Singularity-Free Breit Equation from Constraint Two-Body Dirac Equations

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ORNL-CTP-95-07 hep-ph/9603402

Singularity-Free Breit Equation from ConstraintTwo-Bo dy Dirac

Equations

Horace W. Crater

The University of TennesseeSpace Institute, Tul lahoma, Tennessee, 37388

Chun WaWong

Department of Physics and Astronomy, University of California, Los Angeles, CA 90095-1547

Cheuk-Yin Wong

Oak Ridge National Laboratory, Oak Ridge, TN 37831-6373

Abstract

We examine the relation b etween two approaches to the quantum relativis-

tic two-b o dy problem: (1) the Breit equation, and (2) the two-b o dy Dirac

equations derived from constraint dynamics. In applications to quantum elec-

tro dynamics, the former equation b ecomes pathological if certain interaction

terms are not treated as p erturbations. The diculty comes from singularities

which app ear at nite separations r in the reduced set of coupled equations

for attractive p otentials even when the p otentials themselves are not singular

there. They are known to give rise to unphysical b ound states and resonances.

In contrast, the two-b o dy Dirac equations of constraint dynamics do not have

these pathologies in many nonp erturbative treatments. To understand these 1

marked di erences we rst express these contraint equations, whichhavean

\external p otential" form similar to coupled one-b o dy Dirac equations, in a

hyp erb olic form. These coupled equations are then re-cast into two equiv-

alent equations: (1) a covariant Breit-like equation with p otentials that are

exp onential functions of certain \generator" functions, and (2) a covariant or-

thogonality constraint on the relative momentum. This reduction enables us

to show in a transparentway that nite-r singularities do not app ear as long

as the the exp onential structure is not tamp ered with and the exp onential

generators of the interaction are themselves nonsingular for nite r . These

Dirac or Breit equations, free of the structural singularities which plague the

usual Breit equation, can then b e used safely under all circumstances, encom-

passing numerous applications in the elds of particle, nuclear, and atomic

physics whichinvolve highly relativistic and strong binding con gurations. 2

I. INTRODUCTION

In contrast to the accepted description

[ p + m + V ] =0 (1.1)

of the relativistic quantum mechanics of a single spin-one-half particle moving in an exter-

nal p otential V given by Dirac, a numb er of di erent approaches have b een used for two

interacting spin-one-half particles. A traditional one based on the Breit equation [1], also

known as the Kemmer-Fermi-Yang equation [2],

[ p + m + p + m + V (r )] = E ; (1.2)

1 1 1 1 2 2 2 2 12

contains a sum of single-particle Hamiltonians and an interaction term b etween them. (E

is the total energy in an arbitrary frame.) Although the Breit equation is not manifestly

covariant, it has provided go o d p erturbative descriptions of the p ositronium and muonium

energy levels. However, it is well known that those parts of the Breit interaction

1 1

^ ^

(1 r r); (1.3) V (r )=

1 2 1 2

r 2 2

in the Breit equation b eyond the Coulomb term should not b e treated nonp erturbatively,

but must b e handled only p erturbatively. In other words, a consistent treatment of the Breit

equation in p owers of will generate unwanted terms not present in quantum electro dy-

namics [Refs. 3,4].

A more serious diculty of the Breit equation is that when the interactions are treated

nonp erturbatively, structural p ole singularities could app ear at nite r even when the in-

teractions themselves are singularity-free there [5,8]. Twoofushave found recently that

these p ole singularities o ccur under certain conditions dep ending on well-de ned algebraic

relations among the di erent p otentials that could app ear [6,7]. They lead to unphysical

states or unphysical resonances and therefore must b e strictly avoided [5,8].

The primary purp ose of this pap er is to showhow these p ole singulariti es can b e avoided

from the b eginning so that the Breit equation can b e used without dicultyindiverse 3

applications in particle, nuclear, and atomic physics involving highly relativistic motions

and strong binding p otentials. This is accomplished by relating this older approachtoone

that has b een develop ed much more recently.

Dirac's constraint Hamiltonian dynamics [9] provides a framework for an approach pro-

p osed by Crater and Van Alstine [10,11] that di ers notably from that of the Breit equation.

It gives two-b o dy coupled Dirac equations, each of which has an \external p otential" form

similar to the one-b o dy Dirac equation in that for four-vector and scalar interactions one

has

~ ~

S [ (p A )+m +S ] =0; (1.4a)

1 51 1 1 1 1 1

~ ~

S [ (p A )+m +S ] =0; (1.4b)

2 52 2 2 2 2 2

( ;i=1;2 are the matrices for the constituent particles). Unlike the Breit approach,

5i 5

these equations are manifestly covariant and haveinteractions intro duced by minimal sub-

stitution.

They have common solutions if the op erators S ; S commute. This situation, called

1 2

strong compatibility [9], can b e achieved by the prop er choice of the op erators A and

S [10,11]. The commutator cannot, however, b e made to vanish for more general typ es of

interactions such as pseudoscalar or pseudovector. Nevertheless, under certain circumstances

the commutator can b e reduced to a combination of S and S themselves. The equations

1 2

are then said to b e weakly compatible [9], b ecause this will also ensure that solutions of

S = 0 in the more general cases could b e solutions of S = 0 as well. These compatibility

1 2

prop erties are imp ortant b ecause they guarantee the existence of common solutions to (i.e.

the consistency of ) the constraint equations b efore they are actually solved.

Although the constrainttwo-b o dy (CTB) Dirac equations have b een used far less fre-

quently than the Breit equation, they have imp ortant advantages over the latter. In de-

scribing electromagnetic b ound states [11,12] they yield nonp erturbative and p erturbative

results in agreement with each other. That is, the exact (or numerical) solution pro duces 4

a sp ectrum that agrees through order with that given by p erturbative treatment of the

Darwin, spin-orbit, spin-spin, and tensor terms obtained from the Pauli reduction. In par-

+ 1

ticular, total c.m. energies w for the e e system in the J states is found to satisfy a

Sommerfeld formula [11,12]

t q

w = m 2+2 1+

1 1

2 2 2

n + (J + ) J )

2 2

2 4 4

m m m 11

=2m + + O ( ): (1.5)

2 3 4

4n 2n (2J +1) 64 n

They agree through order with those of the p erturbative solution of the same equation,

and also with those of standard approaches to QED. A recent pap er has numerically extended

this agreement at least to the n =1;2;3 levels for all allowable J and unequal masses [12].

In this pap er, we are concerned with another advantage of the CTB Dirac equations,

namely that no unphysical states and resonances of the typ e discussed in [6,7] haveever

app eared in past applications. We are able to show here that this is in fact true for a general

interaction and that this is a consequence of the exp onential structure of the interactions

app earing in them. This result is obtained by rst reducing the CTB Dirac equations

to a Breit equation and an equation describing an orthogonality constraint. The equivalent

Breit equation is then shown to b e singularity-free, provided that the exp onential interaction

structure is not destroyed by inadvertent approximations and that the op erators app earing

in the exp onent are themselves free of nite r singularities.

The exp onential structure that tames the unphysical singularities turns out to b e a

consequence of a relativistic \third law" describing the full recoil e ects b etween the two

interacting particles. We carefully trace, in the formulation of the CTB Dirac equations,

how this structure arises from the need to make these equations at least weakly compatible.

Compared to the laissez-al ler approach of the Breit equation for whichanyinteraction

seems p ossible, the restriction of the interaction structure needed in the constraint approach

represents a conceptual advance in the problem. For this reason, we take pain to elucidate

its conceptual foundation as we present elements of the CTB Dirac equations needed for our 5

demonstration that they are singularity-free.

This pap er is organized as follows. We start in Sec. I I with a brief overview of the

derivation of the constrainttwo-b o dy Dirac equations for scalar interactions b oth to de ne

the notation used and to describ e the main concepts involved. One of the most imp ortant

prop erties is the compatibility of the two constraints. We remind the reader in Sec. III

howtointro duce general interactions into them that preserve this prop erty. In Sec.IV we

derivea covariantversion of the Breit equation from the constraint equations, intro ducing

the concept of exp onential generators for a wide range of covariantinteractions. In Sec. V

we clarify the structure of this covariant Breit equation by decomp osing it into a matrix form

involving singlet and triplet comp onents of the matrix wave function followed by reduction

to radial form by the use of a vector spherical harmonic decomp osition. This reveals clearly

how the constraint approachavoids the structural p ole singularities that have plagued the

original Breit equation since its intro duction over 65 years ago. In Sec. VI we showby

contrast, how the p ole singularities arises for each of the nonzero generators if one uses the

original Breit interaction. Section VI I contains brief concluding remarks.

II. TWO-BODY DIRAC EQUATIONS OF CONSTRAINT DYNAMICS

Following To dorov [13], we shall use the following dynamical and kinematical variables

for the constraint description of the relativistic two-b o dy problem:

i.) relative p osition, x x

1 2

ii.) relative momentum, p =( p p )=w ;

2 1 1 2

iii.) total c.m. energy, w = P ;

iv.) total momentum, P = p + p ;

1 2

and v.) constituent on-shell c.m. energies,

2 2 2 2 2 2

w + m m w + m m

1 2 2 1

= ; = : (2.1)

1 2

2w 2w

In terms of these variables, wehave 6

^ ^

p = P + p; p = P p; (2.2)

1 1 2 2

where P = P=w.

We start from the (compatible) Dirac equations for two free particles

S =( p +m ) =0; (2.3a)

10 1 1 1 51

S =( p +m ) =0; (2.3b)

20 2 2 2 52

where is just the pro duct of the two single-particle Dirac wave functions. These equations

can b e written as

S =( p+ P +m ) =0; (2.4a)

10 1 1 1 1 51

S =( p+ P +m ) =0; (2.4b)

20 2 2 2 2 52

when expressed in terms of the To dorovvariables. The \theta" matrices

i ;=0;1;2;3 ;i =1;2 (2.5)

i i

i (2.6)

5i 5i

satisfy the fundamental anticommutation relations

[ ; ] = g ; (2.7)

[ ; ] =0; (2.8)

5i +

[ ; ] = 1: (2.9)

5i 5i +

[Pro jected \theta" matrices then satisfy

^ ^

[ P; P] =1; (2.10)

i i + 7

[ P; ] =0; (2.11)

i +

^ ^

where = ( + P P )]. They are mo di ed Dirac matrices [14] which ensure that the

Dirac op erators S and S are the square ro ot op erators of the corresp onding mass-shell

10 20

1 1

2 2 2 2

op erators (p (p + m + m ) and ):

1 2 1 2

2 2

Using the To dorovvariables and the ab ove brackets, the di erence

2 2 2 2 2 2

(p +m p m ) (2.12) (S S ) =0=

1 1 2 2 10 20

leads to an equation

2 2

P p = [w ( ) (m m )] =0: (2.13)

1 2

The physical signi cance of the orthogonality of these two momenta is to put a constraint

on the relative momentum ( eliminating the relative energy in the c.m. frame).

We will also use covariant (c.m. pro jected) versions of the Dirac and matrices here

de ned by

^ ^

= P =2 P; (2.14)

i i 5i i

=2 P; (2.15)

and

= =2 2i P ; i=1;2: (2.16)

5i 5i i ?i

i i

These covariant Dirac matrices take on the simple form =(0; ) and =(0; )inthe

i i

center-of-mass system for which P =(1;0).

If wenowintro duce scalar interactions b etween these particles by naively making the

minimal substitutions

m ! M = m + S ;i=1;2 (2.17)

i i i i

(as done in the one-b o dy equation), the resulting Dirac equations 8

S =( p+ P +M ) =0; (2.18a)

1 1 1 1 1 51

S =( p+ P +M ) =0 (2.18b)

2 2 2 2 2 52

will not b e compatible b ecause

[S ; S ] =[ p; M ]+ [M ; p] = i(@M + @M ) 6=0; (2.19)

1 2 1 2 52 1 51 2 1 1 52 2 2 51

where @ is the four-gradient.

In the earlier work [10,15], compatibility is reinstated by generalizing the naive S and S

1 2

op erators with the help of sup ersymmetry arguments. The pro cedure contains four ma jor

steps:

a) Find the sup ersymmetries of the pseudo classical limit of an ordinary free one-b o dy

Dirac equation.

b) Intro duce interactions of a single Dirac particle with external p otentials that preserve

these sup ersymmetries. For scalar interactions, this requires the co ordinate replacement

x ! x~ x + i : (2.20)

m + S (~x)

(Note that the Grassmann variables satisfy = 0. As a result this self referent or recursive

relation has a terminating Taylor expansion).

c) Maintain the one-b o dy sup ersymmetries for each spinning particle through the re-

placement

(x x ) ! (~x x~ ) (2.21)

1 2 1 2

in the relativistic p otentials S .

These steps lead to the pseudo classical constraints (the weak equality sign means these

equations are constraints imp osed on the dynamical variables)

S =( p+ P +M i@ M =M ) 0; (2.22a)

1 1 1 1 1 51 2 1 2 52 51

S =( p+ P +M + i@ M =M ) 0: (2.22b)

2 2 2 2 2 52 1 2 1 52 51 9

They are strongly compatible under the following two conditions:

i.) the mass p otentials are related through a relativistic \third law"

2 2

@ (M M )=0; (2.23)

1 2

and ii.) they dep end on the separation variable only through the space-like pro jection

p erp endicular to the total momentum

M = M (x ); (2.24)

i i ?

where

^ ^

x =( + P P )(x x ) : (2.25)

1 2

Integration of the \third law" condition yields

2 2 2 2

M M = m m (2.26)

1 2 1 2

with the hyp erb olic solution

M = m coshL + m sinh L; M = m coshL + m sinh L; (2.27)

1 1 2 2 2 1

given in terms of a single invariant function L = L(x ).

The x dep endence of the p otential and the relativistic \third law" lie at the heart

of two-b o dy constraint dynamics. Without these conditions the constraints would not b e

compatible. While the physical imp ortance of the x dep endence lies in its exclusion of the

relative time in the c.m. frame, the \third law" condition relates the mutual interactions

between the particles to the e ective p otentials each particle feels in the presence of the

other in a consistentway. It is useful to show its implications in the simpler case of spinless

particles. The two generalized mass shell constraints that are the counterparts of Eq. (2.22)

for scalar interactions are

2 2

H = p + M 0; i =1;2: (2.28)

i i

The compatibility condition for these two constraints involves the classical Poisson bracket 10

2 2 2 2 2 2

[H ; H ]=[p ;M ]+[M ;p ]+ [M ;M ] 0: (2.29)

1 2

1 2 1 2 1 2

One can see that this is satis ed provided that the \third law" condition Eq. (2.23) and

condition (2.24) are satis ed. (Although the \third law" solution Eq. (2.26) combined with

Eq. (2.24) is the simplest solution, it is not unique [15 ].)

For scalar interactions parametrized by

M = m + S ; i =1;2; (2.30)

i i i

the \third law" condition b ecomes

m S = m S ; (2.31)

1 1 2 2

in the nonrelativistic limit (jS j << m ). This result can also b e obtained from Eq. (2.27)

i i

bykeeping only terms linear in L. The two constraints (2.28) can now b e written as

2 2 2 2 2 2

p + M p +2m S +S +m =0; (2.32)

i i

i i i i i

where wehave used the fact that H H = P p 0 remains unchanged up on the

1 2

intro duction of scalar interaction in Eq. (2.24). Hence the total c.m. energy w = +

1 2

takes on a familiar form in the nonrelativistic limit

w = m + m + + S + O (S ); (2.33)

1 2

where

S =(m +m )S =m =(m +m )S =m : (2.34)

1 2 1 2 1 2 2 1

d) The nal step is to canonically quantize the classical dynamical system de ned by S

and S by replacing the Grassmann variables ; ;i=1;2 with twomutually commuting

2 i 5i

sets of theta matrices, and the p osition and co ordinate variables by op erators satisfying the

fundamental commutation relation

fx ;p g![x ;p ]= i : (2.35) 11

The compatible pseudo classical spin constraints S and S then b ecome commuting quan-

1 2

tum op erators

[S ; S ]= 0: (2.36)

1 2

The resulting CTB Dirac equations for scalar interactions

S =( p+ P +M i@ L ) =0; (2.37a)

1 1 1 1 1 51 2 52 51

S =( p+ P +M + i@ L ) =0; (2.37b)

2 2 2 2 2 52 1 52 51

where

@M @M

2 1

= ; (2.38) @L =

M M

2 1

are then said to b e strongly compatible. This strong compatibility has b een achieved bya

sup ersymmetry which pro duces the extra spin-dep endent recoil terms involving @L. These

extra terms vanish, however, when one of the particles b ecomes in nitely massive (as seen

by the parametrization M = m + S of the scalar p otential) so that we recover the exp ected

i i i

one-b o dy Dirac equation in an external scalar p otential.

Note that the Dirac constraint op erators satisfy [10]

2 2 2 2 2 2

S S = (p +m p m )=P p 0: (2.39)

1 2 1 1 2 2

Thus the relative momentum remains orthogonal to the total momentum after the intro-

duction of the interaction. This also implies that the constituent on-shell c.m. energies

are weakly equal to their o mass shell values (P p ). Notice further that since

i i

[P p; M (x )] P x 0; this constraint do es not violate the requirement of compatibility

? ?

given in Eqs. (2.24-25).

III. A GENERAL INTERACTION FOR TWO-BODY DIRAC EQUATIONS

The previous work [16] has shown how the compatibility problem can b e solved without

having to invent new sup ersymmetries if the scalar p otential is replaced byvector, pseu-

doscalar, pseudovector, or tensor p otentials. That work also relates the sup ersymmetric or 12

\external p otential" approach to the alternative treatment of the two b o dy Dirac equations

of constraint dynamics presented by H. Sazdjian [17].

The \external p otential" form Eqs. (2.37) of the CTB Dirac equations for scalar inter-

action can b e rewritten in the hyp erb olic form [16]

S = (cosh S + sinh S ) =0; (3.1a)

1 1 2

S = (cosh S + sinh S ) =0; (3.1b)

2 2 1

where generates the scalar p otential terms in (2.37) provided that

= L(x ): (3.2)

51 52 ?

The op erators S and S are auxiliary constraints of the form

1 2

S (S cosh + S sinh ) =0; (3.3a)

1 10 20

S (S cosh + S sinh ) =0: (3.3b)

2 20 10

Toverify that the \external p otential" forms Eq. (2.37) result from using Eqs. (3.3) in

Eqs. (3.1), one simply commutes the free Dirac op erator S to the rightonto the wave

function using Eqs. (2.7-2.11), (2.38) and hyp erb olic identities [16]. With this construction,

the interaction enters only through an invariant matrix function with all other spin-

dep endence a consequence of the factors contained in the kinetic free Dirac op erators S

and S .

Even though the form of the contraints Eqs. (3.1) and (3.3) were motivated by exam-

ining world scalar interactions, let us prop ose them for arbitrary and determine their

compatibility requirements. We do this for arbitrary interactions by generalizing arguments

given in Refs.[16-17]. First consider the conditions for the compatibility of Eqs. (3.3a-b).

Multiplying Eq. (3.3a) by S and Eq. (3.3b) by S and subtracting we obtain

10 20

P p (cosh ) =0: (3.4a) 13

Multiplying Eq. (3.3b) by S and Eq. (3.3a) by S and subtracting we obtain

10 20

P p (sinh ) =0: (3.4b)

2 2

Wehave used Eq. (2.2) and =(m m )=w to simplify these equations. Multiplying

1 2

Eq. (3.4a) by sinh , Eq. (3.4b) by cosh, bringing the op erator P p to the right and

subtracting we nd the condition

[P p; ] =0: (3.5)

Multiplying Eq. (3.4a) by cosh, Eq. (3.4b) by sinh , bringing the op erator P p to the

right and subtracting we nd the further condition

P p =0: (3.6)

Notice that this latter condition was previously asso ciated with the \third law" condition

when derived from the \external p otential" forms of the constraints (see Eq. (2.39)). Here

the \third law" condition is built into the constraintbyhaving the same generator for Eqs.

(3.3a) and (3.3b). Thus the two tentative constraints Eqs. (3.3a) and (3.3b) taken together

imply that for arbitrary the orthogonality condition P p 0 has to b e satis ed when

acting on .However, in order to verify that there are no additional conditions b eyond Eqs.

(3.5) and (3.6) wemust check for mathematical consistency by examining the compatibility

condition. We compute the commutator [S ; S ]by rearranging its eight terms and nd that

1 2

[S ; S ]= [S ; cosh]S [S ; cosh]S +[S ; sinh ]S [S ; sinh ]S

1 2 10 2 20 1 20 2 10 1

2 2 2 2

+cosh (S S )sinh sinh (S S )cosh (3.7)

10 20 10 20

do es not in general vanish, unlike Eqs. (2.36) and (2.37). By using Eqs. (2.12) and bringing

the op erator P p to the right, and using the conditions given in Eqs. (3.5) and (3.6) we can

reduce [S ; S ] to only terms involving S and S . Since Eqs.(3.5) and (3.6) follow from

1 2 1 2

combining the constraints S = 0, no further conditions for mathematical consistency need

b e imp osed on the constraints or the wave function. Eq. (3.6) is the quantum counterpart of 14

Eq. (2.39) but for arbitrary interactions. Eq. (3.5) is also satis ed for arbitrary provided

only that the generator satis es

= (x ) (3.8)

generalizing Eq. (2.24).

The weak compatibility of the \external p otential" form of the constraints Eq. (3.1) for

general

[S ; S ] =0 (3.9)

1 2

can b e seen by examining the four commutators in [S ; S ]. The commutator

1 2

[cosh S ; cosh S ] = cosh(cosh[S ; S ]+[S ;cosh]S + [cosh; S ]S ) (3.10)

1 2 1 2 1 2 2 1

and is weakly zero since S = 0 and [S ; S ] = 0. Likewise, wehave

i 1 2

[sinh S ; sinh S ] =0: (3.11)

2 1

The remaining two brackets are

[cosh S ; sinh S ] [sinh S ; cosh S ]

1 1 2 2

=[S ;sinh ()]S sinh()[cosh(); S ]S +(1 ! 2): (3.12)

1 1 1 1

Since they contain the constraints on the rightwe obtain Eq. (3.9) after combining with

Eqs. (3.10) and (3.11).

One nal feature should b e mentioned. Eqs. (3.1) and (3.3) are also applicable for a sum

of the four \p olar" interactions

= + + + ; (3.13)

1 L J G F

where

= L scalar; (3.14)

L 51 52 15

^ ^

= J P P time likevector; (3.15)

J 1 2

= G space likevector; (3.16)

G 1? 2?

^ ^

=4F P P tensor (p olar ): (3.17)

F 1? 2? 51 52 1 2

For the sum

= + + + (3.18)

2 C H I Y

of their axial counterparts

= C=2; pseudoscalar; (3.19)

^ ^

= 2H P P time like pseudovector; (3.20)

H 1 2 51 52

=2I space like pseudovector; (3.21)

I 1? 2? 51 52

^ ^

= 2Y P P tensor (axial ); (3.22)

Y 1? 2? 1 2

the sinh terms in Eq. (3.1) should carry negative signs instead [16]. In contrast, Eq. (3.3)

as written remains valid as is for .For systems with b oth p olar and axial interactions

[16], one uses in (3.1) and + in (3.3). The terms L; J; G ; F ;C;H;I;Y are

1 2 1 2

arbitrary invariant functions of x .

Eq. (3.3) is more convenient to use for the construction of Breit-like equations from the

constraint formalism for general interactions. Eq. (3.1) is more convenient if one aims to

obtain a set of Dirac equations in an \external p otential" form similar to that exhibited in

the one b o dy Dirac equation with the transformation prop erties one would exp ect for such

p otentials. Wehave already seen this for the scalar case in which the scalar interaction

\generator" L in (3.14) in the hyp erb olic form leads to a mo di cation of the mass term. In 16

the case of combined scalar, time- and space-likevector and pseudoscalar interactions, we

use the hyp erb olic parametrization

M = m coshL + m sinh L; (3.23a)

1 1 2

M = m coshL + m sinh L; (3.23b)

2 2 1

E = coshJ + sinh J; (3.24a)

1 1 2

E = cosh J + sinh J; (3.24b)

2 2 1

G = e ; (3.25)

where L; J; and G generate scalar, time-likevector and space-likevector interactions resp ec-

tively, while C generates pseudoscalar interactions. The resultant \external p otential" form

for

= + + + (3.26)

J L G C

S =(G p + E P + M

1 1 1 1 1 51

^ ^

+iG( @ G + @J P P @L + @C=2)) =0 (3.27a)

2 1? 2? 2 1 2 2 51 52 2

S =(G p + E P + M

2 2 2 2 2 52

^ ^

iG( @ G + @J P P @L + @C=2)) =0: (3.27b)

1 1? 2? 1 1 2 1 51 52 1

The scalar generator pro duces the mass or scalar p otential M terms, the time-likevector

generator pro duces the energy or time-like p otential E terms, the space-likevector generator

i 17

pro duces the transverse or space-like momentum G terms, while the pseudoscalar generator

pro duces only spin-dep endent terms. Note that the vector and scalar interactions also

have additional spin-dep endent recoil terms essential for compatibility. The ab ove features

are just what one would exp ect for interactions transforming in this way and are direct

consequences of the hyp erb olic parametrization of the contraints. Other parameterizations

may not have this imp ortant prop erty.

IV. REDUCTION TO A BREIT EQUATION

Wenow derive a Breit equation from the CTB Dirac equations (3.3). Consider the

^ ^

combination 2( P S + P S ). The terms prop ortional to w = + are

1 1 2 2 1 2

^ ^

w [cosh + 2( P P) sinh ] = w exp(D ) (4.1)

1 2

where

^ ^

D =2( P P): (4.2)

1 2

The other terms can also b e written in terms of D , using the relations

^ ^

cosh D = cosh ; sinh D =2 P P sinh ; (4.3)

1 2

^ ^ ^ ^

(2 P p 2 P p) sinh = (2 P p 2 P p) sinh D ; (4.4)

2 1 1 2 1 1 2 2

and

^ ^ ^ ^

(2 P m +2 P m ) sinh = (2 P m +2 P m ) sinh D : (4.5)

2 51 1 1 52 2 1 51 1 2 52 2

^ ^

Then the combination 2( P S + P S ) of the twohyp erb olic equations in Eq. (3.3)

1 1 2 2

takes the simple form

w exp(D ) =(H + H )exp(D ) (4.6)

10 20

where 18

^ ^ ^

H = 2 P p 2 P m = p + m + P p (4.7)

10 1 1 1 51 1 1 ? 1 1

^ ^ ^

H =2 P p 2 P m = p + m P p (4.8)

20 2 2 2 52 2 2 ? 2 2

^ ^

(with the de nition p p + Pp P) are covariant free Dirac Hamiltonians involving the

covariant and matrices given previously in Eqs. (2.14) and (2.15). If we take

= exp(D ) (4.9)

we obtain nally the manifestly covariant Breit-like equation

w exp(2D ) = (H + H ) : (4.10)

10 20

The interactions app ear in the Breit equation in the exp onentiated form exp(2D ), where

2D contains all eightinteractions shown in and . We can even add momentum-

1 2

indep endentinteractions prop ortional to r^ r^ to 2D to give the more general interaction

1 2

2D = J L + C + H + (I + Y + F + G )

1 2 1 2 51 52 1 2 1 2 1 2 51 52

+ r^ r^(N + T + S + R); (4.11)

1 2 1 2 1 2 51 52

where

= ; (4.12)

i i 5i

the covariant is from Eq. (2.16),r ^ = x =jx j, and N; T ; S , and R are arbitrary invariant

? ?

functions of x . Note that each term in 2D involves identical op erators for particle 1 and

particle 2. As a result, they all commute with each other. For example, wehave

[ r^ r;^ ]= 0: (4.13)

1 2 1 2

Hence the a single exp onential function 2D can also b e written as a pro duct of separate

exp onentials.

Before continuing our discussion on the structure of the covariant Breit equation, we

consider the remaining linear combination of the two constraint equations (3.3) involving 19

^ ^

the di erence 2( P S P S ). Using identities in the b eginning of this section we nd

1 1 2 2

that the di erence equation b ecomes

( )exp(D ) =(H H )exp(D ) : (4.14)

1 2 10 20

Transforming to and using the Breit equation (4.10) gives

2 2 2 2

(H H (m 2(H + H )P p ) m )

10 20

10 20 1 2

( ) = = + : (4.15)

1 2

w w w

Combined with Eq. (2.13) this gives

P p =0; (4.16)

con rming the exp ectation that these momenta remain orthogonal after the interaction is

intro duced. This result has also b een obtained recently by Mourad and Sazdjian [18] who

emphasize that this would ensure the Poincaire' invariance of the theory. They further p oint

out that the c.m. energy dep endence of the p otential in the \main equation" (our Eq. (4.10))

ensures the global charge conjugation symmetry [18] of the system, a feature that is missing

from the original Breit equation.

In summary, the constraint equations imply covariant Breit-like equations of a certain

form (4.10) whose wave function also satis es the constraint equation (4.16). Alternatively,

if we start o with a covariant Breit-like equation of the form (4.10) with D = D (x ) and

require simultaneously that P p =0,we can work backward to obtain the two compatible

constraint equations (3.3).

We p oint out nally that the application of the constraint equations to electromagnetic

interactions do es not involve the term r ^ r^ app earing in the Breit interaction Eq. (1.2).

1 2

Nevertheless, it do es pro duce the correct sp ectrum, as shown in [11,12] using the \external

p otential" form in Eqs. (3.27) of the two-b o dy Dirac equations with L = C =0, J = G .

As has b een recently noted [19], in the context of the Breit-like form Eq. (4.10) of that

equation, one obtains the reduction

(H + H + V + V + V + V ) = w ; (4.17)

10 20 1 2 1 2 3 51 52 4 1 2

which contain pseudovector terms in place of the vector Breit term ( r ^ r^).

1 2 20

V. STRUCTURE OF THE BREIT EQUATION

The Breit equation (4.10) can b e written, as usual, as a set of coupled equations for

di erent comp onents of the wave function contained in . Wework in the center-of-mass

system for which P =(1;0), =(0;), andr ^ =(0;r). We b egin by simplifying the general

interaction (4.11) to the more compact form

(1) (2)

^ ^

2D = (A + B + r rC )q q (5.1)

1 2 1 2

where the sup erscripts (1) and (2) lab el the interacting particles 1 and 2. The op erators

(q ;q ;q ;q )=(1; ;i; ) (5.2)

0 1 2 3 5

are de ned so that q ;q and q are analogous to the Pauli matrices , and satisfying

1 2 3 1 2 3

q q = + i q (5.3)

i j ij ij k k

where i; j and k =1; 2, and 3. Eq. (5.1) is in the form of four-scalar pro ducts

^ ^

2D =(A+ B+ r r C) Q (5.4)

1 2 1 2

(1) (2)

involving the \four-vector" Q = q q , and

A =(J;H;C; L);B=(I; G; F ;Y );C=(N; R; S; T ): (5.5)

The wave function in Eq. (4.10) can b e written as a spinor or column vector with two

indices, one for each particle

(1) (2)

= : (5.6)

It is however more convenient to express the content of the wave function in terms of a

new 44 matrix wave function

0 (1) (2)T

= ; (5.7)

(2)

where has b een transp osed and an op erator has b een added on the right, as explained

(1) (2)

b elow. We can represent the op eration of A for particle 1 and B for particle 2 acting 21

on the original spinor wave function in terms of op erations A and B on the new wave

function ,

(1) (2) (1) (2) T 0 0

A B !A [B ] = A B ; (5.8)

where the matrix op erator B is

0 T

B = B : (5.9)

y y

(1) (2)

The arrow in Eq. (5.8) indicates the transformation of the op eration of A B on the wave

0 0

function to the op eration of A and B with resp ect to the new wave function . The

(2)

op erator in Eq. (5.7) insures that op erators suchas acting on the second particle is

represented by

(2) 0 T 0

! = ; (5.10)

y y

where the same negative sign app ears for the di erent comp onents of the op erator .By

using the wave function and Eq. (5.8), an op erator acting on the rst particle app ears

on the left side of the wave function , while an op erator O acting on the second particle

T 0

b ecomes O and app ears on the right side of .

y y

In this matrix notation, the righthand side (RHS) of Eq. (4.10) b ecomes

0 0 0 0

(H + H ) ! p + p + m m RHS : (5.11)

10 20 1 2

The reduction of Eq. (4.10) is facilitated by separating the matrix wave function into two

parts:

0 0 0

q ; (5.12) = + ! = +

0 1 0 1

;=0

where =1, = [20],

0 5

= q ; (5.13a)

and 22

3 3 3

X X X

= q = q : (5.13b)

=0 =0

These parts give di erent results when op erated on by ,

1 2

0 0 0

! = 3 + : (5.14)

1 2

0 1

0 0

This shows that ( ) and ( ) are resp ectively the spin-singlet and spin-triplet parts

0 1

of the wave function. Furthermore, all the op erators in Eq. (4.11) commute. With (

^ ^

r r) = 1, the tensor interaction can b e written as

r^ r^CQ

^ ^

e = cosh(C Q)+ r r sinh(C Q) : (5.15)

1 2

This means that interaction (4.11) alone will not mix spin-singlet and spin-triplet states (as

seen b elow in Eq.(5.34), the kinetic energy term do es mix singlet and triplet states).

The reduction to the matrix form is easier for the spin-singlet wave function

(1 ) b ecause of its simpler spin structure: =

1 2 0

(A+ B + r^ r^ C)Q TQ

1 2

e = e ; (5.16)

0 0

where T = A 3B C .ATaylor expansion of the exp ontial op erator on the ab ove equation

shows that it is necessary to evaluate the basic op eration of the typ e

(1) (2)

T Q = T q q : (5.17)

0 0

In terms of the new wave function of Eq. (5.13a), the ab ove equation can b e represented

in the matrix form:

3 3 3

X X X

T (q q q ) = T S q ; (5.18)

0 0

=0 =0 =0

0 T

where q = q = q , with ( ; ; ; )=(1;1;1;1) and wehaveintro duce the

y y 0 1 2 3

quantity(S ) de ned by

=(S ) q ; (5.19a) q q q

and 23

(S ) = [1 + 2(1 0)(1 )( 1)] : (5.19b)

The 4 4 matrix S is called a signature matrix b ecause its matrix elements can only b e +1

or 1. The rowvectors of S are

S =(1; 1; 1;1);

S =(1; 1;1; 1);

S =(1;1; 1; 1);

S =(1;1;1;1) : (5.20)

Eq. (5.18) can b e applied rep eatedly to give the desired result

T Q n

we (T Q) = w

0 0

n=0

T S

! w q LHS : (5.21) e

0 0

It can also b e used to prove the general result

f (T Q) ! f (T S )q : (5.22)

0 0

The treatment of the spin-triplet expression

2D (A+B )Q

^ ^

e = e [cosh(C Q)+ r r sinh (C Q)] (5.23)

1 1 2 1

is simpli ed by noting that the q and matrices are indep endentofeach other. Hence the

Q dep endences can b e eliminated in favor of the signature vector S with the help of Eq.

(5.19a), which also applies to the q structure of . This leaves the spin-dep endent part

which has the form

^ ^ ^ ^

r r !r r

1 2 1 1

^ ^

= q ( 2 r r ) : (5.24)

Hence, using Eq. (5.15), the spin-triplet part of the left hand side of Eq. (4.10) is 24

(A+B )S C S

^ ^

LHS = w e q [e 2 sinh (C S ) r r ] : (5.25)

The Breit equation (4.10) for the matrix wave function is thus

LHS + LHS = RHS (5.26)

0 1

where the expressions are from Eqs. (5.11), (5.21), and (5.25).

Eq. (5.26) can b e written explicitly as

3 3

X X

W (1 )V r^ r^ q

3 3

X X

; (5.27) p + m q q m q q = q + q

i 1 3 2 3 i i

i=1

where

(A+B +C )S (A3BC)S

; (5.28a) +w(1 )e W = we

0 0

(A+B )S

sinh (C S ) : (5.28b) V =2we

Multiplying the equation from the rightby q and taking traces we nally get a set of 16

coupled equations for the wave function comp onents,

W (1 )V r^ r^

0 0

3 3 3

X X X

0 0 0 0 0 0 0 0 0

= (1 + f f )g g p + (m f m )g ; (5.29)

1 i 1 1 i 1 3 2 3

0 0 0

i=1

=0 =0

where

f = +(1 )(2 1) ; (5.30)

i 0 0 i

and

0 0 0 0 0

g = + +(1 )(1 )i : (5.31)

i 0 i 0 i 0 0 i 25

The structure of these equations b ecomes more transparentby writing them out explicity

in terms of the singlet and triplet wave functions

( ; ; ; )=( ; ; i ; ); (5.32)

00 10 20 30

( ; ; ; )=( ;;i ;): (5.33)

0 1 2 3

[w U ] =2p+(m m ); (5.34a)

1 2

[w U ] =2p +(m +m ) ; (5.34b)

1 2

[w U ] =(m +m ); (5.34c)

1 2

[w U ] =(m m ) ; (5.34d)

1 2

[w V V r^r^] =2p+(m m ); (5.34e)

1 2

[w V V r^r^] =2p +(m +m ) ; (5.34f )

1 2

[w V V r^r^] = 2ip +(m +m ); (5.34g)

1 2

[w V V r^r^] = 2ip +(m m ) ; (5.34h)

1 2

where

(A3B C )S

w U = we ; (5.35a)

(A+B +C )S

w V = we ; (5.35b)

and V is de ned in Eq. (5.28).

In order to see explicitly the distinction b etween the traditional Breit approach and that

of constraint dynamics, we p erform an angular momentum decomp osition. For the spin-zero

wave functions we take

= Y ;=Y ; = Y ;=Y ; (5.36a)

j jm j jm j jm j jm

where Y is an ordinary spherical harmonic (j = l here). For the spin-one wave functions,

we take a form that dep ends on the spatial parity:

= Y + Z ; or = X ; (5.37a)

jy jm jz jm jx jm

= Y + Z ; or = X ; (5.37b)

jy jm jz jm jx jm

= Y + Z ; or = X ; (5.37c)

jy jm jz jm jx jm

= Y + Z ; or = X ; (5.37d)

jy jm jz jm jx jm

where the three orthonormal vector spherical harmonics are

L rp r

q q

X = Y ; Z = i Y ; Y = Y : (5.38)

jm jm jm jm jm jm

j (j +1) j (j +1)

The rst and last vanish for j =0.

To obtain the radial wave equations, we use the following identities

X =0; pX =0; (5.39a)

jm jm

j (j +1)

Z r

X = iZ ; p X = Y + ; (5.39b)

jm jm jm jm

r r r

r 2i

Y = Y ; p Y = Y ; (5.39c)

jm jm jm jm

r r

j (j +1)

Y =0; pY = X ; (5.39d)

jm jm jm

r r 27

j (j +1)

Z =0; pZ = i Y ; (5.39e)

jm jm jm

r r

r X

Z = X ; p Z = : (5.39f )

jm jm jm

r r

The equations so obtained can b e divided into two sets of di erent total parity. One set has

the natural parity solution:

j+1

P =() ; (5.40)

jm jm

and involves the 8 wave functions:

= Y ; = Y ; (5.41a)

j jm j jm

= X ; = i Y + i Z ; (5.41b)

jx jm jy jm jz jm

= X ; = i Y + i Z : (5.41c)

jx jm jy jm jz jm

(Four of the wave functions ; ; and do not app ear for j = 0, b ecause their

jx jz jx jz

angular parts vanish.) The eight simultaneous equations satis ed by them are

j (j +1)

2 +(m +m ) ; (5.42a) (w U ) =2 +

jy jz 1 2 j j

r r

(w V V ) = 2 +(m m ) ; (5.42b)

jy 1 2 jy

(w V ) = 2 +(m m ) ; (5.42c)

jz jx 1 2 jz

j (j +1)

(w V ) =2 + 2 +(m +m ) : (5.42d)

jx jz jy 1 2 jx

r r

(w U ) =(m +m ) ; (5.43a)

j 1 2 j

j (j +1)

(w V ) = 2 +(m m ) ; (5.43b)

jz j 1 2 jz

r 28

(w V ) =(m +m ) ; (5.43c)

jx 1 2 jx

j (j +1)

(w V V ) = 2 +(m m ) : (5.43d)

jy jx 1 2 jy

The second set has the \unnatural" parity

P =() ; (5.44)

jm jm

and involves the 8 wave functions:

= Y ;=Y ; (5.45a)

j jm j jm

= X ; = i Y + i Z ; (5.45b)

jx jm jy jm jz jm

= X ; = i Y + i Z : (5.45c)

jx jm jy jm jz jm

The eight simultaneous equations satis ed by them are

j (j +1)

(w U ) =2 + 2 +(m m ) ; (5.46a)

j jy jz 1 2 j

r r

(w V ) = 2 +(m +m ) ; (5.46b)

jz jx 1 2 jz

j (j +1)

(w V ) =2 + 2 +(m m ) : (5.46c)

jx jz jy 1 2 jx

r r

(w V V ) = 2 +(m +m ) ; (5.46d)

jy 1 2 jy

(w U ) =(m m ) ; (5.47a)

j 1 2 j

(w V ) =(m m ) ; (5.47b)

jx 1 2 jx

j (j +1)

(w V ) = 2 +(m +m ) ; (5.47c)

jz j 1 2 jz

r 29

j (j +1)

(w V V ) = 2 +(m +m ) ; (5.47d)

jy jx 1 2 jy

where the terms w U , w V are given in Eqs. (5.35a-b) and w V V by

(A+B C )S

w V V = we : (5.48)

Each set of 8 equations consists of 4 di erential equations [Eqs. (5.42) or (5.46)] and 4

algebraic equations [Eqs. (5.43) or (5.47)]. The algebraic equations can b e used to express

4ofthe8wave functions in terms of the remaining 4. For example, two of the eliminated

wave functions are

m + m

1 2

; (5.49) =

j j

w U

m m

1 2

= : (5.50)

j j

w U

After the elimination, p ole singulariti es could app ear in the di erential equations at particle

separations r where equations such as Eq. (5.49) have p oles, e.g., where

w U =0; wU =0;m 6=m (5.51)

1 2

provided that the total relativistic center-of-mass energy w is nonzero. These are the well-

known singulariti es that plague the traditional Breit equations.

However, from the p ersp ective of the exp onentiated interactions of constraint dynamics

such as that shown in Eq. (5.35), these structural p oles can app ear only if the p otential

generators in the exp onentgoto 1. In the absence of such singular b ehavior, the constraint

two-b o dy equations, or their Breit analogs, are free of the pathologies describ ed in [6,7].

These pathologies arise b ecause the wave functions app earing on the right-hand side of Eq.

(5.49) must have zeros at the p ole p ositions. These additional b oundary conditions give rise

to spurious resonances in the continuum for suciently strong interactions, and to spurious

b ound states for any nonzero interaction strength at total energies (including rest masses)

which go to zero. 30

We should add, for the sake of completeness, that another class of spurious solutions

app ears when the values of the exp onential generators go to +1. Then the algebraic

equations imp ose the new b oundary conditions that the wave functions app earing on the

left-hand sides of these equations must vanish at these singular p oints.

It is imp ortant to p oint out that if the exp onentiated p otentials given in Eqs. (5.35) and

(5.48) are approximated by nite Taylor approximants, it might b e p ossible to satisfy the

singularity conditions such as (5.50) whenever the approximant has a zero where the true

value of the exp onential function is nonzero. In other words, it is the exp onential structure

of the e ectiveinteraction that protects the CTB Dirac equations from the undesirable

pathology.We trace that exp onential structure to the hyp erb olic forms (3.3) of the constraint

equations which in turn are motivated by the requirements of compatibility. Recall that in

those forms the \generators" pro duce p otential terms in the \external p otential" form of

the constraint equations whichhave the exp ected transformation prop erties, including the

essential recoil corrections.

Our result can also b e interpreted in another way. The exp onential functions app earing

on the left hand side of Eq. (4.10) can b e expanded in terms of hyp erb olic functions. Indeed,

one could express the \e ective p otentials" contained in exp(2D ) in the form given by the

right-hand side of Eq. (4.11). If wenow parametrize these e ective p otentials directly using

functions with no singularity at nite r ,we will nd that under favorable circumstances the

singularity condition (5.50) can still b e satis ed. This is in fact the result of [6,7]. Thus the

regularity of the e ective p otentials app earing in exp(2D ) do es not guarantee the regularity

of the resulting Breit equation. It is the regularity of the basic p otentials app earing in the

exp onent2D that guarantees the regularity of the resulting Breit equation. 31

VI. COMPARISON OF THE CONSTRAINT AND BREIT EQUATIONS FOR

QED

In this section we discuss the implications of our new Breit-like equation in quantum

electro dynamics. Consider rst the original Breit equation whose matrix form is

0 0 0 0 0 0 0

(w + ) + a + b r^ r^ = p + p + m m ; (6.1)

1 2

r r r

where a and b are p otential parameters. By identifying it with the center-of-mass form

of the Breit-like form Eq. (4.10) of our covariant constraint equation, we can solve for the

generators given by Eq. (5.5). Using Eqs. (5.26) or equivalently Eqs. (5.27-28) we obtain

the following twelve equations for the p otentials.

exp[(A 3B C ) S ] = 1 +(1+3a+b); (6.2a)

exp[(A 3B C ) S ] = 1+(1+3a+b); (6.2b)

exp[(A 3B C ) S ] = 1+(1 3a b); (6.2c)

exp[(A 3B C ) S ]= 1+(13ab); (6.2d)

exp[(A + B + C ) S ] = 1+(1 ab); (6.2e)

exp[(A + B + C ) S ] = 1+(1 ab); (6.2f )

exp[(A + B + C ) S ] = 1 + (1 + a + b); (6.2g)

exp[(A + B + C ) S ] = 1 + (1 + a + b); (6.2h)

exp[(A + B C ) S ] = 1+(1 a+b); (6.2i)

0 32

exp[(A + B C ) S ]= 1+(1a+b); (6.2j)

exp[(A + B C ) S ] = 1+(1+ab); (6.2k)

exp[(A + B C ) S ] = 1+(1+ab); (6.2l)

in which = =(wr). The rst four equations come from the singlet wave function ( =0

terms in Eq.(5,28a)), while the last eight arise from the triplet, with the last four coming

from a combination of Eq. (5.28b) and the triplet (or 6= 0) part of Eq. (5.28a). These

twelve algebraic equations can b e solved for the twelve unknown generators shown in Eq.

(5.5). One can readily show that six of these generators vanish (corresp onding to scalar,

pseudoscalar, and tensor interactions):

S = T = F = Y = C = L =0: (6.3)

The remaining generators, J;H;I;G;N and R, are nonzero, but we shall not need them in

our discussion except in the combinations app earing on the left-hand sides of the algebraic

equations, Eqs. (5.43) and (5.47). These are just the six equations shown in Eqs. (6.2c-f )

and (6.2k,l), now expressible as

w U = w U = w exp[J +3I N H +3G +R] = 1+(1 3ab); (6.4)

w V = w V = w exp[J I + N + H + G + R] = 1+(1 ab); (6.5)

w V V = w V V = w exp[J I + N H + GR] = 1+(1+ab): (6.6)

Singularities at nite particle separation r arise, for nonzero w , when the right-hand sides

of these equations vanish (assuming that the numerators in the algebraic equations in which

these o ccur do not). Recalling the context in which these equations app ear, we see that this

o ccurs when

3a + b>1 (6.7) 33

j+1 j

for all P =() states, and all P =() states when m 6= m ;

1 2

a + b>1 (6.8)

j j+1

for all P =() states, all P =() states when m 6= m ; and for j = 0 states when

1 2

m = m and

1 2

b a> 1 (6.9)

j+1

for all P =() states when m 6= m , and for j = 0 states when m = m and all

1 2 1 2

P =() states. This con rms the result rst derived in Ref. [7].

It is worth noting that the original Breit equation corresp onds to a = b =1=2 while

the Barut equation corresp onds to a =1;b = 0 [21]. Both equations are therefore singular

according to the rst of the ab ove three conditions.

From the p ersp ective of the exp onential generators of constraint dynamics, the vanishing

of the right-hand sides of Eqs. (6.4-6.6) is p ossible only when the generators take on the

value 1 at nite r . Hence they can b e avoided by simply using generators not having

such r singulariti es. Indeed, in their constraint approach to QED, Crater and Van Alstine

[11,12] found that

G J 1=2

e = e = (1+2 =(wr)) (6.10)

with all of the remaining ten generators set equal to zero. As a result, no midrange zero

app ears since

w U = w exp(J +3G)= w exp(2G )= (6.11)

1+2 =w r

is nite for p ositive r . Furthermore, the Lorentz nature of the two nonzero generators

matches that of the vector character of the QED interaction at this level. In contrast, the

Breit generators involve pseudovector as well as vector parts and additional vector and pseu-

dovector \tensor terms" corresp onding to the Coulomb gauge ( as opp osed to the Feynman

gauge used in the constraint equations). 34

The conclusions obtained in this section for QED can b e extended to other interactions

(scalar, pseudoscalar, psuedovector and tensor) imp ortant in semiphenomenological appli-

cations in nuclear and particle physics. Thus, one may with safety solve the new Breit

equations numerically with no concern ab out structural singularities whichwould otherwise

render such solutions meaningless.

VI I. CONCLUSIONS AND DISCUSSIONS

Breit equations and constraint Dirac equations of relativistic two-b o dy quantum mechan-

ics have markedly di erent prop erties: Breit equations could have structural singularities at

nite particle separations even when the interactions themselves are nonsingular there, while

constraint Dirac equations seem to b e free of them. CTB Dirac equations are manifestly

covariant, while Breit equations are not, b eing valid only in the center-of-mass frame. We

are able to understand, and to reconcile, their di erences in this pap er.

The constraint Dirac equations were originall y derived for scalar interactions with the

help of sup ersymmetries in addition to Dirac's constraint dynamics. Generalizing this con-

cept to arbitrary interactions leads to a \hyp erb olic" form of the equations. The two

single-particle Dirac equations can then b e recast into two other equations: (1) a covariant

Breit-like equation with exp onentiated interactions, and (2) a covariant equation describing

an additional orthogonality condition on the relative momentum. We use the equivalence

between these twotyp es of equations to show that the constraint Dirac equations are com-

pletely free of the unphysical structural singularities when the exp onential structure of the

interactions are not tamp ered with.

The advantage of the constraint form of the Breit equation is that the structural singular-

ities of the traditional Breit equation are nowmoved entirely to the exp onential generators

of the interaction. As a consequence, they can b e eliminated right from the b eginning by

the simple requirement that these generators themselves b e free of singularities at nite

separations. The resulting Breit equations are then guaranteed to b e free of the undesir- 35

able structural singulariti es that plague traditional Breit equations. These improved Breit

equations, which are dynamically equivalent to the constraint Dirac equations, can nowbe

used in nonp erturbative descriptions of highly relativistic and strong- eld problems such

as those app earing in two-b o dy problems in quantum electro- and chromo-dynamics, an in

nucleon-nucleon scattering.

Of course, the constraint Dirac equations can also b e used, now that we knowhowto

keep them singularity-free. However, in actual applications, they have to b e reduced down

to a set of coupled di erential equations b efore actual solutions can b e attempted. These

di erential equations are transforms (see Eq.(4.9)) of those obtained from the equivalent

Breit equations. So in reality the two di erent formulations havenow b ecome completely

identical to each other.

Acknowledgement: This researchwas supp orted by the Division of Nuclear Physics,

U.S. Department of Energy under Contract No. DE-AC05-84OR21400 managed byLock-

heed Martin Energy Systems. One of the authors (HWC) wishes to acknowledge very useful

discussions with P.Van Alstine, M. Moshinsky and A. Del Sol Mesa on closely related topics 36

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