Relativistic dynamics for N-body systems

Volume 208, number 3,4

RELATIVISTIC DYNAMICS

PHYSICSLETTERSB

21 July 1988

F O R N-BODY SYSTEMS

H. SAZDJIAN Division de Physique Th~orique ~, Institut de Physique Nucl~aire, Universit~Paris XI, F-91406 Orsay Cedex, France Received 11 April 1988

N spin-0 particle systemsare describedby means of N manifestlycovariantrelativistic waveequations. Thesehave the following properties: The relative time evolution laws are determined in a kinematic way. The dynamics is governed by a single 3N- ~dimensional wave equation. An approximate form of separability is satisfied.

Manifestly covariant two-particle relativistic quantum mechanics [ 1-5 ] describes two-particle systems by means of two independent wave equations. This permits the kinematic determination of the relative time evolution law of the system and therefore the reduction of the internal dynamics to a three-dimensional one, besides the contribution of the spin degrees of freedom. It is this feature that makes two-particle relativistic quantum mechanics rather attractive from the practical point of view. In particular, it circumvents the difficulty of relative energy excitations inherent to the Bethe-Salpeter equation [6]. On the other hand, a non-singular transformation establishes the connection of the quantum mechanical equations with the Bethe-Salpeter equation [7]. The aim of the present paper is to generalize, for spin-0 particles, the quantum mechanical relativistic wave equations to the N-body ( N > 2) case, by incorporating as much as possible the separability (cluster decomposition) requirement. It is however known that when the latter condition is taken into account exactly [ 8,9 ], then the interaction potentials do no longer appear in a closed form. Furthermore, one also loses, at least in its explicit form, the factorization property of the motion into a kinematic time dependent part and a dynamical 3 N- ~-dimensional internal part. This is why, for the purpose of practical applications, we shall realize the separability condition in an approximate form, while maintaining the

closed form of the interaction potentials and the factorization property of the motion quoted above. N-body relativistic systems were also considered in the past [ 10-12], but the systems that have been proposed do not possess the correct non-relativistic limit in the unequal mass case, neither are they reducible to a free N-particle system in the absence of interaction. We first consider the case of N free particles and search for an appropriate description to make the kinematic factorization property of the time evolution laws explicit. It is evident that the wave function must satisfy here N independent Klein-Gordon equations:

(p]-m])ga(x,

470

( a = l .... ,N),

(1)

where Pau = ih O/OxU~.We consider wave functions that are eigenfunctions of the total momentum p. We use the notation N

P = a=l ~ Pa,

Pu l~u--(p2)1/2'

XaL =)Ca "1~, PaL =P'Pa,

N

P 2>0'

M=

~" ma,

a=l

Nab =Xa --)CO

(a, b - - l , ..., N), T

P a l t = P a l, - - ( P ' P a ) P l Z , p Ta 2 w ~ '-/ 2 /a --

1 u

Laboratoire associ6au CNRS.

..... XN)=O

(P'Pa) 2 ,

X = ~ ~= X~.

XTblt ~-Xablz - - ( P ' X a b )P,u , ~T2 ~2 .,o~ =.-,,,~-

(6.x~0

~ ,

(2)

0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

Volume 208, number 3,4

PHYSICS LETTERSB

Notice the decomposition of vectors into longitudinal and transverse components with respect to the total momentum p. The longitudinal coordinates XaL play the role of individual covariant time variables (in the CM frame they reduce to the usual time components). Their conjugate variables are the longitudinal momentum operators PaL, which play the role of individual covariant energy variables. The latter satisfy the identity N

paL=(p2) '/2 .

(3)

a=l

the longitudinal momenta PaL solely in terms of p2 and the free masses. Therefore, the present representation is not adequate for this purpose. It is necessary, for N > 2 , to modify the wave function representation in order to solve the above problem. To this end, we introduce a canonical transformation that removes from the right-hand sides of eqs. (5) the transverse momenta and thus leaves the new longitudinal momenta as functions o f p 2 and the free masses only. Let U be the operator of the canonical transformation. It can be written in the form

ex(i

The kinematic factorization property of the time evolution laws means that the XaL dependences of the wave function factorize into exponential functions:

where k.L are functions of the transverse momenta, the total mass squared p2 and the free masses. They also satisfy the condition

T(Xl .... ,XN) = e x p ( - - i a=l ~ XaLPaL) ~l(xTc'''')"

(4) Here the PaL represent well defined eigenvalues. They must be kinematically expressed (i.e., independently from the transverse momenta paT, which are dynamical operators) in terms of the total mass squared eigenvalue p2 and the free masses of the particles. The function q/represents the internal wave function. It depends on N - 1 independent transverse relative coordinates (2). The internal dynamics is therefore 3N-~-dimensional because the transverse coordinates have only three independent components. (The total wave function ~ is also an eigenfunction of the total momentum p, the internal wave function ~udepends on relative coordinates only. ) The factorization (4) occurs in the two-particle case [4]. We shall however show that for N > 2 it fails in the above "Klein-Gordon" representation and that a canonical transformation has to be performed in order to restore it. By subtracting two wave equations ( 1 ) from each other (for different values of the index a) and by introducing the longitudinal and transverse components of the momenta, (2), we get 2 2 (PaL--PbL) ~ = [ ( m 2 - m 2) - (pT2 _p~2)l

( a , b = l .... , N ) .

21 July 1988

(5)

Except in the two-particle case, where pl2 =pT2, it is not possible to express through eqs. (3) and (5)

N

2 kaL=0,

(7)

a=l

which means, according to eq. (3), that the total mass squared is not affected by this transformation. The operator U, (6), transforms the longitudinal momenta P,L into paL: PaL --}PAL =PaL -I- kaL .

(8)

The operators kaL are chosen in such a way that the transformed wave function is an eigenfunction of the longitudinal momenta PaL with eigenvalues satisfying the relations

p2L--P~L=m2--m~

( a , b = l .... , N ) .

(9)

We divide the above relation by PaL'St'PbL,sum all the relations with respect to the b's for fixed a and use relation (3). We end up with the N (non-independent) equations

NPaL--

ma--mb --(p2) 1/2 ( a = l ..... N ) .

b= 1 PaL "t-PbL

(10) (The sum of these equations with respect to a yields the identity (3). ) Except for the two-body case, these equations cannot be solved analytically in a closed form. However, an approximate solution can be found which can be used for successive iterations. This will be presented below. 471

Volume 208,

number

3,4

PHYSICS LETTERS B

On the other hand, the operators kaL must satisfy the relations (PaL "3i-kaL ) 2 - (PbL "3f"kbL ) 2

(p2)1/2 PaL

~T2~

t~'a --Pb I+(PaL--PbL)(kaL +kbL)

PaL +Pbe + k,L + kbL

( a = l ..... N ) .

=0 (12)

An approximate solution for the k's can be obtained by retaining in eqs. (12) those terms which have contributions in the non-relativistic limit. In this limit P,L and pT2 behave as c 2 (c is the velocity of light). Therefore, the kaL behave as c °. We can then neglect in the denominators of the second terms of eqs. (12) the k's. We get a system of linear equations for the k's that can be solved exactly. The solution is ~T2

va - w

k~=b=l

( a = l ..... N ) .

~T2 \ ] [ N

Ill~

x 1

) (13)

One can use these expressions to get improved approximations of the solutions of eqs. (12) by an iterative procedure. Notice that to the next order of the approximation the k's cease to be quadratic functions of the transverse momenta. The above procedure can also be used to find approximate expressions for the PAL'S. These should yield the correct non-relativistic limit up to order c °. In this limit PaL behaves as maC 2 + ha. By using such an expansion in eqs. (10) one gets linear equations for the h's that can be solved exactly. These solutions serve to obtain covariant versions for the expressions of the PAL'S. These in turn are replaced in the denominators of the second terms of eqs. (10). One finally gets the approximate solutions

472

-

N

ma--m

( a = 1.... , N ) .

(11)

(cf. eqs. (5) and (8)). We proceed similarly as above. We divide eqs. ( 11 ) by PaL+PbL'+kaL+kbL, sum all the equations with respect to the b's for fixed a, use eqs. (9) and ( 1O) and end up with the equations

h= ~

-

N b=~ l + [ ( P 2 ) W Z - - M ] / ( 2 m a m b ~ , U = ~ l / 2 m ¢ )

(a,b=l,...,N)

t'~T2

--

+!

= ( m ] - m 2) - (pV2 _pT2)

Nk, L +

21 July 1988

(14)

These expressions are also well defined in the ultrarelativistic case, when one or some of the masses vanish. In the two-particle case, expressions (14) reduce to the exact solutions ofeqs. (10). As for the k's, the approximate solutions (14) can be used in an iterarive procedure in eqs. (10) to improve the approximation (for N > 2). Thus, we have transformed the free wave equations ( 1 ) by means of the canonical transformation (6) in such a way that the longitudinal momenta have eigenvalues given by eqs. (10). This means that the covariant time dependences, XaL, (2), of the transformed wave function are defined through exponential functions as in the decomposition (4). The Nnon-independent eigenvalue equations (10) can be considered as N wave equations defining the covariant time evolution laws of the system. There remains to define an Nth (independent) wave equation, which should govern the dynamics of the system. This equation is given by the sum of the wave equations ( I ). Upon using transformations (8), one gets the equation satisfied by the internal wave function ~,, (4), N a=l

(P]L+2p~LkaL +kaL+P~ 2 T2 --m~)N(x~b 2 T .... )

=0,

(15)

which fixes the eigenvalue o f p 2. Since the exact expressions of kaL, (12), are not simply quadratic functions of the transverse momenta, we conclude that the wave equation (15) is not a second-order differential equation in the spacelike coordinates, but has a more complicated structure. This is the price that is paid for the factorization property (4) of the wave function. If we use the approximate expressions o f k , L, (13), and neglect the quadratic terms in the k's in eq. (15), we get a differential equation which is of second order in the spacelike coordinates:

Volume 208, number 3,4

Pa/

PHYSICS LETTERSB

PaL

2

body potentials. In this case V~ is given by a sum of two-body potentials Vab:

T

~ ( pZL+N~N-T2"21/2PbL ma ) gt(X~b, ...) =0.

a~l

21 July 1988

b=l

(16) This equation could be considered as the zeroth-order approximation of the exact equation (15). The corrective terms might be treated as perturbations to improve the result about the eigenvalue ofp 2. The k's are functions of the differences between the individual "kinetic energies" in the CM frame. In the mean these quantities might be expected to be small in bound states. We now turn to the interacting case. The final wave equations we obtained in the free case suggest the way interaction should be introduced. The N non-independent wave equations (10) have a kinematic nature and should not be modified by the presence of interactions. The factorization (4) of the wave function remains therefore unchanged. It is the dynamical equation ( 15)-(16) which should feel the presence of interactions. Particle a, say, feels an interaction potential Va which enters additively into its "kinetic energy" term --pav2. The latter, therefore, undergoes the modification

_p~2__, _p~2 + vo.

(17)

Eq. (16) then becomes

~ (pZL +N (pvJ-Va)/2paL --m~)¥(x~b,.)2

-r

ZN=I 1/2pbL

a=l

=0.

(18)

The modifications (17) should also be used in the exact equation (15) through the expressions of the operators k~e. In order that the wave equation (18), or the analog ofeq. ( 15 ), be compatible with the other wave equations (10) and the factorization property (4), it is necessary that the potentials V a be independent of the longitudinal coordinates XbL ( b - l , ..., N) (2). Therefore they depend on the transverse relative coordinates X~b, (2), and eventually on the momenta. (Translation invariance implies that only relative coordinates appear in V, while Lorentz invariance implies that the Va are Lorentz invariant functions of their arguments. ) In the following we shall confine ourselves to two-

N

vo= Z Va~.

(19)

b=l

bv~a

The two-body potentials V~b satisfy the symmetry property [4]

Vab=Vba ( a , b = l .... , N ) ,

(20)

and are functions of the variables of particles a and b only:

g a b = Vab ( X Tab,Pa, Pb, PaL, PbL ) .

(21)

For two-body "central" potentials one has

Vab-I ~T2, PaL,PbL) • -- V~b~.~b

(22)

In the non-relativistic limit the latter should behave as

lim Vab= 2rn~m~ Vo~b(X~b).

c~o~

ma + mb

(23)

(This limit is also obtained from field theoretic expressions in the relativistic instantaneous approximation [ 7 ]. ) Eq. (18) possesses a weak form of separability. When there are non-interacting clusters the internal wave function q/factorizes into independent wave functions. The term "weak" refers to the fact that each independent wave function will still feel the total mass squared eigenvalue of the system through the coefficients PaL. This phenomenon is unavoidable in the present description, where, from the start, we favored a representation related to the total momentum of the system. This form of separability is lost with the exact equation (15) (with potentials); the "kinetic energies" and the potentials appear in complicated non-linear expressions and do not lead to a factorization of the internal wave function corresponding to non-interacting clusters. However, if eq. ( 18 ) is a good approximation to the exact equation (15 ), then the system will display an approximate form of separability. In the non-relativistic limit, eq. (18), which is the relevant one in this limit together with expressions (14), yields a Galilei invariant and separable hamiltonian: 473

Volume 208, number 3,4 p2 H = ,=1 + ~-~m Z

,,