable from the stationary action principle

dS@c#5dE d4xLR50 ~8!

with the Lagrangian density

LR5c~x!~ igm]m2m2egmA0m~x!!c~x!2

1

2E d4x8 j m~x8!Dmn~x2x8! j n~x! ~9!

provided that the Green function is symmetric in the sense that

Dmn~x2x8!5Dmn~x82x! and Dmn~x2x8!5Dnm~x2x8!. ~10!

One can proceed to do conventional covariant perturbation theory using the reformulatedQED Lagrangian~9!. The interaction part of~9! has a somewhat modified structure from that ofthe usual formulation of QED. Thus, there are two interaction terms. The last term of~9! is a‘‘current–current’’ interaction which contains the photon propagator sandwiched between the fer-

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mionic currents. As such, it corresponds to Feynman diagrams without external photon lines. Theterm containingA0

m corresponds to diagrams that cannot be generated by the term containingDmn ,including diagrams involving external photon lines~care would have to be taken not to doublecount physical effects!. However, we shall not pursue covariant perturbation theory in this work.Rather, we shall consider a variational approach that allows one to derive relativistic few-fermionequations, and to study their bound and scattering solutions.

III. HAMILTONIAN OF THE QUANTIZED THEORY IN THE EQUAL-TIME FORMALISM

We consider this theory in the quantized, equal-time formalism. To this end we write down theHamiltonian density corresponding to the Lagrangian~9!, with the term for the freeA0

m(x) fieldsuppressed since it will not contribute to the results presented in this paper. The relevant expres-sion is

HR5H01HI , ~11!

where

H05c†~x!~2 i a•¹1mb!c~x!, ~12!

HI51

2E d4x8 j m~x8!Dmn~x2x8! j n~x!. ~13!

We construct a quantized theory by the imposition of anticommutation rules for the fermionfields, namely,

$ca~x,t !,cb†~y,t !%5dabd3~x2y!, ~14!

while all other vanish. In addition, ifA0mÞ0, there would be the usual commutation rules for the

A0m field, and commutation of theA0

m field operators with thec field operators.To specify our notation, we quote the usual Fourier decomposition of the field operators,

namely,

c~x!5(sE d3p

~2p!3/2S m

vpD 1/2

@bpsu~p,s!e2 ip•x1dps† v~p,s!eip•x#, ~15!

with p5pm5(vp ,p), and vp5Am21p2. Dirac spinorsu and v for free particles of massm,where (gmpm2m)u(p,s)50, (gmpm1m)v(p,s)50, are normalized such that

u†~p,s!u~p,s!5v†~p,s!v~p,s!5vp

mdss , ~16!

u†~p,s!v~p,s!5v†~p,s!u~p,s!50. ~17!

The creation and annihilation operatorsb†, b of the ~free! fermions of massm, andd†, d forthe corresponding antifermions, satisfy the usual anticommutation relations. The nonvanishingones are

$bps ,bqs† %5$dps ,dqs

† %5dssd3~p2q!. ~18!

IV. VARIATIONAL PRINCIPLE AND FERMION–ANTIFERMION TRIAL STATES

Unfortunately we do not know how to obtain exact eigenstates of the Hamiltonian~11!.Therefore we shall resort to a variational approximation, based on the variational principle

d^cuH2Euc& t5050. ~19!

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For a fermion–antifermion system, the simplest Fock-space trial state that can be written down inthe rest frame is

ucT&5(s1s2

E d3pFs1s2~p!bps1

† d2ps2

† u0&, ~20!

whereFs1s2are four adjustable functions. We use this trial state to evaluate the matrix elements

needed to implement the variational principle~19!, namely,

^cTu:H02E:ucT&5(s1s2

E d3pFs1s2* ~p!Fs1s2

~p!~2vp2E! ~21!

and

^cTu:HI :ucT&5e2m2

~2p!3 (s18s28s1s2

E d3pd3p8

vpvp8Fs1s2

* ~p!Fs18s

28~p8!

3S 2u~p,s1!gmu~p8,s18!Dmn~p2p8!v~2p8,s28!gnv~2p,s2!

1u~p,s1!gmv~2p,s2!Dmn~p1p8!v~2p8,s28!gnu~p8,s18! D , ~22!

wherep5(vp ,p), p85(vp8 ,p8), with p1p850 ~i.e., p1p85(2vp,0)) in the rest frame, and

Dmn~x2x8!5E d4k

~2p!4 Dmn~k!e2 ik•(x2x8). ~23!

We have normal-order the entire Hamiltonian, since this circumvents the need for mass renor-malization which would otherwise arise. Not that there is difficulty with handling mass renormal-ization in the present formalism~as shown in various earlier papers; see, for example, Ref. 10, andcitations therein!. It is simply that we are not interested in mass renormalization here, since it hasno effect on the two-body bound state energies that we obtain in this paper. Furthermore, theapproximate trial state~20!, which we use in this work, is incapable of sampling loop effects.Thus, the normal-ordering of the entire Hamiltonian does not ‘‘sweep under the carpet’’ loopeffects, since none arise at the present level of approximation, that is with the trial stateucT&specified in Eq.~20!.

The variational principle~19! leads to the following equation:

(s1s2

E d3p~2vp2E!Fs1s2~p!dFs1s2

* ~p!

2m2

~2p!3 (s1s2s1s2

E d3pd3q

vpvqFs1s2

~q!~2 i !Ms1s2s1s2~p,q!dFs1s2

* ~p!50, ~24!

whereMs1s2s1s2(p,q) is an invariant ‘‘matrix element,’’ which contains two terms:

Ms1s2s1s2~p,q!5Ms1s2s1s2

ope ~p,q!1Ms1s2s1s2

ann ~p,q!, ~25!

where

Ms1s2s1s2

ope ~p,q!52u~p,s1!~2 iegm!u~q,s1!iD mn~p2q!v~2q,s2!~2 iegn!v~2p,s2!,

~26!

Ms1s2s1s2

ann ~p,q!5u~p,s1!~2 iegm!v~2p,s2!iD mn~p1q!v~2q,s2!~2 iegn!u~q,s1! ,

~27!

correspond to the usual one-photon exchange and virtual annihilation Feynman diagrams.



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At this point it is worthwhile to make a few comments about our Eq.~24! and to compare itsgeneral features with other two-fermion equations, particularly field-theory based approaches.Firstly we note that the present variational derivation leads to momentum-space Salpeter-typeequations, with at most four independent componentsFs1s2

(p). The equations have only positive-energy solutions, as is evident from Eq.~24! with the interaction turned off, in which case onlyE52vp.0 is obtained. This is in contrast to the BS equation, which is a 16-component equationand contains both positive, negative and mixed energy solutions.

The interaction kernels, represented by the covariantM-matrices, result from the variationalderivation, that is, they are not put in by hands. This is in contrast to two fermion equations, whichare not derived from a underlying quantum field theory, such as various two-body generalizationsof the one-body Dirac equation. There are many such equations on the market, for example theeight component two-fermion equation of Pilkuhn.11 In these treatments QFT effects, such as thevirtual annihilation interaction@Eq. ~27!# do not arise naturally but need to be added in.

The fact that only the lowest order~‘‘tree level’’ ! diagrams appear in our Eq.~24! is areflection of the fact that we have used the simplest possible variational ansatz~20!. Even so, it isimportant to note that, because of the reformulation discussed in Secs. II and III, their derivationdoes not require additional Fock-space terms in the variational state~20! as is the case in tradi-tional ~nonreformulated! treatments~e.g., Refs. 12–14!.

In the nonrelativistic limit, the functionsFs1s2can be written as

Fs1s2~p!5F~p!Ls1s2

, ~28!

where the nonzero elements ofL i j for total spin singlet (S50) states areL1252L2151/&,while for the spin triplet (S51) states the nonzero elements areL1151 for ms511, L125L21

51/& for ms50, andL2251 for ms521. We use the notation that the subscripts 1 and 2 ofLcorrespond toms51/2 andms521/2 ~or ↑ and↓! respectively. Substituting~28! into ~24!, thevariational procedure, after multiplying the result byLs1s2

and summing overs1 ands2 , gives theequation

~2vp2E!F~p!51

~2p!3 E d3q K~p,q!F~q!, ~29!

where

K~p,q!52 im2

vpvq(

s1s2s1s2

Ls1s2Ms1s2s1s2

~p,q!Ls1s2. ~30!

To lowest-order inupu/m ~i.e., in the nonrelativistic limit!, the kernel~30! reduces toK5e2/up2qu2, and so~29! reduces to the~momentum-space! Schrodinger equation

S p2

2m2« DF~p!5

1

~2p!3 E d3 qe2

up2qu2 F~q!, ~31!

where«5E22m andm5m/2. This verifies that the relativistic two-fermion equation~24! has theexpected nonrelativistic limit.

In the relativistic case we do not complete the variational procedure in~24! at this stage to getequations for the four adjustable functionsFs1s2

, because they are not independent in general.Indeed we require that the trial state be an eigenstate of the total angular momentum operator~inrelativistic form!, its projection, parity and charge conjugation, namely, that



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F J2

J3

PCG ucT&5F J~J11!

mJ

PC

G ucT&, ~32!

wheremJ5J,J21, . . . ,2J as usual. We present explicit forms for the operatorsJ2, J3 in Appen-dix A. The form forJ2, Eq.~109!, in particular, is not readily found in standard texts and referencebooks.

The functionsFs1s2(p) can be written in the general form

Fs1s2~p!5 (

,s1s2

(ms1s2

fs1s2

,s1s2ms1s2~p!Y

,s1s2

ms1s2~ p!, ~33!

whereY,s1s2

ms1s2(p) are the usual spherical harmonics. Here and henceforth we will use the notation

p5upu, etc.~four-vectors will be written aspm). The orbital indexes,s1s2andms1s2

depend on the

spin indexess1 ands2 and are specified by equations~32!. The radial coefficientsfs1s2

,s1s2ms1s2(p) in

the expansion~33! also depend on the spin variables.Substitution of~33! into ~20! and then into~32! leads to two categories of relations among the

adjustable functions, as shown in Appendixes A and B. It follows that, for trial states of the form~20!, the total spin of the system is a good quantum number, and the states of the system separateinto singlet states with the total spinS50 ~parastates! and into triplet states withS51 ~ortho-states!. We should point out that this phenomenon is characteristic of the fermion antifermionsystems, which are charge conjugation eigenstates, and does not arise for systems likem1e2.

A. The singlet states

In this case,s1s2[,5J, m115m2250 and m125m215mJ . The nonzero components of

Fs1s2(p) areF↑↓(p)[F12(p), F↓↑(p)[F21(p) and have the form

Fs1s2~p!5 f s1s2

(sgl)J~p!YJ

ms1s2~ p!, ~34!

where the relations betweenf 12(sgl)J(p) and f 21

(sgl)J(p) involve the Clebsch–Gordan~C–G! co-

efficientsCJmJ

(sgl)Jms1s2 , that is

f s1s2

(sgl)J~p!5CJmJ

(sgl)Jms1s2f J~p!, ~35!

as is shown in Appendix A. We see that the spin and radial variables separate for the singlet statesin the sense that the factorsf s1s2

(sgl)J(p) have a common radial functionf J(p). Thus, for the singlet

states we obtain

Fs1s2~p!5C

JmJ

(sgl)Jms1s2f J~p!YJmJ~ p!. ~36!

The C–G coefficientsCJmJ

(sgl)Jms1s2 have a simple form:CJmJ

(sgl)Jm115CJmJ

(sgl)Jm2250, CJmJ

(sgl)Jm125

2CJmJ

(sgl)Jm2151 ~see Appendix A!. Therefore for the singlet states we can write expression~20! in

the explicit form

ucT&5E d3pf J~p!YJmJ~ p!~bp↑

† d2p↓† 2bp↓

† d2p↑† !u0&. ~37!



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These states are characterized by the quantum numbersJ,mJ parity P5(21)J11 and chargeconjugationC5(21)J. As we can see, the quantum numbers, ~orbital angular momentum!, andtotal spinS are good quantum numbers for the singlet states as well. The spectroscopical notationis 1JJ .

B. The triplet states

The solution of the system~32! for S51 leads to two cases~Appendix A!, namely,,s1s2

[,5J, for which

Fs1s2~p!5 f s1s2

(tr)J~p!YJ

ms1s2~ p!, ~38!

and,s1s2[,5J71, for which

Fs1s2~p!5 f s1s2

J21~p!YJ21

ms1s2~ p!1 f s1s2

J11~p!YJ11

ms1s2~ p!, ~39!

where

m115mJ21, m125m215mJ , m225mJ11. ~40!

The expressions forf s1s2

, (p) in both cases involve the C–G coefficientsCJmJ

(tr),ms for S51 listed in

Appendix A, that is

f s1s2

(tr),~p!5CJmJ

(tr),msf ,~p!, ~41!

where the indexms is defined as

ms511, when ms1s25m11,

ms50, when ms1s25m125m21, ~42!

ms521, when ms1s25m22.

Thus, for the triplet states with,5J,

Fs1s2~p!5CJmJ

(tr)Jmsf J~p!YJ

ms1s2~ p!. ~43!

These functions correspond to states, which can be characterized by the quantum numbersJ,mJ ,parity P5(21)J11 and charge conjugationC5(21)J11. The orbital angular momentum,, aswell as the total spinS51, are good quantum numbers in this case. The spectroscopic notation forthese states is3JJ .

For the triplet states with,5J71 we obtain the result

Fs1s2~p!5CJmJ

(tr)(J21)msf J21~p!YJ21

ms1s2~ p!1CJmJ

(tr)(J11)msf J11~p!YJ11

ms1s2~ p!, ~44!

which involves two radial functionsf J21(p) and f J11(p) corresponding to,5J21 and,5J11. This means that, is not a good quantum number. Such states are characterized by quantumnumbersJ, mJ , P5(21)J, charge conjugationC5(21)J and spinS51. In spectroscopic no-tation, these states are a mixture of3(J21)J and3(J11)J states.

The requirement that the states be charge conjugation eigenstates@the last equation of~32!# isintimately tied to the conservation of total spin. Indeed, a linear combination of singlet and tripletstates like



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Fs1s2~p!5C1f s1s2

(sgl)J~p!YJ

ms1s2~ p!1C2f s1s2

(tr)J~p!YJ

ms1s2~ p!, ~45!

satisfies the first three equations of~32!. However, it is unacceptable for describing a fermion–antifermion system because the first and the second terms in~45! have different charge conjuga-tion. For a system of two particles of different mass~such asm1e2) charge conjugation is notapplicable, so that the total spin would not be conserved.

V. THE RELATIVISTIC RADIAL EQUATIONS AND APPLICATION TO POSITRONIUMLIKESYSTEMS

We return to Eq.~24! and replace the functionsFs1s2(p) by the expression~36! for singlet

states and by~43! and~44! for triplet states. The variational procedure then leads to the followingresults:

For the singlet states,5J, P5(21)J11, C5(21)J, the radial equations are

~2vp2E! f J~p!5m2

~2p!3 E q2dq

vpvqK (sgl)~p,q! f J~q!, ~46!

where the kernel

K (sgl)~p,q!52 i (s1s2s1s2mJ

E dp dq CJmJ

(sgl)s1s2s1s2Ms1s2s1s2~p,q!YJ

mJ* ~ p!YJmJ~ q! ~47!

is defined by the invariantM -matrix and the coefficients

CJmJ

(sgl)s1s2s1s2[CJmJ

(sgl)JmsCJmJ

(sgl)JmsY (n1n2mJ

~CJmJ

(sgl)Jmn!2. ~48!

Here we have summed overmJ , because of the (2J11)-fold energy degeneracy.For the triplet states, we obtain different equations for the,5J, and,5J71 cases. Thus for

the states with,5J, P5(21)J11, C5(21)J11 the result is

~2vp2E! f J~p!5m2

~2p!3 E q2dq

vpvqK (tr)~p,q! f J~q!, ~49!

where the kernelK (tr) is formally like that of~47!, namely,

K (tr)~p,q!52 i (s1s2s1s2mJ

CJmJ

(tr)s1s2s1s2E dp dq Ms1s2s1s2~p,q!Y

J

ms1s2* ~ p!YJ

ms1s2~ q!. ~50!

However it involves different C–G coefficients, namely,

CJmJ

(tr)s1s2s1s25CJmJ

(tr)JmsCJmJ

(tr)JmsY (n1n2mJ

~CJmJ

(tr)Jmn!2. ~51!

For the triplet states with,5J71, we have two independent radial functionsf J21(p) andf J11(p). Thus the variational equation~24! leads to a system of coupled equations forf J21(p)

and f J11(p). It is convenient to write them in matrix form,

~2vp2E!F~p!5m2

~2p!3 E q2dq

vpvqK~p,q!F~q!, ~52!

where



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F~p!5F f J21~p!

f J11~p!G , ~53!

and

K~p,q!5FK11~p,q! K12~p,q!

K21~p,q! K22~p,q!G . ~54!

The kernelsKi j are similar in form to~47! and ~50!, that is

Ki j ~p,q!52 i (s1s2s1s2mJ

CJmJi js1s2s1s2E dp dq Ms1s2s1s2

~p,q!Y, j

ms1s2~ q!Y, i

ms1s2* ~ p!. ~55!

However the coefficientsCJmJi js1s2s1s2 are defined by expression

CJmJi js1s2s1s25CJmJ

(tr), jmsCJmJ

(tr), imsY (s1s2mJ

~CJmJ

(tr), ims!2, ~56!

where,15J21, ,25J11 andmS is as defined in Eq.~42!. The system~52! reduces to a singleequation forJ50 sincef J21(p)50 in that case.

Our Eq. ~24!, or its radial components~46!, ~49!, ~52!, contain the relativistic two-bodykinematics~kinetic energy, recoil effects! exactly, but the dynamics are included approximatelydue to the limited nature of our trial state~20!. This limitation is reflected in the fact that theinteraction kernels of our equations contain only ‘‘tree-level’’ Feynman diagrams. Neverthelessour Eqs.~46!, ~49!, ~52! have no negative-energy solutions, in contrast to the BS equation. Theyare variationally derived, hence the energy eigenvalues obtained from them will give meaningfulvalues for any strength of the coupling.

To our knowledge, it is not possible to obtain analytic solutions of the relativistic radialmomentum-space equations~46!, ~49!, and ~52!. Thus one must resort to numerical or otherapproximation methods. Numerical solutions of such equations are discussed, for example, in Ref.10, while a variational approximation has been employed in Ref. 5. However, in this paper we willconcentrate on perturbativeO(a4) solutions, since it is important to verify that our equations yieldthe correct fine structure for systems like positronium.

Our equations will yield energies which are incomplete beyondO(a4), because our varia-tional trial state~20!, as mentioned, reflects only ‘‘tree-level’’ Feynman diagrams, that is noradiative corrections are incorporated. One could, of course, augment them by the addition ofinvariant matrix elements corresponding to higher-order Feynman diagrams~including radiativecorrections! to the existingM-matrices in the kernels of our equations, as is done in the BSformalism. Indeed, such an approach has been used in a similar, though not variational, treatmentof positronium by Zhang and Koniuk.15 These authors show that the inclusion of invariant matrixelements corresponding to single-loop diagrams yields positronium energy eigenstates which areaccurate toO(a5,a5 ln a). However such augmentation of the kernels ‘‘by hand’’ would be con-trary to the spirit of the present variational treatment, and we shall not pursue it in this work.

VI. SEMIRELATIVISTIC EXPANSIONS AND THE NONRELATIVISTIC LIMIT

For perturbative solutions of our radial equations, it is necessary to work out expansions of therelevant expressions to first order beyond the nonrelativistic limit. This shall be summarized in thepresent section. We perform the calculation in the Coulomb gauge, in which the photon propagatorhas the form16

D00~k!51

k2 , D0 j~k!50, Di j ~km!51

kmkmS d i j 2

kikj

k2 D , ~57!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

wherekm5(vp2vq ,p2q).To expand the amplitudesM of ~26! and~27! to one order of (p/m)2 beyond the nonrelativ-

istic limit, we take the free-particle spinors to be

u~p,i !5F S 11p2

8m2D~s•p!

2m

G w i , v~p,i !5F ~s•p!

2m

S 11p2

8m2D G x i , ~58!

as discussed in Appendix C. In this approximation the photon propagator takes on the form

D00~p2q!51

~p2q!2 , Di j ~p2q!.21

~p2q!2 S d i j 2~p2q! i~p2q! j

~p2q!2 D . ~59!

Corresponding calculations give for the orbital part of theM-matrix

Ms1s2s1s2

ope(orb) ~p,q!5 ie2H 1

~p2q!2 11

m2 S 1

41

q•p

~p2q!2 1~p3q!2

~p2q!4D J ds1s1ds2s2

. ~60!

The terms of the expansion linear in spin correspond to the spin–orbit interaction:

Ms1s2s1s2

ope(s–o)~p,q!53e2

4m2 ws1

† xs2

† ~s(1)2s(2)!•~p3q!

~p2q!2 ws1xs2

. ~61!

Here s(1) and s(2) are positron and electron spin matrices, respectively, defined as follows:s(1)ws1

xs25(s(1)ws1

)xs2, s(2)ws1

xs25ws1

(s(2)xs2). The quadratic spin terms or spin–spin

interaction terms are

Ms1s2s1s2

ope(s–s) ~p,q!5ie2

4m2 ws1

† xs2

† H 2~s(1)

•~p2q!!~s(2)•~p2q!!

~p2q!2 1s(1)•s(2)J ws1

xs2.

~62!

Lastly, the virtual annihilation contribution is given by

Ms1s2s1s2

ann ~p,q!52ie2

4m2 ws1

† xs2

† $s(1)•s(2)%ws1

xs2, ~63!

where we have excluded a divergent term, which appears in the Coulomb gauge calculation. Thisdivergence is an artifact of the Coulomb gauge. It does not arise, for example, in the Lorentzgauge, where only expression~63! is obtained. However the Lorentz gauge is not convenient forobtaining all otherO(a4) corrections because it contains spurious degrees of freedom~longitudi-nal polarization! of the photon.

We have used expressions~60!–~63! to obtain the corresponding radial kernels. Details of thecalculations can be found in Appendix D. We use the notationz5(p21q2)/2pq, andQl(z) is theLegendre function of the second kind.17 The contributions of the various terms to the kernel are asfollows:Singlet stateswith ,5J (J>0), P5(21)J11, C5(21)J.

Orbital term

K (sgl)(o)~p,q!52pe2

pqQJ~z!1

pe2

m2 S 2J23

2 S p

q1

q

pDQJ~z!1~J11!QJ11~z!22dJ,0D .

~64!

Spin–orbit interaction



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

K (sgl)(s–o)~p,q!50. ~65!

Spin–spin interaction

K (sgl)(s–s)~p,q!52pe2

m2 dJ,0 . ~66!

Triplet states with ,5J (J>1), P5(21)J11, C5(21)J11.Orbital term

K (tr)(o)~p,q!52pe2

pqQJ~z!1

pe2

m2 S 2J23

2 S p

q1

q

pDQJ~z!1~J11!QJ11~z! D . ~67!


K (tr)(s–o)~p,q!523pe2

m2

1

2J11$QJ11~z!2QJ21~z!%. ~68!


K (tr)(s–s)~p,q!5pe2

2m2 S p

q1

q

pDQJ~z!2pe2

m2

1

2J11$JQJ11~z!1~J11!QJ21~z!%. ~69!

Triplet states with ,5J21 (J>1), ,5J11 (J>0), P5(21)J, C5(21)J.The off-diagonal elements of the kernel matrix@Eqs. ~52!–~54!#, K12 and K21 which are

responsible for mixing of states with,5J21 and,5J11, get a nonzero contribution from thespin-spin interactions only:

K12~p,q!5K21~p,q!5pe2

5m2

AJ~J11!

~2J11! S p

qQJ11~z!1

q

pQJ21~z!22QJ~z! D . ~70!

The contributions to the diagonal elements of the kernel matrix are the following:Orbital terms

K 11(o)~p,q!5

2pe2

pqQJ21~z!1

pe2

m2 S 2J24

2 S p

q1

q

pDQJ21~z!1JQJ~z!22dJ21,0D , ~71!

K 22(o)~p,q!5

2pe2

pqQJ11~z!1

pe2

m2 S 2J22

2 S p

q1

q

pDQJ111~J12!QJ12D . ~72!


K 11(s–o)~p,q!5

3pe2

m2

J21

2J21~QJ~z!2QJ22~z!!, ~73!

K 22(s–o)~p,q!52

3pe2

m2c2

J12

2J13~QJ12~z!2QJ~z!!. ~74!


K 11(s–s)~p,q!5

pe2

2m2

1

2J11 S S p

q1

q

pDQJ21~z!22QJ~z! D , ~75!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

K 22(s–s)~p,q!5

pe2

2m2

1

2J13 S S p

q1

q

pDQJ11~z!22QJ12~z! D . ~76!

Annihilation term

K ann~p,q!522pe2

m2 dJ21,0. ~77!

We note that in the nonrelativistic limit only the first terms of the orbital part of the kernelssurvive. They have the common form 2p ie2Q,(z)/pq, hence all radial equations reduce to theform

~2vp2E! f ,~p!5m2e2

pvpp E0

`

dqq

vqQ,~z! f ,~q!. ~78!

Recalling, also, that

vp5Am21p2.mS 111

2 S p

mD 2D , ~79!

we obtain, in the nonrelativistic limit, the momentum-space Schro¨dinger radial equations

S p2

2m2« D f ,~p!5

a

p

1

p E0

`

dq q QJ~z! f ,~q!, ~80!

wherea5e2/4p, m5 m/2 , «5E22m.

VII. ENERGY LEVELS: FINE AND HYPERFINE STRUCTURE

The relativistic energy eigenvaluesEn,J can be calculated from the expression

EE0

`

dp p2f J~p! f J~p!5E0

`

dp p2 2vpf J~p! f J~p!

2m2

~2p!3 E0

` dpp2

vpE

0

`

dqq2

vqK (sgl,tr)~p,q! f J~p! f J~q! ~81!

for the singlet and,5J triplet states.For the,5J71 triplet states the corresponding result is@see Eq.~52!#

EE0

`

dp p2F†~p!F~p!5E0

`

dp p2 2vpF†~p!F~p!

2m2

~2p!3 E0

` dpp2

vpE

0

`

dqq2

vqK~p,q!F†~p!F~q!. ~82!

To obtain results forE to O(a4) we use the forms of the kernels expanded toO(p2/m2) @Eqs.~64!–~77!# and replacef ,(p) by their nonrelativistic~Schrodinger! form ~see~D10!, Appendix D!.The most important integrals that we used for calculating~81! and~82!, are given in Appendix D.In Appendix E we show that the contribution of kernelsK12 andK21 in ~82!, is zero atO(a4).Thus, the energy corrections for the triplet states with,5J21 and,5J11 can be calculatedindependently.

The results will be presented in the formD«5E22m1a2m/4n2.



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

A. Singlet states „øÄJ „JÐ0…, PÄ„À1…J¿1, CÄ„À1…J…

The kinetic energy corrections

D«K(sgl)52

a4m

8 S 1

2J11

1

n3 23

8

1

n4D . ~83!

The potential energy corrections

D«P(sgl)(o)52

a4m

8 S S 3

2J1122dJ,0D 1

n3 21

n4D , ~84!

D«P(sgl)(s–o)50, ~85!

D«P(sgl)(s–s)52

a4m

4

dJ,0

n3 . ~86!

The total energy corrections

D« (sgl)52a4m

8 S 4

2J11

1

n3 211

8n4D . ~87!

B. Triplet states „øÄJ „JÐ1…, PÄ„À1…J¿1, CÄ„À1…J¿1…


D«K(tr)52

a4m

8 S 1

2J11

1

n3 23

8

1

n4D . ~88!


D«P(tr)(o)52

a4m

8 S S 3

2J1122dJ,0D 1

n3 21

n4D , ~89!

D«P(tr)(s–o)52

a4m

8

3

J~J11!~2J11!

1

n3 , ~90!

D«P(tr)~s– s)5

a4m

8

1

J~J11!~2J11!

1

n3 . ~91!


D« (tr)52a4m

8 S S 4

2J111

2

J~J11!~2J11! D 1

n3 211

8

1

n4D . ~92!

C. Triplet states „øÄJÀ1 „JÐ1…, PÄ„À1…J , CÄ„À1…J…


D«K(tr)(J21)52

a4m

8 S 1

2J21

1

n3 23

8

1

n4D . ~93!




132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

D«P(tr)(o)(J21)52

a4m

8 F S 3

2J2122dJ,1D 1

n3 21

n4G , ~94!

D«P~ tr!(s–o)(J21)5

a4m

8

3~12dJ,1!

J~2J21!

1

n3 , ~95!

D«P~ tr!(s–s)(J21)52

a4m

8 S 12dJ,1

J~2J11!~2J21!2

2

3dJ,1D 1

n3 , ~96!

D« (ann)5a4m

4

1

n3 dJ,1 . ~97!


D«~ tr!(J21)52a4m

8 S S 4

2J212

2~3J11!

J~2J11!~2J21!22dJ,1D 1

n3 211

8

1

n4D . ~98!

D. Triplet states „øÄJ¿1 „JÐ0…, PÄ„À1…J , CÄ„À1…J…


D«K(tr)(J11)52

a4m

8 S 1

2J13

1

n3 23

8

1

n4D . ~99!


D«P(tr)(o)(J11)52

a4m

8 F 3

2J13

1

n3 21

n4G , ~100!

D«P~ tr!(s–o)(J11)52

a4m

8

3

~J11!~2J13!

1

n3 , ~101!

D«P~ tr!(s–s)(J11)52

a4m

8

1

~J11!~2J13!~2J11!

1

n3 . ~102!


D«~ tr!(J11)52a4m

8 S 2

2J13 S 213J12

~J11!~2J11! D 1

n3 211

8

1

n4D . ~103!

These results are in agreement with the well-known positronium fine structure results.18,19

VIII. CONCLUDING REMARKS

We have considered a reformulation of electrodynamics, in which covariant Green functionsare used to solve the field equations for the mediating electromagnetic field in terms of the fermionfield. This leads to a reformulated Hamiltonian with an interaction term in which the photonpropagator appears sandwiched between fermionic currents.

The variational method within a Hamiltonian formalism of quantum field theory is used todetermine approximate eigensolutions for bound relativistic fermion–antifermion states. The re-formulation enables us to use the simplest possible trial state to derive a relativistic momentum-space Salpeter-type equation for a positroniumlike system. The invariantM matrices correspond-ing to one-photon exchange and virtual annihilation Feynman diagrams arise directly in theinteraction kernel of this equation.



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

The trial states are chosen to be eigenstates of the total angular momentum operatorJ2 andJ3 , along with parity and charge conjugation. A general relativistic reduction of the wave equa-tions to radial form is given. For givenJ there is a single radial equation for total spin zero singletstates, but for spin triplet states there are, in general two coupled equations. We show how theclassification of states follows naturally from the system of eigenvalue equations obtained with ourtrial state.

It is not possible, as far as we know, to obtain analytic solutions of our relativistic radialequations nor the resulting eigenvalues of the particle–antiparticle system described. However, itis possible to obtainO(a4) corrections analytically for all states using perturbation theory. Theresults agree with well known results for positronium, obtained on the basis of the Bethe–Salpeterequation,19 which lends credence to the validity of our variationally derived equations.

The method presented here can be generalized to include effects higher order in alpha byusing dressed propagators in place of the bare propagators. This shall be the subject of a forth-coming work.

ACKNOWLEDGMENT

The financial support of the Natural Sciences and Engineering Research Council of Canadafor this work is gratefully acknowledged.

APPENDIX A: TOTAL ANGULAR MOMENTUM OPERATOR IN RELATIVISTIC FORM

The total angular momentum operator is defined by expression

J5E d3x:c†~x!~ L1S!c~x!:, ~A1!

where L5 x3p and S5 12s are the orbital angular momentum and spin operators. We use the

standard representation for the Pauli matrices

s5Fs 0

0 sG , ~A2!

s15F0 1

1 0G , s25F0 2 i

i 0 G , s35F1 0

0 21G . ~A3!

Using the field operatorc(x) in the form ~15!, after tedious calculations we obtain

J15E d3qS Lq1~bq↑† bq↑1bq↓

† bq↓1dq↑† dq↑1dq↓

† dq↓!

1 12 ~bq↑

† bq↓1bq↓† bq↑1dq↓

† dq↑1dq↑† dq↓!

D ,



† dq↓!

1i

2~2bq↑

† bq↓1bq↓† bq↑2dq↑

† dq↓1dq↓† dq↑!D , ~A4!



† dq↓!

1 12 ~bq↑

† bq↑2bq↓† bq↓1dq↑

† dq↑2dq↓† dq↓!

D .

Here Lq is the orbital angular momentum operator in momentum representation:

~ Lq! i[Lqi52 i ~q3¹q! i . ~A5!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

Note that these expressions are valid for anyt, since the time-dependent phase factors of the formeivqt cancel out.

For the operatorJ25 J121 J2

21 J32 we have

J25E d3qS ~ Lq21 3

4!~bq↑† bq↑1bq↓


† dq↓!

1Lq2bq↑† bq↓1Lq1bq↓

† bq↑1Lq2dq↑† dq↓1Lq1dq↓

† dq↑

1Lq3~bq↑† bq↑2bq↓


† dq↓!

D

11

2 E d3q8d3q

¨

2Lq8•LqS bq8↑† bq8↑dq↑

† dq↑1bq8↑† bq8↑dq↓

† dq↓1bq8↓

† bq8↓dq↑† dq↑1bq8↓

† bq8↓dq↓† dq↓

D1 1

2 ~bq8↑† bq8↑dq↑

† dq↑2bq8↑† bq8↑dq↓

† dq↓!

2 12 ~bq8↓

† bq8↓dq↑† dq↑2bq8↓

† bq8↓dq↓† dq↓!

1bq8↑† bq8↓dq↓

† dq↑1bq8↓† bq8↑dq↑

† dq↓

1Lq81S bq8↑† bq8↑dq↓

† dq↑1bq8↓† bq8↓dq↓

† dq↑1bq↓

† bq↑dq8↑† dq8↑1bq↓

† bq↑dq8↓† dq8↓

D1Lq82S bq8↑

† bq8↑dq↑† dq↓1bq8↓

† bq8↓dq↑† dq↓

1bq↑† bq↓dq8↑

† dq8↑1bq↑† bq↓dq8↓

† dq8↓D

1~ Lq831Lq3!~bq8↑† bq8↑dq↑

† dq↑2bq8↓† bq8↓dq↓

† dq↓!

2~ Lq832Lq3!~bq8↑† bq8↑dq↓

† dq↓2bq8↓† bq8↓dq↑

† dq↑!

©, ~A6!

where

Lq15Lq11 i L q2 , Lq25Lq12 i L q2 . ~A7!

The requirement that the trial state~20! be an eigenstate ofJ2 and Jz leads to the system ofequations

~ L311!F115mJF11,

L3F125mJF12,

L3F215mJF21, ~A8!

~ L321!F225mJF22,

~J~J11!2L22222L3!F115L2~F121F21!,

~J~J11!2L221!F125F211L1F111L2F22,

~J~J11!2L221!F215F121L1F111L2F22, ~A9!

~J~J11!2L22212L3!F225L1~F121F21!.

Substitution of the expressions~33! for Fs1s2and use of Eq.~A8! gives

m125m215mJ , m115mJ21, m225mJ11, ~A10!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

,115,225,125,215,, ~A11!

and

~J~J11!2,~,11!22mJ! f 11, ~p!5A~,2mJ11!~,1mJ! f 12

, ~p!1A~,2mJ11!~,1mJ! f 21, ~p!,

~J~J11!2,~,11!21! f 12, ~p!5 f 21

, ~p!1A~,1mJ!~,2mJ11! f 11, ~p!

1A~,2mJ!~,1mJ11! f 22, ~p!,

~A12!

~J~J11!2,~,11!21! f 21, ~p!5 f 12

, ~p!1A~,1mJ!~,2mJ11! f 11, ~p!

1A~,2mJ!~,1mJ11! f 22, ~p!,

~J~J11!2,~,11!12mJ! f 22, ~p!5A~,1mJ11!~,2mJ! f 12

, ~p!1A~,1mJ11!~,2mJ! f 21, ~p!.

The singlet states correspond to the solutionf 11, (p)5 f 22

, (p)50, f 12, (p)52 f 21

, (p) of thissystem with,5J (J>0).

For the triplet states the solutions aref 12, (p)5 f 21

, (p)[ f ,(p), and, for,5J21 (J>1):

~J2mJ! f 11J21~p!5A~J2mJ!~J1mJ21! f J21~p!, ~A13!

~J1mJ! f 22J21~p!5A~J1mJ!~J2mJ21! f J21~p!, ~A14!

for ,5J (J>1):

mJf 11J ~p!52A~J1mJ!~J2mJ11! f J~p!, ~A15!

mJf 22J ~p!5A~J2mJ!~J1mJ11! f J~p!, ~A16!

for ,5J11 (J>0):

~J111mJ! f 11J11~p!52A~J2mJ12!~J1mJ11! f J11~p!, ~A17!

~J112mJ! f 22J11~p!52A~J2mJ11!~J1mJ12! f J11~p!. ~A18!

It is convenient to introduce the table of coefficientsCJmJ

(tr),ms:

ms511 ms50 ms521

,5J21 A~J1mJ21!~J1mJ!

J~2J21!A~J2mJ!~J1mJ!

J~2J21!A~J2mJ21!~J2mJ!

J~2J21!

,5J2A~J1mJ!~J2mJ11!

J~J11!

mJ

AJ~J11!A~J2mJ!~J1mJ11!

J~J11!

,5J11 A~J2mJ11!~J2mJ12!

~J11!~2J13!2A~J2mJ11!~J1mJ11!

~J11!~2J13!A~J1mJ12!~J1mJ11!

~J11!~2J13!

These coefficients coincide with the usual Clebsch–Gordan coefficients forS51 except for afactor 2 in the denominator, which we absorb into the normalization constant.



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

APPENDIX B: PARITY AND CHARGE CONJUGATION

We consider the application of the parity operator to the trial state~20!:

PucT&5(s1s2

E d3pFs1s2~p!Pbps1

† d2ps2

† u0&5(s1s2

E d3pFs1s2~p!Pbps1

† P21Pd2ps2

† P21Pu0&.

~B1!

Making use of the properties

Pbps1

† P215hPb2ps1

† , Pd2ps2

† P2152hPdps2

† , Pu0&5u0&, ~B2!

wherehP is the intrinsic parity ((hP)251), it follows that

PucT&5(s1s2

E d3pFs1s2~p!Pbps1

† d2ps2

† u0&

52(s1s2

E d3pFs1s2~2p!bps1

† d2ps2

† u0&

5P(s1s2

E d3pFs1s2~p!bps1

† d2ps2

† u0&, ~B3!

where the parity eigenvalueP depends on the symmetry ofFs1s2(p) in different states:

For the singlet states (,5J) we get from ~36! Fs1s2(2p)5(21)JFs1s2

(p), so that P

5(21)J11.For the triplet states with,5J we get from ~38! Fs1s2

(2p)5(21)JFs1s2(p), henceP

5(21)J11.For the triplet states with,5J61 we get from~39! Fs1s2

(2p)5(21)J11Fs1s2(p), therefore

P5(21)J.Charge conjugation is associated with the interchange of the particle and antiparticle. Apply-

ing the charge conjugation operator to the trial state~20! we get

CucT&5(s1s2

E d3pFs1s2~p!Cbps1

† d2ps2

† u0& ~B4!

5(s1s2

E d3pFs1s2~p!Cbps1

† C21Cd2ps2

† C21Cu0&. ~B5!

Using the relations

Cbps1

† C215hCdps1

† , Cd2ps2

† C215hCb2ps2

† , Cu0&5u0&, ~B6!

where (hC)251, we obtain

CucT&5(s1s2

E d3pFs1s2~p!Cbps1

† d2ps2

† u0&

52(s1s2

E d3pFs2s1~p!bps1

† d2ps2

† u0&

5C(s1s2

E d3pFs1s2~p!bps1

† d2ps2

† u0&, ~B7!

where the charge conjugation quantum numberC depends on the symmetry ofFs1s2(p) in differ-

ent states:For the singlet states (,5J) we get from ~36! Fs1s2

(2p)5(21)J11Fs1s2(p), henceC

5(21)J.



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

For the triplet states with,5J we get from~38! Fs1s2(2p)5(21)JFs1s2

(p), thereforeC

5(21)J11.For the triplet states with,5J61 we get from~39! Fs1s2

(2p)5(21)J11Fs1s2(p), so that

C5(21)J.

APPENDIX C: EXPANSION OF THE SPINORS

We recall the form of the particle spinors:

u~p,i !5NpF 1

~s•p!

vp1mGw i , ~C1!

where

w15F10G , w25F01G , Np5Avp1m

2m. ~C2!

The antiparticle or ‘‘positron’’ representation for thev i(p) spinors has the form

v~p,i !5NpF ~s"p!

vp1m

1Gx i , ~C3!

where

x15F01G , x252F10G . ~C4!

The normalization is

u~p,i !u~p, j !5d i j , v~p,i !v~p, j !52d i j . ~C5!

Expanding in powers ofp/m and keeping the lowest non-trivial order terms,

~s•p!

vp1m.

~s•p!

2m, ~C6!

Np5Avp1m

2m.11

p2

8m2 , ~C7!

we obtain the result

u~p,i !.S 11p2

8m2D F 1

~s•p!

2mGw i5F S 11

p2

8m2D~s•p!

2m

G w i , ~C8!

v~p,i !.S 11p2

8m2D F ~s•p!

2m

1Gx i5F ~s•p!

2m

S 11p2

8m2D G x i . ~C9!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

APPENDIX D: SOME USEFUL IDENTITIES AND INTEGRALS

The following identity is useful for evaluating theM matrices:

~~p2q!•p!2

~p2q!4 5p2

~p2q!2 2~p3q!2

~p2q!4 . ~D1!

The angular integration in~47!, ~50!, ~55! involves the following integrals:

E dp dq F~ p"q!YJ8

mJ8~ q!YJmJ* ~ p!52pdJ8Jdm

J8mJE d~ p•q!F~ p•q!PJ~ p•q!, ~D2!

E d~ p•q!p•q

~p2q!2 PJ~ p•q!51

upuuqu S J11

2J11QJ11~z!1

J

2J11QJ21~z! D , ~D3!

E d~ p•q!~p3q!2

~p2q!4 PJ~ p•q!5~J11!~J12!

2~2J11!QJ11~z!2

J~J21!

2~2J11!QJ21~z!, ~D4!

whereF(p•q) is an arbitrary function ofp•q, PJ(x) is the Legendre polynomial, andQJ(z) is theLegendre function of the second kind of orderJ.

The following integrals are needed for the calculation of the relativistic energy corrections:

E0

È0

`

dp dq p2q2f J~p! f J~q!52pS am

n D 3

dJ,0 , ~D5!

E0

È0

`

dp dq pq fJ~p! f J~q!QJ~z1!5pam

n2 , ~D6!

E0

È0

`

dp dq p2q2f J~p! f J~q!QJ~z1!5E0

È0

`

dp dq p3q fJ~p! f J~q!QJ~z1!

5pS am

n D 3S 4

2J112

1

nD , ~D7!

E0

È0

`

dp dq p2q2f J~p! f J~q!QJ21~z1!5pS am

n D 3S 2

J2

1

nD , ~D8!

E0

È0

`

dp dq p2q2f J~p! f J~q!QJ11~z1!5pS am

n D 3S 2

J112

1

nD . ~D9!

Here f J is the nonrelativistic hydrogenlike radial wave function in momentum space19

f J~p![ f nJ~p!5S 2

p

~n2J21!!

~n1J!! D 1/2nJ12pJ22(J11)J!

~n2p211!J12 Gn2J21J11 S n2p221

n2p211D , ~D10!

whereGn2J21J11 (x) are Gegenbauer functions.

APPENDIX E: K12 , K21 KERNELS FOR øÄJÂ1 STATES

The contribution of the kernelK12 to the energy correction is

E dp dq p2q2K12~p,q! f J21~p! f J11~q!, ~E1!



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

where

K12~p,q!5 (s1s2s1s2

CJmJ12s1s2s1s2E dp dq Ms1s2s1s2

ope(s–s) ~p,q!YJ11

ms1s2~ q!YJ21

ms1s2* ~ p!. ~E2!

This requires the following integral:

(s1s2s1s2

CJmJ12s1s2s1s2E d3p d3q f J21~p!Y

J21

ms1s2* ~ p!Ms1s2s1s2

ope(s–s) ~p,q! f J11~q!YJ11

ms1s2~ q!. ~E3!

We calculate this form in coordinate space. The Fourier transform ofMs1s2s1s2(p,q) is

Ms1s2s1s2~p,q!5E d3r d3r 8Ms1s2s1s2

~r ,r 8!e2 i (p2q)•(r2r8), ~E4!

where theMs1s2s1s2(r ,r 8) matrix is a local operator in general,16 that is

Ms1s2s1s2~r ,r 8!5Ms1s2s1s2

~r !d~r2r 8!. ~E5!

We apply this transformation to theMs1s2s1s2

ope(s–s) (p,q) matrix @see Eq.~62!#. Because of the angular

integration in~E2!, only the first term in~62! survives. The Fourier transformation of that term is

~s(1)•~p2q!!~s(2)

•~p2q!!

4m2~p2q!2 → 3~s(1)

•r !~s(2)•r !

16pm2r 5 . ~E6!

Furthermore,

E d3pf J21~p!YJ21

ms1s2* ~ p!e2 ip•r5RnJ21~r !Y

J21

ms1s2* ~ r !, ~E7!

E d3qf J11~q!YJ11

ms1s2* ~ q!e2 iq•r5RnJ11~r !Y

J11

ms1s2~ r !, ~E8!

where

Rn,~r !52

2

n2A~n2,21!!

~~n1, !! !3 e2r /nS 2r

n D ,

Ln1,2,11S 2r

n D . ~E9!

The associated Laguerre functionLlm(r) is related to the confluent hypergeometric function by

Llm~r!5~21!m

~l! !2

m! ~l2m!!F~2l1m,m11;r!. ~E10!

The generating function for the Laguerre function is

Um~r,u![~21!mum

~12u!m11 expS 2ur

12uD5 (l5m

` Llm~r!

l!ul, ~E11!

hence



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

(s1s2s1s2

CJmJ12s1s2s1s2E d3p d3q f J21~p!Y

J21

ms1s2* ~ p!Ms1s2s1s2

ope(s–s) ~p,q! f J11~q!YJ11

ms1s2~ q!

5 (s1s2s1s2

CJmJ12s1s2s1s2E d3r Rn

J21~r !YJ21

ms1s2* ~ r !3S 3a~s(1)

•r !~s(2)•r !

16pm2r 5 DRnJ11~r !Y

J11

ms1s2~ r !

53a

16pm2 E dr r 21

r 3 RnJ21~r !Rn

J11~r !

3 (s1s2s1s2

CJmJ12s1s2s1s2E dr Y

J21

ms1s2* ~ r !~s(1)• r !~s(2)

• r !YJ11

ms1s2~ r !. ~E12!

It follows that

(s1s2s1s2

CJmJ12s1s2s1s2E drY

J21

ms1s2* ~ r !~s(1)• r !~s(2)

• r !YJ11

ms1s2~ r !51

15

AJ~J11!

2J11, ~E13!

but

E0

`

dr r 21

r 3 RnJ21~r !Rn

J11~r !50. ~E14!

The last expression can be proved in the following way. Let us consider the more general case

E0

`

dr r b12Rn,~r !Rn

,8~r !. ~E15!

The generating function forRn,(r ) is

Gn,~r ,u!522

n2A~n2,21!!

~~n1, !! !3 e2r /nS 2r

n D ,

~21!2,11u2,11

~12u!2,12 expH 2u

12u

2r

n J .

~E16!

Then we consider the expression

E0

`

drr b12Gn,~r ,u!Gn,8~r ,v !5E0

`

drr b124

n4A~n2,21!! ~n2,821!!

~~n1, !! !3~~n1,8!! !3 e22r /nS 2r

n D ,1,8

3u2,11v2,811

~12u!2,12~12v !2,812expH 2S u

12u1

v12v D 2r

n J5

4

n4A~n2,21!! ~n2,821!!

~~n1, !! !3~~n1,8!! !3

u2,11v2,811

~12u!2,12~12v !2,812

3E0

`

drS 2r

n D b121,1,8expH 2S 11

u

12u1

v12v D 2r

n J .

~E17!

It is well known that



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

E0

`

drrbe2r5G~b11!, ~E18!

therefore

E0

`

drS 2r

n D b121,1,8expH 2S 11

u

12u1

v12v D 2r

n J5S n

2D b13S ~12u!~12v !

12uv D b131,1,8G~b131,1,8! ~E19!

and

E0

`

drr b12Gn,~r ,u!Gn,8~r ,v !522b21

n2b11A~n2,21!! ~n2,821!!

~~n1, !! !3~~n1,8!! !3

3u2,11v2,811~12u!b112,1,8~12v !b111,2,8

~12uv !b131,1,8

3G~b131,1,8!. ~E20!

We expand this expression in a series,

E0

`

drr b12Gn,~r ,u!Gn,8~r ,v !5(hh8

Chh8~n,b,,,,8!uhuh8. ~E21!

It is not difficult to show20 that the coefficientCn1,,n1,8 represents the integral

Cn1,,n1,8~n,b,,,,8!5E0

`

drr b12Rn,~r !Rn

,8~r !. ~E22!

Simple but tedious calculations show that this coefficient is zero forb523, ,5J21, ,85J11. Thus the kernelK12 does not contribute to the energy corrections toO(a4). The same resultis obtained for the kernelK21.

1J. W. Darewych, Ann. Fond. Louis Broglie23, 15 ~1998!.2J. W. Darewych, inCausality and Locality in Modern Physics, edited by G. Hunteret al. ~Kluwer, Dordrecht, 1998!, p.333.

3J. W. Darewych, Can. J. Phys.76, 523 ~1998!.4M. Barham and J. W. Darewych, J. Phys. A31, 3481~1998!.5B. Ding and J. Darewych, J. Phys. G26, 907 ~2000!.6J. D. Jackson,Classical Electrodynamics~Wiley, New York, 1975!.7A. O. Barut,Electrodynamics and Classical Theory of Fields and Particles~Dover, New York, 1980!.8W. T. Grandy,Relativistic Quantum Mechanics of Leptons and Fields~Kluwer, Dordrecht, 1991!.9A. O. Barut, inGeometrical and Algebraic Aspects of Nonlinear Field Theory, edited by S. De Filippo, M. Marinaro, G.Marmo, and G. Vilasi~Elsevier, New York, 1989!, p. 37.

10J. W. Darewych and L. Di Leo, J. Phys. A29, 6817~1996!.11H. Pilkuhn, J. Phys. B28, 4421~1995!; Relativistic Quantum Mechanics~Springer, New York, 2003!.12J. W. Darewych and M. Horbatsch, J. Phys. B22, 973 ~1989!; 23, 337 ~1990!.13W. Dykshoorn and R. Koniuk, Phys. Rev. A41, 64 ~1990!.14T. Zhang and R. Koniuk, Can. J. Phys.70, 683 ~1992!.15T. Zhang and R. Koniuk, Can. J. Phys.70, 670 ~1992!.16V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevski,Quantum Electrodynamics~Pergamon, New York, 1982!.17G. Arfken and H. Weber,Mathematical Methods for Physicists~Academic, New York, 2001!.18M. A. Stroscio, Phys. Rep. C22, 5 ~1975!.19H. A. Bethe and E. E. Salpeter,Quantum Mechanics of One and Two-Electron Atoms~Springer-Verlag/Academic, New

York, 1957!.20M. Mizushima,Quantum Mechanics of Atomic Spectra and Atomic Structure~Benjamin, New York, 1970!.



132.174.255.116 On: Sat, 29 Nov 2014 21:12:21

variational derivation of relativistic fermion–antifermion wave equations in qed

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