Two-body Dirac equations

ANNALS

OF PHYSICS

148, 57-94 (1983)

Two-Body

Dirac Equations

HORACE W. CRATER Department

of Physics, Universiry of Tennessee Tullahoma, Tennessee 37388

Space Institute,

AND PETER VAN ALSTINE* Department

of Physics and Astronomy, Nashville, Tennessee

Vanderbilt 37235

University,

Received August 4, 1982; revised November 29, 1982 P. A. M. Dirac took the matrix square root of the Klein-Gordon equation to obtain his relativistic wave equation for a single spin-one-half particle. In this paper, we use Dirac’s constraint mechanics and supersymmetry to perform the same operation on the relativistic description of two spinless particles to obtain consistent descriptions of two interacting particles, either or both of which may have spin one-half. The resulting coupled quantum wave equations correctly incorporate relativistic kinematics as well as heavy-particle limits to one-body Dirac or Klein-Gordon equations. The 16.component wave equations for the system of two spin one-half particles separate exactly into four decoupled four-component equations for the analogs of “upper” and “lower” components of the Dirac equation. Perturbative treatment of our equations through O(a”) automatically reproduces the appropriate fine structure. Furthermore, like the decoupled forms of Dirac’s equation, the two-body versions have spin-dependent pieces that make non-perturbative quantum-mechanical sense. This feature eliminates the need for extra smoothing parameters in the potential or finite particle size in phenomenological applications.

I. INTRoDUCT10~ In 1928, Dirac devised a relativistic wave equation for a spin-one-half particle by “taking the square root” of the Klein-Gordon equation for a spinlessone [ 11. He hit on a first-order wave equation whose peculiar matrix coefficients had just the right algebraic properties to regenerate the Klein-Gordon equation from which he had started. In this paper, we show how an analogous procedure applied to a system of relativistic wave equations for two interacting spinlessparticles leads to consisttnt systems of quantum wave equations for two interacting spin-one-half particles or alternatively for one spin-one-half and one spinlessparticle. The traditional treatment of interacting pairs of spinning particles began with * Present address: Physics Dept., Indiana University, Bloomington, Indiana 47405. 51 0003-4916183 $1.50 Copyright All righis of

0 1983 by Academic Press, Inc. reproductionin anyform reserved.

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Gregory Breit’s suggestion for the spin-dependent forces given in his paper of 1929 [2]. Breit formed a single effective Hamiltonian by adding together two free Dirac Hamiltonians plus an interaction patterned after that in the Darwin Lagrangian for two spinless particles [3]. Breit’s procedure gave a weak-coupling O((u/c)‘) approximation to an unknown underlying exact theory. In 195 1, Bethe and Salpeter formulated the exact relativistic problem in terms of an integro-differential eqation derived from the full-blown quantum field theory 14 1. Their equation in turn justified the original Breit interaction by reproducing it in weak-coupling O((v/c)‘) perturbation theory [ 5 1. Although the Bethe-Salpeter equation has since produced results that are in good quantitative agreement with experiments in quantum electrodynamics, its interpretation as a relativistic boundstate equation has turned out to be obscure. Nakanishi has found that the BetheSalpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time [ 61. These “ghost” states have spoiled the naive interpretation of the Bethe-Salpeter equation as a quantummechanical wave equation. Faced with this dilemma, some authors have recast the field-theoretic information so as to expose other effective wave equations buried in it. For example, Todorov and his co-workers 171 have used field-theoretic scattering amplitudes to determine a “quasi-potential” for use in bound-state calculations. The success of their methods in electrodynamics argues for simple relativistic dynamical structures, at least through order a5 and (r5 In a in perturbation theory [8-lo]. In practice, though, their methods may lose information in the various truncations necessary to bring their equations to tractable form. There is, however, another route by which we can enter the same territory (relativistic bound-state equations), that through relativistic quantum mechanics. If a quantum mechanics of interacting relativistic particles exists, the correspondence principle suggests in turn the existence of a classical relativistic mechanics that would reproduce the quantum version by canonical quantization. At first glance, the CurrieJordan-Sudarshan “No-Interaction” theorem of 1963 forbids such an approach Ill]. However, beginning in 1976, Todorov 1121, Kalb and Van Alstine I13 1, and others [ 141 succeeded in constructing just such a consistent covariant, canonical mechanics for two interacting spinless particles, by using Dirac’s constrained Hamiltonian formalism [ 151. Essentially their descriptions simultaneously evaded the C.J.S. theorem and exorcised quantum ghosts by controlling the relative time in a fully relativistic manner. The dynamics of such a system is given by a set of constraints that act both to constrain the motion in phase-space and (as Hamiltonians) to generate it in the first place. The constraints must be mutually compatible in the sense that each is conserved along the phase-space motion. Compatibility restricts the dependence of the interaction on the relative time as well as requires a relation between constituent potentials, a relativistic “third law.” Canonical quantization of the constraints leads in each case to a system of compatible coupled Klein-Gordon equations (one for each particle). These equations have the same sort of firstquantized validity for the two-particle system as the Klein-Gordon equation with

TWO-BODY

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59

external potential has for a single particle. Since they stem from a classical mechanics that preserves the relativistic nature of both the constituents and the composite whole, the system they define properly describes both bound and unbound motions. Hence, when the potential turns off, they reproduce correct relativistic two-body kinematics, while when the potential acts. they degenerate to the ordinary one-body KleinGordon equation with external potential when either particle is given an infinite mass. Whatever structural complexity they possess seems to be the minimum necessary for a consistent extension of the Klein-Gordon equation to the relativistic system of two bodies. Since their initial discovery, the constraint descriptions for spinless particles have been generalized to many (parametric) time versions and to systems of three (or more) particles [ 161. The two-body versions have been applied in practical calculations using potentials suggested by QCD to predict bound states of quark and anti-quark (in vector mesons, by the authors [ 171) and bound states of gluons (in glueballs, by Lichtenberg et al. [ IS]), although at the cost of neglecting spin. Of course, any attempt to apply the constraint methods to the composite systems of Nature must deal with quantum spin. Since the spinless constraint dynamics led to systems of compatible coupled Klein-Gordon equations, one would rightly guess that the introduction of spin would lead to systems of compatible coupled Dirac equations or mixed Dirac and Klein-Gordon equations. Although in principle one could try to write such quantum descriptions directly, one quickly discovers that without some guiding principles, the construction of mutually compatible constraints is a game of guesswork [ 191. Even if we start with an interaction for which we already know how to treat spinless particles, an ad hoc introduction of gamma matrices destroys the compatibility that we had worked so hard to achieve. The essential difficulty is that when spin is present, we immediately demand quantum compatibility, yet in the spinless case, we first build a consistent classical system before making such a demand. The remedy then seems to be to construct a consistent classical version of spin, then quantize. We could go back to classical mechanics and introduce a miniature gyroscopic spin [20] that could as easily be bosonic as fermionic when we quantize. We would then have to arrange the mechanics to reproduce the Dirac equation and hope for a close connection between the classical and quantum descriptions. But, what we really want is a simplified description of spin that turns directly into Dirac’s version of spin-one-half upon quantization. To produce one, we would need to represent quantum spin classically, an impossibility. However, if we take a less strict correspondence limit of the Dirac equation, one that stops just short of losing spin altogether, we can represent the Dirac matrices as anticommuting numbers-elements of a Grassmann algebra [21]. The resulting “pseudoclassical” mechanics contains just enough spin information to reproduce the Dirac equation in its entirety upon canonical quantization. Put crudely, if Dirac had known the appropriate pseudoclassical mechanics in 1928, he could have reached his equation by direct operator substitution into a classical constraint [just as one reaches (0 + m’) v/ = 0 from p2 + m2 z 01.

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Moreover, we discover that the free Dirac equation contains a hidden symmetry under a transformation that mixes Dirac matrices with momentum operators and that in the correspondence limit, this transformation becomes a supersymmetry transformation mixing ordinary phase-space coordinates with the Grassmann spin-variables [22]. The fermionic and bosonic structures that are degenerate under this onedimensional supersymmetry are the constraints that turn into the Dirac equation and its Klein-Gordon square, respectively. In spinless constraint mechanics, Dirac tells us that compatibility is a consequence of symmetry. Hence, if we build the classical description so as to retain supersymmetry even in the presence of interaction, we stand a good chance of inheriting pseudoclassical compatibility. By now many authors have treated the one-body problem with such methods [ 23 ]. However, in that case compatibility is so immediate that one does not have to pay attention to the role of supersymmetry to achieve it. For each one-body interaction, quantization of the classical constraints leads to the usual system of the Dirac equation with external potential along with its appropriate quantum square. In a previous paper the present authors showed how to extend such descriptions to the two-body case [24]. We found that if we built a two-body classical mechanics in such a way as to preserve the supersymmetric structures for each spinning constituent, all the spin complications dropped away, leaving compatibility problems that were exactly those of the corresponding spinless system. Once we disposed of these in the usual (bosonic) way, we obtained a consistent system of two coupled Dirac constraints for two spinning particles or one Dirac and one Klein-Gordon constraint for one spinning and one spinless particle. For the case of the scalar interaction, we were able to do all this by demanding that the interaction depend on position only through a special supersymmetric invariant position variable, I (This variable appears, although hidden, in the usual one-body Dirac equation with external scalar potential.) We then canonically quantized the constraints to produce a system of compatible coupled wave equations on a single wave function-two Dirac equations for the case of two spin-one-half particles, one Dirac and one Klein-Gordon equation for the case of one spin-one-half and one spin-zero particle. What we were able to do amounted to taking the pseudocorrespondence limit on the one-body Dirac equation, abstracting the special properties of the pseudoclassical one-body problem, ensuring that these properties were incorporated consistently into a two-body classical description, and then requantizing to obtain a first-quantized two-body system. In the present paper, we first review in detail the one-body problem as well as the use of the classical two-body construction to obtain quantum wave equations. Then, we extrapolate textbook methods usually applied to the one-body Dirac equation to rewrite our two-body equations as sets of second-order equations on spinors of low dimension. This automatically leads in the spin-zero spin-one-half case to decoupled equations on two-component spinors, while, surprisingly, this also leads in the case of two spin-one-half particles to decoupled equations on four-component spinors. The decoupling seems to be a consequence of the existence of one constraint and one supersymmetry for each spinning particle in the pair. The resulting equations are just

TWO-BODY

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61

the two-body analogs of the (one-body) Dirac equation’s second-order “upper” and “lower” component equations. Since they were derived from a fully covariant dynamics, they reproduce correct relativistic two-body kinematics when the potential shuts off and degenerate to a one-body Klein-Gordon equation (for a spinless constituent) or a pair of decoupled upper and lower component equations from a onebody Dirac equation (for a spin-one-half constituent) when either particle is given an infinite mass. Moreover, since our two-body equations have swallowed up at least one one-body Dirac equation, they imitate well-known features of the one-body description, but correct these to results appropriate to a two-body description. For example, whereas the Dirac equation automatically yields the correct perturbative fine structure for a single particle in a static scalar Coulomb potential, the two-body equations automatically yield the correct perturbative fine structure for two spin-one-half or one spin-one-half and one spinless particle in relativistic scalar interaction. Better still, the two-body equations incorporate the exact Dirac equation, so they inherit its nonperturbative structure as well. The decoupled upper and lower component equations of the ordinary Dirac equation have potential-dependent denominators in their Darwin and i . 2 terms that temper these forces in regions of strong potential. The two-body equations produce these as well as corresponding denominators for more general spin-dependent terms. If these denominators were not present, certain singular potentials (e.g., l/r) could only be used perturbatively. In two-body descriptions that lack such denominators, other authors have been forced to resort to ad hoc smoothing procedures or particle-fattening to avoid the singularities [25]. Our twobody equations (just like the exact one-body Dirac equation) evade this problem by possessing denominators that “divide out” the singularities. In both the one- and twobody cases, the denominators preserve the point-like nature of each spinning particle, while in the two-body case, the denominators also work to ensure the correct static limits to one-body systems. In order to see more clearly the peculiar structure of our equations, we compare them with the analogous equations given by Todorov’s quasi-potential approach. (Todorov has shown that for spinless particles, quantization of the constraints leads to his local homogeneous quasi-potential equation [ 12, 17, 261.) We show that while both procedures agree on the line structure, they disagree non-perturbatively over the presence or absence of the potential-dependent denominators. Since these denominators are necessary for the correct static limits to exact one-body Dirac equations, we conclude that our equations are to be preferred. For the reader who wishes to go directly to the relativistic quantum wave equations that are the most important results of our paper, we have provided a special summary (following the conclusion) containing the equations for both the spin-one-half spinzero system and the spin-one-half spin-one-half system. From canonical quantization of the classical constraints, the structure of these equations is virtually dictated by the one-body classical supersymmetry of Eq. (64) and the supersymmetric invariant position variable x’ (Eq. (67)] appropriately extended to the two-body problem in Eq. (98) and Eq. (127). For the reader unfamiliar with relativistic constraint

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mechanics for spinless particles, we have provided a review in Section II. In Section III, we study the pseudoclassicalmechanics of a single free spinning particle that emergesas a “correspondence” limit of the Dirac equation. In Section IV, we detail the role played by supersymmetry when the spinning particle is put into an external scalar field. In Section V, we use the constraint technique, classical spin description, and supersymmetry to derive compatible classical constraints and associated quantum wave equations for a spin-one-half particle in scalar interaction with a spinlessone. In Section VI, we do the same for the case of two spin-one-half particles in mutual scalar interaction.

II. RELATIVISTIC

CONSTRAINT

DYNAMICS

FOR Two SPINLESS PARTICLES

In this section, we derive a two-body relativistic Schrodinger equation from quantization of classical generalized mass shell constraints. We also introduce the relevant kinematical variables and give the conditions on the constituent mass-potentials needed for classical consistency. Our treatment here is closely related to those of Todorov [ 121 and of Kalb and Van Alstine [ 131. A free particle is described by the (parameter invariant) arc length action

From p = BL/&t = mu^, where u”= i/p, particle’s four momentum:

follows the mass shell constraint on the

,X=p2+m2z0.

(2)

We introduce interactions with an external scalar field by letting m + M = m + S(x) in (1) and (2). A similar Lagrangian for two spinlessparticles would lead to the two mass shell constraints

q=p:+A4;ao,

Jst”z=p;+M:~o,

(3)

where Mi = mi + Si. In our discussionsof the two-body system we consider more generalized mass potentials Mi than are implied by our particular Lagrangian. In fact, we generalize the Mi to include momentum as well as relative coordinate dependence. Since the Dirac Hamiltonian is R= A,< + A,<, a sufficient condition for 4 and 3 to be conserved in r is that the constraints be first class, {~,&)zO.

(4)

The weak equality signs in (2k(4) mean that the constraints are to be imposed only after working out the Poisson brackets. The condition (4) then confines the motion to the constraint hypersurface defined by (3). The right-hand sides need only vanish on that surface but may vanish identically (strongly).

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The fundamental

brackets

DIRAC

among the constituent {xy,p;}

63

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variables are

= g??,.

(5)

As in non-relativistic mechanics, we introduce canonical relative position and momentum variables. In order to ensure the correct relativistic kinematics, we require that the relative momentum variable collapse to the usual expression in the c.m. rest frame. Then, x=x,-x,,

P=$E*P,

-&,P,),

(6)

where p=p,

p2 = -WI,

+p2;

(7)

and P*pzO. The last equation momentum. The {x“, P”} = 0 and total energy is a

(8)

just says that on the constraint hypersurface, p is the usual relative requirement {x“,p’} = g“” forces Ei = Ei(P*) and E, + E, = W. Since the & depend only on a relative x, the c.m. (F= 0) value w of the constant of the motion. In these variables, condition (4) becomes

{&,A7}=-2p1 = -2P.

d4-2p,&f:+ q&f:

(M:,M:}

+ 44:) - 2p. qh4; - MI) + {Mf , M;}.

If the Si depend on x only through its component x1. perpendicular

(9)

to P, that is,

where x"l = (g""

then condition

(4) is satisfied strongly

- P@PyP*, x,.,

(11)

if [ 121

M:-Mz=m:-m:. That is, we need the relativistic further implies that

counterpart

(12) of Newton’s

-iZ”;-L~=2P~p+m~-m~+(~z-~F,)w~0. Equations

third law. Equation

(12)

(13)

(13) and (8) can only agree if 1 cl-E2=--(mf-mm:).

(14)

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Thus our procedure completely determines the canonical variables. Because of Eq. (8) and P . x1 = 0, two of the variables, the relative energy and relative time in the c.m. system, have effectively disappeared. The choice Eq. (14) implies 3 -6 = 2P . p z 0. The remaining independent combination of the constraints then becomes

where the last two equalities result from P . p z 0. In Eq. (15), b2 (the on-shell value of the relative momentum) has the following equivalent invariant forms [where L(a, b, c) is the triangle function]:

b2(w)=+m;=+m; (16)

= A(w2, mf, m:)/4w2. Thus (12) implies

@=2m,S,+Sf=2m2S2

-IS:.

(17)

The form (15) (also that of Todorov’s quasi-potential Hamiltonian) incorporates the correct relativistic two-body kinematics. Todorov defines the other useful variables

F,,,= (w’ - rnf - m:)/2w,

m, = m, m,/w,

interpreted as the reduced mass and energy of the fictitious motion. In terms of them,

(18) particle

b2 = E:. - rn:, reinforcing

which form

this interpretation.

of relative

(19)

If we define

in the c.m. system is just (E,,P?), then (15) assumes the Klein-Gordon-like R=/r2t(m,tS)2=/f2+m~.t@,

(21)

where S is an effective scalar potential related to those in (17) by

Mt = rnf $2m,S

+ S* E (m,$, + S)* qt,

(22)

M: = rnz + 2m,S + S* = (m, + S)’ vi.

(23)

For spinless particles M, and M, always

appear squared in the constraints

[ 27 ].

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65

In the quantum version of the above we use the brackets (24) to verify that the commutator counterparts to (3),

are compatible become

conditions

[q,

&]

vanishes. This guarantees that the quantum

qW=O,

(25)

&v/=0,

(26)

on the wave function.

The alternative

P *ply=o, (/I’ + (m,. + S)‘) l/l = 0.

forms (8) and (21)

(27) (28)

The latter acts as a relativistic Schrodinger equation which in the c.m. system [where (27) implies p’ty =$*I+Y] has the eigenvalue form ($2 + 2m,,S + S2)y = b2(w)y/. In this frame (24) may be consistently

(28)’

replaced by

[Xj, pj] = iSii

(29)

(with x0 and p” disappearing from our description). Equation (28) [or (28)‘] has the computational simplicity of the non-relativistic Schriidinger equation, is fully relativistic, and in the static limit m, + co (or m, + co) reduces to the Klein-Gordon equation for a particle in an external scalar potential. It incorporates correct two-body relativistic kinematics through A(w’, m:, mi) and hence gives a bona fide covariant description that is more than just an 0(1/c’) approximation to some other equation (e.g., the Bethe-Salpeter equation). The closely related quasi-potential equation has been successfully appiied to calculations of the relativistic corrections to positronium and unequal mass bound states in Q.E.D. The version of Eq. (28) that also includes time-like vector interactions is the wave equation that the authors and others have successfully applied to quark model calculations of the vector meson masses and the gluonium system [ 17, 181. Both of these applications assumed spinless quarks. Before we can trackle the problem of spinning quarks in a realistic potential we must learn how to introduce spin without destroying the relativistic structure of our quantum wave equation (28). In order to show how this may be accomplished with a minimum of complexity we will restrict our attention in this paper to the case of scalar interactions.

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CRATER AND VAN ALSTINE III.

A PSEUDOCLASSICAL DESCRIPTION OF SPIN USING GRASSMANN VARIABLES

In the past decade many authors have augmented conventional classical mechanics by introducing Grassmann degrees of freedom that produce spin upon quantization [21-231. In the resulting “pseudoclassical” mechanics the Grassmann variables are a sort of semiclassical representation of spin. For the reader unfamiliar with Grassmann variables or this particular application of them, we will deduce their algebraic properties by a generalized correspondence principle applied to the Dirac equation. This correspondence principle stops just short of the classical world (which contains no internal spin) in the pseudoclassical world which does. We begin our discussion with Dirac’s equation for a single free particle

(P, YM+ m) v = 0, which can be written

(30)

in the form (31)

where

(32)

e5= i J $5 satisfy

p, ey + = -hgy [e,,P]+ =o, [O,,S,]+ =-A These anti-commutators,

(33)

together with IXU, p“] _ = ihg””

(34)

and [xP,P_=O=

[pC,8QlP,

a=O,

1,2,3,5,

(35)

define the algebraic properties of our dynamical variables. Equations (33F(35) divide the basic variables into two distinct classes: (a) those whose defining quantum brackets are exclusively commutators (call these even); (b) those that participate in fundamental anti-commutators (call these odd). Clearly the (bosonic) x and p variables are even and the (fermionic) 8 variables are odd.

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EQUATIONS

For dynamical variables A, and A, that have well-defined we can write the generalized quantum bracket, IA,J,lL.,,=4Jg

character

(odd or even)

(36)

- r,ddo.

where qao = (-) ‘a”~. The variable E, is 0 if A, is even and 1 two even variables, or one odd and one even, -v,~ = commutator. For two odd variables, -‘loo = + and the commutator. We define the product quantum bracket such that

if A, is odd. Thus, for and the bracket is a bracket is an antiA,A, with A, is

14A.A,l-‘lnynoy.

(37)

This implies that the product of an odd with an odd is an even, the product of an even with an odd is an odd, and that the product of an even with an even is an even. Using the definition in (36) (38) The product rule, of course, implies a Jacobi condition. For the basic variables the correspondence principle bracket leads to the (pseudo)classical bracket

applied to the generalized

(39) Thus the fundamental

non-vanishing

brackets

become

(8”, et’} = ig”“,

(40)

{e,,@,} = i, (x’I,p”I

= gU’..

(41)

We assume that our classical 8’s are real (as are x and p). In ordinary quantum mechanics the correspondence limit leads to commuting and p. When we use the (pseudo)classical version of the product rule (38) kLA,v$l=4~A,.Ayl

+ ~&LA,lA~

x

(42)

we not only recover the commutativity of x and p 1281, but also both the anticommutativity of the 8’s and the commutativity of the B’s with x and p. For example, (43) and (44)

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imply 8”8” + tl”tP = 0.

(45)

ev, + e5efl = 0, 8: = 0.

(46)

Similarly,

In the correspondence

limit, the quantum Jacobi condition

becomes (47)

A differential realization provided by (211

of (40) and (41) in terms of the various x, p, and B’s is

a’a’ -7 =----!---!L+i-L-L+i$C, axu ap, ap ax, i i

--

-’

II

5 5

(48)

Having established the algebraic structure for our (pseudo)classical dynamical variables we now construct the necessary tools for describing the dynamics. First notice that the Dirac equation (31) becomes a constraint on the dynamical variables:

.~‘=~.e+me,~o.

(49)

This is not the only constraint. Since .Y’ is odd, use of (40) allows us to find another constraint. the mass shell condition:

The Jacobi condition (47) gives us a very simple way -i(. 5‘ , {.V , ,Y’} } = 0. Thus we obtain a closed algebra canonical constraints. Associated with this algebra is an important symmetry of constraint .ic in (49) and the Dirac equation in (31). Use of to the invariance of (49) and (31) under the transformation

68” =

-i&p”,

68, =

--i.z J-p2

z -i&m,

to see that (.P ,, W} = of only two consistent both the pseudoclassical the constraint (50) leads 6p” = 0,

(51)

where E, like P, is an odd variable. The transformation (5 1) on pseudoclassical variables is a supersymmetry transformation. We expect that the effects of this supersymmetry will persist when interactions are included. We can discover the same constraint system from a singular Lagrangian that possesses this symmetry. We have seen in (1) and (2) how a Lagrangian that is homogeneous of first degree in the velocities leads to a mass shell condition of the

TWO-BODY

For the Grassmann

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EQUATIONS

variables, the appropriate

kinetic form to accompany

(52) which is homogeneous of first degree in the velocities and Lorentz invariant. This particular linear combination is left invariant by the transformations (5 1) when we make use of (49). The rule for complex conjugation of Grassmann variables is (8,B,)* = (e,e,). Thus the combination in (52) is real. Hence for a free particle with pseudoclassical spin we take

(53) The canonical

momentum is then

(54) where u^= a/&?

has unit length. Thus (49) leads to

p.e=mu^.e+$.e=-me,. A sufficient condition

(55)

for this is that we have the Lagrangian constraint

u^.e+e,=o

(56)

s.e=o.

(57)

and that

The part L’ must contain the configuration

L’ = -im J-i2

space constraint

(56). Thus we choose

~(a . e + e,).

With v an odd real Lagrange multiplier, L’ is even, real, and first-order in velocities. Hence our total singular Lagrangian is [24]

L=-mp[l The resultant canonical

+iv(u^.e+8,)]+~~.e+~BIe,.

(58) homogeneous

(59)

momentum is

p” = g

u

= mP( 1 + id,)

- imvP.

(60)

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Both Dirac and mass shell constraints follow from this and the Lagrangian constraint (56). The corresponding action is parameter invariant so that its Dirac Hamiltonian is just the sum of the constraints (49) and (50). The equations of motion that follow from (59) are, in addition to p = const. and the constraint (56), fp = mjyv

(61)

and

The action .d= is left invariant

by the supersymmetry

‘Ldt J

transformation

(63) 1241

. 6x” = E(P -pe,>;

60” = -icpp;

68,=

ic m;

6v = j$

, (64)

where fi” =p”/@. These are just the transformation (51) supplemented additional transformations on x and U. The Noether generator for them is 1241

Y=p.&~B,.

by

(65)

That is, &I” = {A”, e.V’}, and &d = i l dr d/dz(c.F) = 0. For local transformations (but not for global ones) we need to use the constraint (49) to show the symmetry of the action. One of the special consequences of our .% is that it leaves the three separate pieces of L separately invariant for solutions of the equations of motion. In particular, although mixing fermionic and bosonic variables in the fashion typical of a supersymmetry, it leaves invariant J-i’. This property of (64) allows us to separate problems associated with the introduction of spin from those that would arise even in a purely bosonic system. Note that 3’ has a strongly vanishing bracket with itself, and hence can be termed an abelian supersymmetry generator. Thus two supersymmetry transformations generated by .%’ do not produce a reparameterization of the world line (in contrast to those generated by .Y). The invariance of .w’ under the transformation (51) can be immediately verified as a consequence of (.F’, .? } ZiF=:O.

(66)

The supersymmetry (64) for our free-spinning particle plays its major role when interactions are added. If we require that our interactions maintain this supersymmetry, we determine the spin dependence of the forces. This requirement is easily satisfied through the use of a special supersymmetric invariant position variable [ 29 1.

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71

This variable for the free particle, (67) satisfies (.3,.f”}zO.

(68)

The equality is weak because the result vanishes with F’. This variable, unlike x, has a linear proper time development for the free-spinning particle. We can see this by using (60) (in the proper time gauge)

and (70) to find x”L.

L( (71)

m

Thus the u-dependent pseudoclassical Zitterbewegung has been subtracted out of the motion for x to give that for 2. The fact that x’ is left invariant by .Z means that we can preserve our supersymmetry by restricting the x dependence of L to be only that contained in 2. The further requirement that our action maintain its parameter invariance restricts the ways in which interactions may be introduced. We confine ourselves in this paper to the case of scalar interactions. We introduce these just as we did in Section II by modifying the mass in m (wherever it appears) to a mass potential M(x) = m + S(x), then by replacing x by its supersymmetric counterpart x’. Thus for our scalar interaction we have L=-iv(f)\/-i-2(1

+iu(u^. e+e5j)+&Le+8,eI).

This means that even the m in the denominator we have a self-referent definition for F, that is,

(72)

of (67) is modified to M(x? so that

(73) In the case of the single spinning particle there is no extra spin dependence arising from this prescription because 0: = 0. Because of the use of x’ in place of the position variable x, (72) retains the supersymmetries of (59).

CRATER AND VAN ALSTINE

72 IV. CLASSICAL

CONSTRAINTS FOR A SPINNING PARTICLE

AND QUANTUM WAVE EQUATIONS IN AN EXTERNAL SCALAR FIELD

From the Lagrangian (72) we can derive equations of motion for 1 Our primary goal in this paper, however, is to construct and examine two-body quantum constraints for interacting spinning particles. The main purpose for introducing the pseudoclassical description of spin with Grassmann variables is to facilitate this procedure. Nevertheless, study of the pseudoclassical equations of motion is interesting in its own right. We will, however, put off to a separate paper a thorough description of the orbits for x’ arising from scalar interactions. The main tools that we will use to reach our pseudoclassicaldescription of scalar interactions will be the constraints on the canonical variables and the properties of the variable x’. From the Lagrangian (72) we find that pP = Aaq 1 + iU0,) - AIMp,

(74)

where @ & M(Z). The form of the Lagrangian constraint (56) is not affected by the presenceof the scalar interaction. Hence, from (74) 6. 8= 0, 19:= 0 imply

p.e=-tie,=-Me,.

(75)

Notice that in this case since 0: = 0, the Grassmann Taylor expansion of A?S, terminates with the zeroth-order term. Hence the constraint is

.~=~.e+~e,~o.

(76)

The corresponding quantum wave equation is .‘s+=(p.

e+hqx)e,)W=O,

(77)

which with the identification given in (32) is equivalent to the Dirac equation (multiplied by yS) for a single particle under the influence of a scalar potential. The quantization procedure to be followed here needs some clarification. The starting point is the pseudoclassical constraint (76). The quantization procedure is applied only after the terminating Taylor expansion has been carried out. For example, one does not quantize the equivalent unexpanded form p . 8 + A?tl, z 0, which would lead to the equation (p . e +

M(X + iee,p?) e,) w = 0.

(78)

This quantum equation would not give back the Dirac equation since the matrix Taylor expansion of fi does not terminate. The quadratic constraint on our pseudoclassical system can be derived directly from (76) and the basic Poisson brackets (including (42)): (79)

TWO-BODY

DIRAC

On the other hand, from the Lagrangian,

13

EQUATIONS

using (74) and (56) we find

F=p2+A22%o. The equivalence of (80) and (79) follows

(80)

immediately a4 .

IGzM(f)=M(x)+iT.

from I@S dependence on x’:

ee,

(81)

Here we have an example of a Grassmann Taylor series expansion terminating, this time to first order. This shows that the constraint (79) is just the one particle generalized mass shell or Klein-Gordon form of Section II except with x replaced by 2. The quantum counterpart to (80) or (79) is simply ~~=(p2+M2+2i~M~ee,)~=0.

(82)

This is the same as the squared version of the Dirac equation for a particle external scalar potential. The correspondence (32) gives us (fi = 1): (p2+M2+iyeSf)y=0.

in an

(83)

Note again, one does not quantize the unexpanded form of (80). The correspondence between the classical and quantum algebras is exact for the expanded forms. That is, from the quantum form of .Y‘ (77) with the aid of (38) we find [,U..‘r

1, =--.r,

(84)

where this R is that appearing in (82). Thus we see that the use of x’ in the pseudoclassical mechanics plus the rule that one expands before quantization lead to the correct Dirac equation for a single particle in an external scalar potential. Although we have assumed that use of .? gives supersymmetry in the case of interactions, we should explicitly demonstrate that this is so. We start with the same .‘t as in (65) and compute {.‘F, P}, where, as pointed out in the last section, x’@ =x’

iefl 8, + -=x@+A4

it? es A4

(85)

We obtain

and use the 3

constraint

(79) to expand G, {,Y, 2” } x 0.

thus arriving at (87)

74 Application

CRATER

of the .;1%”constraint

AND

VAN

ALSTINE

also shows that I.%..? } zo.

The fact that the dependence of CZ on x is only through x’ implies that X supersymmetric: {3, 2q x 0.

(88) is also

(89)

Thus our interacting system is supersymmetric. The system of quantum wave equations (77) and (83) resulting from the use of 2 is just the ordinary one of the Dirac equation with an external scalar potential and its squared version. Hence we could use any one of a number of well-known methods to decompose and solve the system. We take this opportunity to remind the reader of a particularly straightforward decomposition whose generalization will be useful in the relativistic two-body problem. The standard decomposition of the (one-body) Dirac wave function into its upper and lower components produces a set of decoupled second-order wave equations that we shall call Pauli-forms of the Dirac equation. For the purpose of a later comparison with two-body wave equations we shall write out explicitly the Pauli-forms of the Dirac equation (77) with a central, static mass potential M = m + S(Z). Writing the wave equation (83) in the eigenvalue form (j?‘+2mS+S2ti~~~S)y=(~*-m2)~,

(90)

we notice that the spin-dependent term couples the upper and lower parts of the fourcomponent spinor I+U= (i). The Dirac equation (77) provides a mechanism for decoupling this equation. Its form (77) W.F-YsYO~+YsW=O can be rewritten

as

iy=(&yO-M);‘y’.p’y. Substitution demonstrate form :

(91)

(92)

of this into the spin-dependent part of (90) and use of a diagonal y” explicitly that decoupling takes place, leading to the following Pauli-

~2+2mS+S2t

is’.? . p’ S’ M t &y” - M t &y”

(93)

where S’ = (l/r)(&S/&) and b2 = E’ - m *. Note that this b2 is just the static (heavyparticle) limit of the kinematical variable defined in Eq. (16). In practice the f (JJ” = 1) equation is the easier of the two to solve for positive energy solutions. One can then find the g wave function directly from the f function by using (92).

TWO-BODY DIRAC EOUATIONS

75

V. CLASSICAL CONSTRAINTS AND QUANTUM WAVE EQUATIONS FOR THE SPIN-ZERO SPIN-ONE-HALF SYSTEM In this section we derive wave equations for a system of a spin-zero and spin-onehalf particle in scalar interaction and decompose them into their Pauli-forms. The resulting equations correspond to Eq. (28)’ for the completely spinless system. As in the previous section we begin by constructing the constraints for the appropriate pseudoclassical mechanics. At the very least we expect that the conditions for compatibility of the resulting constraints will include those already encountered in the bosonic system. Namely, the potentials must obey qM:

- M;) = 0

(94)

and Mj = M,(xJ,

(95)

where x”l = i

gW”‘ + J$)

(x, -x&.

(96)

The conclusions of Section III lead us to introduce pseudoclassical spin in a way that preserves supersymmetry for the spinning particle. Thus (96) must be replaced by (97) where, with 12?,= M,(*Tu,), (98) For the fermionic

particle, the odd constraint

is (99)

The squared version of this constraint

is just

Since particle two is spinless, it has no i ; its even constraint Fz=&+ti2=p;+Mf+

2iM, aM, . M,

8, e,,

takes the form z 0,

(101)

76

CRATER

AND

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whose dependence on particle one’s spin in the last term comes from the supersymmetric x’ dependence of the interaction. The potential restriction (94) further simplifies this to

&=p:+M:+2iLM,.8,BS,.

(102)

The dynamical system defined by (99) and (102), though an interacting one, retains the supersymmetry of the free case. Since only one of the particles has spin, we have just a single supersymmetry generator:

Its brackets with x’, , I,, ,5‘;, X;, and & all vanish (weakly). This makes plausible that the three constraints ,U;, 8, and q will turn out to be compatible. We already know (.< , T} = 0 (as in the single-particle case) but we must check that (.Y, , .&) is zero. (We assumethat there is no momentum dependencein the potential except through x1 .) We find that {.U;,~}=a(M:-M:).8,-22aM,

.PB,,=O.

(104)

The conditions (94) and (95) on the potential imply that the right-hand side of (104) vanishes strongly. The one remaining compatibility condition {‘q ,$} = 0 can be evaluated using the Jacobi identity:

(,~*,X;}=-i(~~,(~U;,.U;}}=2i{.Y,,(.i,,

e}}=O.

(105)

Both ST and ,& have the same spin-dependentpart so that just as in the spinless case

Both *q and -3 are weakly equivalent to each other and to

,X.=p’-b2+@=Q+~$p0,

. (107)

W

where (with M, = m, + S,) ~=2m,S,+S:+2i~S,.e,e,,

= 2m,S, + Sf + 2iB,.

(108)

e,e,,.

Use of the variables m,, E,, and /t defined in (18) and (20) and the mass-potential forms of (17) and (22) give

3 = j’ + (m, + S)’ + 2i

as . e,e,, VI

z 0,

TWO-BODY

DIRAC

EQUATIONS

77

where from (22) we deduce 1271 q,=

rnf - mZ,. 1+ J (m,. + S)’ ’

(110)

In the static limit m, + co, we find m,,, + m, , q, + 1, /r +p, , and

j;"=p:+(m,tS)2t2ias.e,e,,~Oo,

(111)

which is equivalent to the one-body constraint (79). In the other static limit m, -+ co, we find q, --t co, m, + m,, /I +p2 so that the spin term disappears giving X=p:

t (m, t S)’ z 0,

(112)

equivalent to the same static limit of (21). We are now in the position to consistently quantize our constraints. That is, .4/,, 4, and & become quantum operators. Because of the correspondence (39) between the quantum and classical brackets, the proof of compatibility is rather straightforward, the steps almost identical to those of the classical case. In computing [Yi,&], one uses the quantum product rule (38) in place of the classical product rule (42) [30] with the result that extra quantum contributions of order h exactly cancel. Thus we find [Yi , &] _ = 0 just as in the classical case. Compatibility of these two quantum operators, together with -.q = ],~i*;, <4”;] + , implies that the same four-component spinor satisfies .i,y=o,

(113a)

qy=o,

(113b)

,qy=o.

(113c)

Using the quantum versions of (106)-(109) we may replace the above system of wave equations by the equivalent (consistent) system 9; y = 0. P*piy=O, Fly = 0.

(114a) (114b) (114c)

When (114b) is satisfied, then 4 w = 0 is equivalent to 4 w = 0 and they in turn are equivalent to 3~ = 0. Thus, because of the relation between q and .rJ,, the spectrum that we would find for w by solving A?+w= 0 is equivalent to the one we would obtain by solving 9, w = 0. We are now in a position to combine these three conditions on -I+Yto obtain two-component Pauli-forms analogous to those for a

18

CRATER

AND

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particle in an external field, (93). In the c.m. system the argument of the potential becomes ?2/2 and (114b) makes p* y =p”w so that (114~) takes the form ji” + 2m,,.S + S2 +

“,p”

) y = bZW,

As in the case of the static limit equation given in the previous section, the upper and lower components of the four-component spinor w = ([) are coupled by the spindependent term of this equation. The condit@n (114b) implies that the term p, . 8, y of the Dirac equation (114a) becomes (p’s 0, - E, 0;)~ in the c.m. system. Thus

where c, = &-

(w’ + rnf - m:), (117)

il4: = rn: + 2m,,S + S2. Once again the expression for R

i P’2+2mJ+S2+

can be brought to the decoupled Pauli-form iS’x’- p’ q(Elyy+M,)

(where b2 = E: - rni, mW = m,m,/w)

si . 5, - rjqE,y:+M,)

i

ty=b*y/

(118)

by using (116) in the form

w=(wsK-%p’y.

(119)

In the static limit m2 + co, we have m,,, + m,, M, + m, + S, r,r, + 1, and (118) goes into the Pauli-form (93) of the Dirac equation. In the other static limit m, --L CL), we find m,-t m,, r, + 00 and the spin terms go away leaving us with the Klein-Gordon equation (p’*+2m2S+S2)y/=b2~.

(120)

As do the Pauli-forms of Dirac’s equation, (118) gives both positive and negative total energy (w) solutions. In this paper, we discard the negative w solutions, always staying in the branch that is connected to the non-relativistic limit. For positive energies one solves the f equation (y” = 1) because its solutions are naturally connected to the non-relativistic limit. In practice, if one needs to find g it is easier to obtain it from (119) than to solve the g equation (y” = -1). The Pauli-form (118) (with y” = 1) is quite similar to a two-component form of Todorov’s quasi-potential equation for relativistic bound states of the spin-zero spinone-half system: p’*+2mM,S+S2-3

G’S 2w m, + E,

-m2

S’L ’ z, Nm, + E,)

ty=b’y.

(121)

TWO-BODY

DIRAC

EQUATIONS

79

The derivation of Eq. (121) is analogous to that of Todorov, Rizov, Krapchev, Karchev, and Aneva for vector interactions. Direct comparison of (118) and (121) shows that the main difference is the appearance of the mass potential M, in (118) in place of the free mass m, in (121) [31]. Thus (118) differs from (121) for strong potentials. On the other hand, for a weak potential they both produce the same spectrum. As an explicit example, when S = -a/r, we can show that both approaches yield the same perturbative fine structure through terms of order c?. To lowest order in the potential, M, and 11~are replaced by m, and w/ml [see Eq. (1 lo)]. Then the only difference between (118) and (12 1) is the difference between is’.?. p’ in (118) and -G’S/2 in (121). The latter term is simply the hermitian part of the former. The remaining non-Hermitian part, i/2(S’, .F{, gives zero if evaluated in first-order perturbation theory and thus has no spectral contribution to order a4. Todorov has discovered that his quasi-potential approach gives a technically simpler method for computing the spectrum than does the Fermi-Breit equation or other 0( l/c’) approximations to the Bethe-Salpeter equation that use Foldy-Wouthuysen-like transformations. Local energy-dependent potentials (e.g., 2m,.S) and the 6’ kinematical term in the quasi-potential approach give the same extra fine structure corrections as the more complicated momentum-dependent and non-local terms in the 0(1/c’) methods [see (161) vs. (167) below]. The similarities between the quasi-potential result (121) and the constraint result (118) leave all the advantages intact in the newer approach. Just like the quasi-potential approach, the constraint method results in a bona fide relativistic description, not just an 0( l/c’) approximation. However, as we discovered above, there is a major difference between (118) and (12 1) when potentials become strong. The appearance in the denominator of the mass potential in place of the free mass leads to physically acceptable behaviors in the static limit and strong field regions. In our equations the static limit m, + co reproduces the well-known Pauli-forms of the Dirac equation in which the potential S appears explicitly in the denominators of the Darwin and x . a’, terms. In the strong field region of the general two-body problem our equation (118) has a spin-dependent structure that has a well-defined non-perturbative behavior. Certain potentials can be used consistently in perturbative approaches in Eq. (121), the Fermi-Breit equation, and 0( l/c’) approximations to the Bethe-Salpeter equation but are ill defined in a non-perturbative sense. For example, if S = -a/r, then the coefficients 5” = a/r3 and V’S = 4xa&r) for t . a’, and the Darwin term, respectively, would prevent boundstate solutions for which those terms are attractive. This problem does not appear in the case of the unapproximated Dirac equation as the Pauli-form (93) indicates. It also does not appear in our Pauli form (118). With 5’ = -a/r,

VI=

13 M,-+m,.---,

a r

and

SL . a’, -ai . f?, I+-L. r3(e1 + m,,.) ar2 r2 VICE, + M,) +

if,,

which will never be attractive enough to overcome the centrifugal barrier term. The quantum mechanical wave equation of our spin-one-half spin-zero system is in the end no more formidable than the ordinary Dirac equation. In fact, the two-

80

CRATER AND VAN ALSTINE

component Pauli-forms are just as simple as the non-relativistic Schrodinger equation for a spinning particle in an external field. Of course, our equation (as in the spinless case) is not an eigenvalue equation in the ordinary senseof the word. It is a “nonlinear eigenvalue equation” since the eigenvalue appears in the potential. This is not a severe difftculty in non-perturbative calculations, however, since the same simple iteration schemenormally employed in the spinlesscase converges well.

VI. CLASSICAL

CONSTRAINTS FOR SPIN-ONE-HALF

AND QUANTUM SPIN-ONE-HALF

WAVE EQUATIONS SYSTEMS

We now turn to the most complex two-body problem, that of two interacting spinone-half particles. We saw in the previous section that the quantum mechanical wave equation (116) for the spin-one-half spin-zero system had a spin structure no more complicated than the ordinary Dirac equation. On the other hand, standard treatment of the spin-one-half, spin-one-half system using the 16-component Bethe-Salpeter wave equation leads us to believe that we are about to confront a formidable task. In fact, we shall see that this is not so. The spin-one-half spin-one-half case takes advantage of the full power of our method and results in a covariant wave equation that possesses not only an acceptable spin structure but also an immediate reduction without approximation to four-component form. Furthermore, the wave equation retains a potential-dependence in the denominators of the spin-dependent terms resembling that of the one-body Dirac equation and, consequently, is well behaved in regions of strong potential. Once again we return to pseudoclassicalmechanics. For two spin-one-half particles our description will need two sets of Grassmann variables 19,,, 19,~; a = 0, 1,2,3,5. Although the 8’s for a given particle must anti-commute among themselves, we shall assumethat they commute with the 8’s belonging to the other particle. That is, each particle has an independent Grassmann space. With the methods of Section III, this structure results from the application of the pseudoclassicalcorrespondenceprinciple to the standard description of two free spin-one-half particles with commuting setsof Dirac matrices. All the fundamental dynamical variables we deal with have definite even or odd character with respect to each space. The oddness or evenness is expressedby the relation

AJ, = %&,A,.

(122)

As we have seen, for a system involving one set of Grassmann variables, rlao = (-)“d”. However, for a system involving two independent sets of Grassmann variables, (%l$3I+%k’~nd (123) r ,0=(-l . Thus, for example, 19,~is odd in space 1 and even in space 2 so that E,, = 1, E,~ = 0. Likewise t& is odd in its own spaceand even in the other so that co, = 0, cqZ= 1. As

TWO-BODY

DIRAC

EQUATIONS

81

a consequence 19,~& = (-)” Bzoela = 8,,8,, 132). For this system, both the product rule (42) and the pseudoclassical Jacobi identity (47) retain their form with the appropriate v (123). The only new Poisson bracket is {e,,, 6z4} = 0. The differential realization of Poisson bracket is a straightforward extension of (48) to the combined system. We may now formulate the constraints in terms of these variables. With no interaction, the two first-order constraints

3; =p, . 8,+ m,e,,z 0, -4C,=p,~e,+m,e,,~o

(1241

are trivially compatible. To maintain compatibility and supersymmetry in the case of mutual scalar interaction, we must modify the x dependence to an x’ dependence for each variable, where

x’y=xL;+-,ieye,, M,

(125)

with ai = Gi(ZL), where p=

p+i

PUPL’ -P2

(f,,, - f2,).

(126)

Since there are now two sets of 8’s, our x” dependence in the M’s in the self-referent definition (125) does not terminate as quickly as it did in (98). Instead, we have

so that our direct interaction

takes place via

(128) where prime denotes derivative with respect to argument. Furthermore, since i@,=M,+iF.

3M, (129)

fi,=M,,i$. 1

82

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the first-order

constraints

AND

VAN

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become dM, . 02 $52 4 1 M z 0,

.u,=P1.e,+~,e,,=p,.e,+M,e,,-i

2

(130) .y2 =P2 * 8, + AT,O,, =p2 .8, + M,tl,,

+ i

%M,+95,~,2~o

M,

Note that in either static limit they reduce to the one-body second-order constraints are

constraint

(76). The

(131)

Even for the system of two spinning particles, these constraints depend on the squares of mass potentials. Once again 6 and .q have the same spin-dependence so that using the potential restriction, we find .$ -$ = 2P . p z 0. just as in the spinless case. We must now check whether the constraints (130) and (13 1) form a compatible system. Just as in the simpler case we first check that this use of ZL to generate interaction has left the supersymmetry intact. Our two supersymmetry generators are 3;=pi. Direct computation

ei+vqTft&,

i= 1,2.

using the expansion (127) (and &z p;, Zj} =: 0, i,

j=

1. 2.

(132)

0) shows that

(133)

Even though each zZi depends in a complicated way on both sets of spinning variables, each remains supersymmetric-invariant under the action of both .%“s. As a consequence of (133) the dynamics is left invariant by the supersymmetry transformation generated by the 59”s. Thus (,q, q}

z 0,

i, j = 1, 2,

(134)

and (
z 0.

(135)

The fact that our constraints have common supersymmetries makes it plausible that they will form a compatible system. At first sight, the check of compatibility appears to involve the computation of a large number of brackets. However, the Jacobi identity reduces the condition for compatibility to the single bracket {YI, ,Y2} = 0. A trivial application of the Jacobi

TWO-BODY

DIRAC

EQUATIONS

83

identity implies {. Y, , CT} = 0 = (. Yi, & }. The other brackets involve a slightly more complicated application of the Jacobi identity. For example, use of (13 1) and steps similar to (105) lead to

We see that all such brackets will vanish if (
(137)

strongly, so that all the other compatibility conditions [e.g., (136)) are satisfied as well. Thus we see that just as in the spin-zero spin-one-half system, the supersymmetry eliminates all spin complications and reduces (pseudoclassical) consistency problems to those of the purely bosonic spin-zero spin-zero system. Just as in the spin-one-half spin-zero case .q and Lq have the same spin-dependent structure. Thus they are weakly equivalent to each other and to

where from (131) and (129) @=2m,S,+S:tZ

= 2m,S, + S: + C. The term L is the spin-dependent

Z= 2iaM,

(139)

piece:

. e,es, - 2iaikfL . e,e,,

Use of the variables E,,, m,, and /r for the effective particle of relative motion (as defined in (18) and (20)] along with the mass-potential forms (17), (22), and (23) gives

F=

/z’ + (m, f S)’ + 2i as. 04, VI

+e, z 0.

4

1

( v1 v2(m, + 9

e,a

_ 2i as . e2e5, )I2

1

e,,e,,+e,.a

1 ( yI, q2(m,. t S) ” * as 1 e5’e52 (141)

84

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In the static limit m, + co, we have m,+ m,, q2 + co, q, + 1, /r +p,, reduces to .i%O=p:+(M,+S)*+2ias.e,8,,~00,

so that .X (142)

which is equivalent to the one-body constraint (79). The other static limit leads to an analogous expression. We are ready now to prove the compatibility of the operator versions of .P, , .i,, ,q, and (6. All we have to do is to verify that [.U; , .Yi] _ = 0. That this is a commutator and not an anti-commutator follows from the fact that though .Y, and CU; are odd in their own spaces, they are even with respect to each other [see (36) and (123)]. Th e computation of this commutator is almost identical to that of the pseudoclassical bracket (.Uj , .V;} [30,32] so that [
=o.

(143)

Since this fundamental compatibility condition is satisfied, all the others involving .4”;, -q, ,!?i, and CT vanish as a consequence of -
[9;,P;]+ = p;,

zoo,

,9;;1+ z 0,

(144)

and the Jacobi condition. In the language of Dirac [ 151, our constraints are “first class” at both the pseudoclassical and quantum levels. Although ordering problems could have destroyed compatibility of the constraints in the transition from the pseudoclassical to the quantized system, such was not the case. The extra terms of order fi cancel exactly. Apparently in a system of this type, the quantum remnant of our pseudoclassical supersymmetry takes care of all quantum spin complications reducing ordering problems to those of a spinless system. The quantum constraints simultaneously act on a 16.component wave function: .Y, I//= 0,

(145a)

.Y,yl=O,

(145b)

~ql//=

0,

&y/=0.

(145c) (145d)

The quantities 4 and ;F; given by (144) each contain an extra quantum piece (&M,/M,)*@, f?:, = (aM,/M,)* f?:,@, that vanishes in the classical limit when the B’s become Grassmann variables. Use of the quantum version of (138) and the difference ,e -4 permits us to replace the above equations by 9; yJ = 0,

(146a)


(146b)

P.ply=O, Aq

= 0.

(146~) (146d)

TWO-BODY

DIRAC

85

EQUATIONS

The spectra of 6 and & will be the same as that of ,r when P 1py = 0. Because of (144) .U; will have the same spectrum as X;, ,Y1 will have the same spectrum asX1, and thus LYr will have the same spectrum as ,Yz. This implies that the spectrum of F will be the same as those of Y1 and %U;. This fact will permit us to use the constituent Dirac equations (146a-b) to simplify the .X equation, which we will then use to determine the spectrum. We shall see that the 16-component 3 equation separates into four uncoupled 4-component Pauli-forms. Equation (146~) along with B,,=i L/z ysi’

i= 1,2,

(147)

and (22) imply that (146a) and (146b) become the following [where Si = S,(?/2)] :

where D,=

forms in the c.m. system

1 (150)

VI r/2(%. + s> .

When m2 -+ co, (148) goes into the Dirac equation (77) for a single particle in a static external scalar field. That is, the extra recoil-dependent spin term vanishes since qz + co. In the other static limit, r7, --t co and (149) goes into the Dirac equation (77) only this time for particle 2. The A++ equation (146d) involves a coupling of all 16 components of I+V.The use of (146~) leads to the c.m. form ~2+2m,S+S2+i~+ $yi . ?(D,y’,

yi . Gs VI . ?S)

.yi * vs I~ + fD;(k)’

+ a?, . G(Dsy2 . VS)

v2

I// = b’ly.

(151)

The decoupling of (151) requires as a first step alternate forms of (148) and (149):

v=(&2Y;4w1

-i;,+f+D,yt,

. GS

(153)

86

CRATER

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Partial decoupling results when these two terms are substituted fifth terms on the left-hand side of (15 I ). We find

We choose matrices:

The direct function

the following

product

representation

structure

implies

for our

into the fourth and

16.dimensional

[y, i, y,,] _ = 0. The

Dirac

16.component

gamma

wave

(157)

is a column vector of four four-component spinors. In this representation, while JJ~ and yy are diagonal, the form ~7 yS, $ yS2 appearing in (154) is not. In fact,

(158)

Thus Eq. (154) is equivalent to four 8-component Pauli-forms with y, coupled to vq and w2 coupled to v/, . The first line and the first term of the second line of (154) are diagonal while the coupling to the &component form is due to the rest of the second and the third lines. The crucial advantage of the constraint approach is that it provides two constituent Dirac equations. We have not yet made full use of this fact. When we do, as we shall see below, complete decoupling of (154) will take place in such a way that we obtain

TWO-BODY

DIRAC

87

EQUATIONS

four four-component Pauli forms. We start with the forms (152) and (153) of our Dirac equations and combine them to give

We then substitute (152) into the right-hand side of (159) and (153) into the left-hand side of (159). Performing simplifying manipulations leads to

=-iz.

(a’,xa’,)(2m,S+S*-b*)(~,y~-M,)-‘(~~y~-M~)~’

y(:ys,y~ys2~.

(160)

This can be further reduced by multiplying both sides by i . (a’, - C?J. The result is an expression for v/ that provides an extra coupling between v/, and w,, and between v/2 and v~. Substituting this expression for w into the second and third lines of (154) gives an eigenvalue equation for ,+Ythat involves no coupling among the subspinors w, , w2, wj, and w4. We write the resulting equation as p” + 2m,.S + S* + aDf.(?S)*

1 si rll(&IYY + M,)

+ i

w,?4

V2(&2Yi

1

S’T. p’

+ M,) + ‘Izb-2Y1 + M2)

+

1

. a’, -

where S’ = (l/r)(aS/ar)

1

Si’a,+G(y:,y;)

M2)

I

v=b’v,

(161)

and

x

[f(2m,.S+s*-b*)(&,y~+M,)~‘(E*~~+M,)~’~.

(162)

(The plus sign holds for spin singlet states and the minus sign for spin triplet states.) Equation (161) is equivalent to a set of four uncoupled four-component wave equations since yy and yi are diagonal in our representation. In either static limit (m, + co or m, -+ IX), the G term vanishes and (161) reduces to the correct one-body Pauli-form (93). Although we have four separate wave equations we do not need to solve all of them to obtain a complete solution for the wave functions and eigenvalues. In practice one solves the w, eigenvalue equation (yy = 7: = 1) and uses (152) (153), and (160) to find w2, v/~, and v/~ in terms of w,. Note that the spin-dependent

88

CRATER

AND

VAN

ALSTINE

terms in (16 1) possess potential-dependent denominators just as they do in the spinone-half spin-zero case and in the Pat&forms of the original Dirac equation. We now compare our equation with Todorov’s quasi-potential result for scalar interactions. His method leads to the four-component equation ~2+2m,S+S’-;

i c2

m,

+

m2

+ m2 5 +m,

?‘S --psr.

S’i+-;2)+R

.s

I

y=b2y/,

1 (163)

where

s= s’,+ T2= f(f?,+iT2).

(164)

Our Pat&form that is most like (163) is the v, form of (161). Both (163) and this form of (16 1) will produce the same spectrum for a weak potential such as S = -a/r for small a. In a perturbative calculation of line structure corrections through order a4 for this potential R consists of complicated spin-dependent terms that will not contribute. To this order, w, E,, and E, can be replaced by m, + m,, m, , and m,, respectively, in the remaining spin-orbit and Darwin terms. Thus for fine structure results through terms of order a4, the quasi-potential equation reduces to p” i- 2m,.S + S2 -

2(m,:m2j

~~-~)S’I.(S;-~~)~y=b2~.

(165)

To obtain from the v, form of our equation (16 1) a comparable equation valid through O(a4), we drop the squared gradient term and the G terms since they (like the R term) have a fine structure contribution beginning at order 06. In the remaining spin-dependent and Darwin terms we replace M, , E, , and M,, &2 by m, and m2, respectively. Thus, for w, , (16 1) takes the form

Si Direct comparison they produce the the Darwin terms Note that both relativistic wave

. (s’, - si)

I//, = b2y/, .

(166)

of our (166) and the quasi-potential equation (165) shows that same fine structure through terms of order a4 for S = --a/r, since differ only in higher order. our equation (161) and Todorov’s equation (163) are bona fide equations, not 0(1/c’) approximations. Hence their simple

TWO-BODY

DIRAC

EQUATIONS

89

momentum dependence holds up to all orders of u/c. This is in stark contrast to 0(1/c*) truncations of the Bethe-Salpeter equation such as 1341 W=(w-m,

H- F’ p” ‘K-m2m,

p”

-m,)y/,

(167)

p’”

1 s7. as’,-2m:

x ({p’. sg-j?.

JS * r’.p’}}

(168)

In this sort of truncation, the kinetic corrections get successively more complicated as one goes from one order to the next. As in the spin-one-half spin-zero case both our constraint approach (166) and the quasi-potential result (165) provide a technically simpler method for computing the perturbative fine structure corrections to the spectrum than does the truncation. Furthermore, our Pauli form (161) and (163) represent relativistic descriptions, not 0( l/c*) approximations. Although all of these formulations agree on the line structure they differ drastically non-perturbatively. As in the considerably simpler spin-zero spin-one-half case, the major difference between our Pauli-forms and the corresponding Fermi-Breit and Todorov equations lies in our potential dependent denominators. The denominators in (161) reproduce the corresponding denominators of the exact one-body Dirac equation (93) in the static limit.Equations like Todorov’s in (163) and the FermiBreit form (167) lack these denominators and therefore do not reproduce the Dirac equation (93) in the static limit. The denominators in our Pauli-forms have an even more important effect on the dynamics of the two-body problem (not shared by the Todorov [3 l] and Fermi-Breit-like equations). In regions of strong potential our Pauli-forms have a spin-dependent structure that has a quantum-mechanically welldefined non-perturbative behavior. This is not true of (163) or a number of FermiBreit-like forms that have appeared in the literature in connection with phenomenological quark potentials similar to -a/r. The typical remedy used by several authors is to introduce smoothing parameters (-a/r-+ -a/(? + a*)“*) or finite quark sizes [25]. From our point of view this remedy is not only unnecessary but also obscures the point-like nature of the field-theoretic constituents (quarks), which is a standard assumption of QCD. Our equations provide an important improvement if this denominator structure persists in generalized Dirac equations with both vector and scalar potentials based on QCD (for small r behaviors -8n/27r In Jr).

90

CRATER AND VAN ALSTINE

VII. CONCLUSION The most important results of this paper are the compatible wave equations (146a,b) for two spin-one-half particles, and Eqs. (113a,c) for one spin-one-half and one spinlessparticle-the two-body Dirac equations of the title. These equations are the operator versions of the classical constraints given in Eqs. (130) and Eqs. (99) and (101). If we trace the spin structure of the wave equations and their Pauli-forms back to the classical mechanics, we seethat it is virtually dictated by the dependence of classical interaction on the supersymmetric position variable, 2. As we have seen, x’ and its supersymmetry are essentialingredients even in the classical mechanics that underlies Dirac’s own one-body equation with external scalar potential. Consequently, by properly extending 2 to the case of two particles, our procedure ultimately leads to wave equations that stretch both perturbative and non-perturbative structures of Dirac’s to the two-body problem. Hence our equations possesstwo-body kinematics, static limits to one-body relativistic wave equations, correct line structure, and non-perturbative quantum-mechanical meaning. Their surprising separation into decoupled Pauli-forms makes them particularly attractive for phenomenological application. The methods detailed in this paper for the scalar interaction can be extended (with the required complications) to time-like and space-like vector interactions and to interacting systems possessingall three. With realistic potentials, we may eventually be able to apply our methods to electrodynamic or strongly interacting systems of relativistic spinning particles. In particular, we hope to seewhether our use of a single relativistic potential for both light- and heavy-quark systems [ 171 holds up under the introduction of spin-dependentforces.

VIII.

SUMMARY

OF IMPORTANT

EQUATIONS

AND DEFINITIONS

The kinematical variables relevant to the constraint description of the relativistic two-body problem are: 6) (ii)

relative position, x, - x2 ;

(iii)

total c.m. energy, w = g;

(iv)

total momentum, P = pI +pz ; constituent c.m. energies,

(VI

relative momentum, p = 1/w(c2p, - c, pz);

E, = (4 motion,

w2+mf-m~ zw

w'+mi--mm: 9

62

=

zw

;

relativistic reduced mass and energy of the fictitious particle of relative

m I(' =- mlm2 w '

E,. =

w'-mf-rn: 2w

;

TWO-BODY

(vii)

DIRAC

91

EQUATIONS

on-shell value of the relative momentum squared,

b*(w) = E:,.- mt,. = E; - rnf = EI - rni = --$ Scalar interactions

(w” - 2w2(mf + rni) + (rn: - mi)‘).

are introduced

through constituent

mass potentials

i= 1,2,

Mi = m, + S,(x_).

(S-l)

where (S-2) They are not independent but are related through

Mf = rn: + 2m,,.S + S* = (m,,. + S)’ VT,

(S-3)

Mi = rni + 2m,,.S + S* = (m,,. + S)’ vi.

(S-4)

where S is an effective scalar potential. For reference, we give the c.m. form of the two-body

Klein-Gordon

equation

(p” + 2m,,.S + S’) v/ = b2(w) y which results directly from quantization

of the compatible constraints

q =p: + kqx,) & For a spin-zero spin-one-half equation (in the c.m. frame) is

p” +2m,.S+S*

+

(S-5 )

=p: + Mf(x,) system,

is’.?..6 tldw~f~,)

zz 0,

(S-6)

z 0.

(S-7)

the Pauli-form

s/7, . a’, - vh4+Wf

of the two-body

IJ/= b’(w)y,

Dirac

(S-8)

where S’ = (l/r)(aS/&) and r = a. Th e wave function I+Vhas four components, but since yy is diagonal this equation reduces to an uncoupled set of two twocomponent wave equations. Equation (S-8) results directly from quantization of the compatible supersymmetric constraints ci”; = 81 *p, + e,,M,(f,)

4 = p: + kq-f,)

=: 0,

Jq = p: + M:(x’,) z 0,

z5 0,

(S-9) (S-IO) (S-l 1)

92

CRATER

AND

VAN

ALSTINE

where

x’y= g@”- 7) i

(XI,- X2)r

(S-12)

with iO,85, XI = Xl + -&p-J’

(S-13)

The 8’s are Grassmann variables and are related to gamma matrices upon quantization. For a spin-one-half spin-one-half system, the Pauli-form of the two-body Dirac equation is j’ + 2m,S sz

t t S2

ADS’

+

. c?,

rll@f, + El YY>-

sz 112w2

t

iS’i?afi rl,w, + w4

+ t * CT2

G(y:, 7;)

E2Y3

is’.?. p’ vzw, + E2YP> I

I = b’(w)w.

(S-14)

This c.m. form involves a 16-component wave function. The function G is defined by Eq. (162) in the text and for weak scalar interactions provides corrections of order ah (i.e., smaller than fine structure). Since all terms (including G) depend only on the diagonal gamma matrices ri and ri, this equation reduces to an uncoupled set of four four-component wave equations. Equation (S-14) directly results from quantization of the compatible supersymmetric constraints 2; = 8, ‘p, t 8,,M,(f,)z

0,

(S-15)

9; = 8, ’ p2 + B,,M,(2,)

z 0,

(S-16)

-;it; =p; + M;(q

z 0,

(S-17)


z 0,

(S-18)

where (S-19) with . i4 es1 Xl = XI + M,(q’

(S-20)

M

(S-2 1)

x2=x2+$qg.

iQ20, 2

TWO-BODY DIRAC EQUATIONS

93

ACKNOWLEDGMENTS We wish to thank Ingram Bloch for helpful discussion of all aspects of his work and ours on relativistic dynamics and wave equations for spinning particles. We especially wish to acknowledge Michael King for communicating to us his unpublished work along parallel lines.

REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16.

17. 18. 19. 20. 21. 22.

P. A. M. DIRAC, Proc. Roy. Sot. Sect. A I1 7 (1928), 6 10. G. BREIT, Phys. Rev. 34 (1929), 553. C. G. DARWIN, Philos. Mag. 39 (1920), 531. E. E. SALPETER AND H. A. BETHE, Phys. Rev. 84 (1951) 1232. H. A. BETHE AND E. E. SALPETER, “Q uan turn Mechanics of One and Two Electron Atoms.” Springer-Verlag, Berlin (1957); C. ITZYKSON AND J. ZUBER, ‘iQuantum Field Theory,” McGrawHill, New York (1980), especially Chap. 10. N. NAKANISHI, Suppl. Prog. Theor. Phys. 43 (1969), 1. I. T. TODOROV, in “Properties of the Fundamental Interactions” (A. Zichichi, Ed.), Vol. 9, Part C. pp. 953-979, Editrice Compositori, Bologna (1973); Phys. Rev. D 3 (1971) 235 1. B. L. ANEVA. N. I. KARCHEV. AND V. A. RIZOV, Bulg. .I. Phys. 2 (1975) 409. V. A. RIZOV, I. T. TODOROV, AND B. L. ANEVA, Nucl. Phys. B 98 (1975). 447. A. ATKINSON AND H. CRATER, Phys. Rev. D 11 (1975), 2885. D. G. CURRIE, T. F. JORDAN, AND E. C. G. SUDARSHAN,Rev. Mod. Phys. 35 (1963), 350, 1032. I. T. TODOROV, JINR Report No. E2-10175, Dubna (1976); I. T. TODOROV. Ann. Inst. H. Poincare 28 (1978). 207. M. KALB AND P. VAN ALST~NE, Yale Reports, COO-3075-146 (1976), COO-3075-156 (1976): P. VAN ALSTINE, Ph.D. dissertation, Yale Univ. (1976). PH. DROZ-VINCENT, Phys. Rev. D 19 (1979), 702; A. KOMAR. Ph.w. Rev. D 18 (1978). 1881, 1887. 3617; F. ROHRLICH, Ann. Phys. (N.Y.) 117 (1979), 292; M. KING AND F. ROHRLICH, Phys. Rev. Left. 44 (1980), 621; T. TAKABAYASI, Prog. Theor. Phys. 54 (1979). 1235: D. DOMINICI. J. GOMIS. AND G. LONGHI, Nuovo Cimento B 48 (1978), 152. P. A. M. DIRAC, Canad. J. Math. 2 (1950), 129; Proc. Roy. Sot. Secf. A 246 (1958) 326; “Lectures on Quantum Mechanics.” Belfer Graduate School of Science, Yeshiva Univ.. New York (1964). L. P. HORWITZ AND F. ROHRLICH, Phys. Rev. D 24 (1981) 1928; F. ROHRLICH. Phys. Rev. D 23 (1981). 1305; H. SAZDJIAN, Nucl. Phys. B 161 (1979) 469, and “Relativistic and Separable Classical Hamiltonian Particle Dynamics,” Orsay preprint IPNO/TH 8 l-4 (198 1); V. M. PENAFIEL AND K. RAFANELLI, “Canonical Formalism for Relativistic Dynamics,” Queens College preprint, Oct. 3. 1981; I. T. TODOROV, “Constraint Hamiltonian Mechanics oi Directly Interacting Particles.” Barcelona Workshop on “Relativistic Action at a Distance-Classical and Quantum Aspects,” June 1981. H. CRATER AND P. VAN ALSTINE, Phvs. Lett. B 100 (1981). 166. D. B. LICHTENBERG, W. NAMGUNG, AND J. G. WILLS, Phys. Lett. B 113 (1982) 267. This guesswork can be dangerous. For example, the two-particle interaction proposed by Ph. Droz Vincent in Lettere al Nuovo Cimento, 30, 375 (1981), is pure gauge. H. C. CORBEN. “Classical and Quantum Theories of Spinning Particles,” Holden-Day. San Francisco (1968); A. J. HANSON AND T. REGGE. Ann. Phys. (N.Y.) 87 (1974), 498; K. RAFANELLI. J. of Math. Ph.w. 9 (1968). 1429. R. CASALBUONI, Phys. Letf. B 63 (1976) 49, Nuovo Cimento A 33 (1975) 389: F. A. BEREZIN AND M. S. MARINOV, JETP Left. 21 (1975) 678, Ann. Phys. (N.Y.) 104 (1977), 336. Our particular supersymmetry transformation [Eq. (64)] is to be contrasted with those studied by C. A. P. GALVAO AND C. TEITELBOIM. J. Math. Phys. 21 (1980), 1863. A. BARDUCCI. R. CASALBUONI.

94

23. 24.

25. 26.

27. 28.

29.

30.

31.

32. 33. 34.

CRATER

AND

VAN

ALSTINE

AND L. LUSANNA, Nuovo Cimento A 32 (1976). 377, AND F. RAVNDAL. Phvs. Rec. D 21 (1980). 2823. In addition to those listed above. see also L. BRINK, P. DIVECCHIA. AND P. HOWE, Nucl. Phys. B 118 (1977). 76, and more recently L. BRINK AND J. SCHWARTZ. Phys. Lett. B 100 (1981). 310. P. VAN ALSTINE AND H. CRATER, “Scalar Interactions of Supersymmetric Relativistic Spinning Particles,” J. Math. Phys. 23 (1982) 1697. The one-body action of this work and its consequences are modifications of those given by Galvao and Teitelboim, op. cit. D. B. LICHTENBERG AND J. G. WILLS. Phys. Rec. Lett. 35 (1975). 1055: D. P. STANLEY AND D. ROBSON. Php. Rec. D 21 (1980). 3180. In fact some of the earlier success of the quasi-potential method in treating the quantum problem inspired a version of the relativistic two-body problem based on the h --t 0 limit of Todorov’s quasipotential approach. The resultant equation for the effective “Hamiltonian” resembles that developed later with the aid of constraint methods. H. CRATER AND J. NAFT. Phys. Rev. 120 (1975). 1115. The variables vi and n2 defined by these equations are needed to preserve the correct sign structure in the static limits of the two body spin constraints of Sections V and VI. The commutativity of x” and p” is not immediately obvious in the presence of Grassmann variables with the units of (action)“‘. Candidates for [.P.p”] such as i(S”0’ - 8’8”) are ruled out since they do not have zero bracket with ~9”. a property that [x”,p“] has. Similar comments apply to the anticommutativity of the 0’s and the commutativity of the B’s with x and p. An .f-like variable for spinning systems without Grassmann variables has been used by M. H. L. PRYCE. Proc. Roy. Sot. 199 (1948). 62: F. HALBWACHS. Nuoco Cimento 176 (1964). 832; AND K. RAFANELLI. Canad. J. Ph.vs. 50 (1972). 2489. Other x’s have been used with Grassmann variables by Barducci. Casalbuoni. and Lusanna, op. cit., and F. Ravndal, op. cit. lthe latter’s version with an independent Grassmann variable 19 replacing B,(r)l. Recently, Brink and Schwartz, op. cit., used an I to arrive at a quantum mechanics for a single spinning particle in an external potential. Note that in the proof of the pseudoclassical compatibility the order of the variables is to be kept in a form dictated by the product rule (42). This makes the transfer from the pseudoclassical to quantum proof unambiguous (at least for potentials independent of p3. This difference can be traced to the use of free particle spinors in the construction of the quasipotential. Apparently. our two-body Dirac equation contains off-shell information that is missing from the quasi-potential approach. It is just this information that permits our equations to agree with the Dirac equation in the static limit. See appendix to our preprint “Two Body Dirac Equations,” University of Tennessee Space Institute and Vanderbilt University preprint. July 1981. Oddly enough all of the properties demonstrated below hold even if the Grassmann variables belonging to different particles anti-commute. Use of the product rules shows that the quantum and classical consistency conditions are algebraically identical if the squares of the B’s for each particles are set equal I24 I. D. GROMES. Nuci. Phys. 5 131 (1977), 80: H. J. SCHNITZER, Phw. Rev. D 13 (19761, 74.