Relativistic dynamics on a null plane

ANNALS OF PHYSICS 112, 94-164 (1978)

Relativistic Dynamics on a Null Plane*

H. LEUTWYLER

Institut fur Theoretische Physik, der Universitiit Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

AND

J. STERN

Division de Physique Thtorique,? Institut de Physique Nuclhaire, Universitt de Paris XI, 91406 Orsay, France

Received June 8, 1977

In view of possible applications to the quark model and to hadron spectroscopy, we investigate relativistic Hamiltonian quantum theories of finitely many degrees of freedom. We make use of the fact that if null planes are used as initial surfaces, the structure of the theory closely resembles nonrelativistic quantum mechanics: the inner variables that describe the structure of the system uncouple from the motion of the system as a whole. The dynamical content of such a theory resides in the operators M, 1 of mass and spin that act in the space carrying the inner degrees of freedom. Relativistic invariance is equival-

ent to the requirement that M and / generate a unitary representation of U(2). In contrast to this requirement, the condition that the wavefunctions of the system transform covariantly strongly restricts the dynamics. It is proven that for systems containing two constituents, covariance is equivalent to an algebraic relation that involves M and 3 - the angular condition. A class of solutions of the angular condition is provided by a particular type of local manifestly covariant wave equations. One nontrivial solution of this class, a relativistic oscillator is given in detail. Confinement models of this type represent an interesting alternative to the solutions of the angular condition that result from the perturbation expansion of a local field theory through the threedimensional quasipotential versions of the Bethe-Salpeter equation.

CONTENTS

1. Introduction

2. Kinematics and dynamics

A. Poincare algebra in a null plane basis B. Stability group of a null plane C. Unitary irreducible representations of the stability group

*Work supported in part by the Swiss National Science Foundation. + Laboratoire associe’au C.N.R.S.

94 00034916/78/1121-0094$05.00/0 Copyright 0 1978 by Academic Press, Inc. AU rights of reproduction in any form reserved.

RELATIVISTIC DYNAMICS ON A NULL PLANE 95

D. Hamiltonians, mass and spin E. Null plane position operators F. Summary: Kinematical and dynamical algebra

3. Elementary systems and covariant wavefunctions

4. Composite systems

A. Systems of two localizable particles B. Two noninteracting particles C. A general class of composite systems D. Relativistic models of confinement

5. Covariance and angular conditions

A. Covariance for spin 0 constituents B. Angular condition for spin 0 constituents C. Spin 4 constituents D. SU(2)s 0 0(3)19 symmetry

6. Manifestly covariant wave equations

A. Relativistic oscillator B. What else?

7. Nonrelativistic internal motion

A. Free particles B. Interaction

8. Fields

9. Summary and concluding remarks

Appendix I. : Canonical covariant basis for elementary systems.

Appendix II. : Equivalence of angular conditions and global covariance

Appendix III. : An expansion theorem for Hermite polynomials

References

1. INTRODUCTTON

A complete theory of strong interactions presumably involves infinitely many degrees of freedom carried by local fields, strings, or bags-the gauge theory of colored quarks and vector gluons is a good candidate for such a complete theory. The success of the constituent quark model suggests however that the main characteristics of single hadron states involve only a finite subset of these degrees of freedom; the remainder appears to be frozen in. The spectrum of baryons and mesons, their static properties, transitions, etc. indicate that only the degrees of freedom of a fixed number of quarks are excited-there is no trace of gluon excitations.

In this paper we investigate general properties of relativistic systems with a finite number of degrees of freedom. We do not attempt to base this discussion on an underlying field theory, nor do we start from a given classical model subject to

96 LEUTWYLER AND STERN

quantization. We rather insist on the general Hamiltonian implementation [l] of relativistic invariance [2] and analyze the dynamical properties of quantum systems that are described by a particular set of degrees of freedom.

An analogous procedure is extensively used in solid state theory: many properties of a solid are intimately related to its symmetry group; the specific form of the interaction among its constituents is of secondary importance. In that case, of course, the reason why the degrees of freedom of the electromagnetic field that keeps the solid together are frozen in is well known. This certainly is not the case for the strongly interacting fields that keep a hadron together. Quantum field theory has so far provided a clear and predictive treatment of composite systems only if the effective coupling is weak and/or the dynamics is nonrelativistic. There are well-known systems such as solids, molecules, atoms, positronium, nuclei for which these conditions are satisfied. Although the quark model suggests that hadrons, too, are effectively described by a fixed number of constituents, these are strongly bound and presumably have a substantially relativistic internal motion.

Since Dirac [1] set the basis for a general Hamiltonian formulation of relativistic dynamics, many authors have investigated Poincare invariant theories of N interacting particles following this line [3-51. The central problem of the Hamiltonian approach is to realize the 10 generators of the Poincart group in terms of a given set of dynamically independent variables, such as coordinates, momenta, spins, etc. Causal- ity suggests a natural choice of variables that are dynamically independent: the variables associated with a hypersurface Z in Minkowski space that does not contain timelike directions. In classical physics suitable variables on Z constitute a complete dynamically independent set. We assume that the same holds true in quantum theory: there is a complete set of measurements performed at the “instant” Z that allows us, at least in principle, to unambigously characterize the state of the system. (Note that we do not require Z to be spacelike everywhere-it may contain lightlike directions. In classical physics this may lead to problems if the theory contains massless particles. We exclude massless physical states throughout this paper.) The most familiar example of a surface with the above property is the time instant x0 = 0. We will give a number of alternatives shortly.

The PoincarC generators expressed in terms of the dynamical variables specify the dynamics of the system completely. Additional concepts such as the Lagrangian or the action may turn out to be useful, but they are not of primary importance:

The subgroup of the Poincare group that maps Z onto itself is called its stability group Gr . The construction of a dynamical theory involves two steps. First, one specifies the generators of the stability group G, , i.e., the kinematics inside the surface Z: The form of these generators is independent of the interaction; the dynamical variables and the physical states transform in a simple manner under GZ . Next, one considers the Poincart generators that do not belong to the stability group, but transform the original surface 2 into some other surface Z’. In particular, these generators describe the evolution of the system as a function of time. They contain all the information about the dynamics of the system; Dirac calls them Hamiltonians.

The‘ same procedure’ may of course be applied to nonrelativistic dynamics; it


suffices to replace the PoincarC group by the Galilei group. In this case only one type of initial surfaces has the property to intersect every world line once and only once: the surface x0 = constant. Consequently there is only one possible way to split the Galilei group into a kinematical and a dynamical part. The kinematical Galilei subgroup that leaves invariant a time instant has 9 parameters and there is a single Hamiltonian-the generator of time displacements.

Dirac has stressed that this uniqueness of the nonrelativistic Hamiltonian description is lost in the relativistic case. The reason is that Einstein’s causality considerably restricts the family of world lines and, consequently, allows for a larger class of initial surfaces 2. In principle, any surface that does not contain timelike directions may be used. It is clear, however, that if this surface does not have any symmetries, the stability group GZ is empty-every one of the 10 generators of the PoincarC group is a Hamiltonian. In this case, the description does not contain any interaction independent, kinematical part. The larger the stability group of Z, the smaller the dynamical part of the problem. We therefore consider only surfaces 2, for which the stability group is at least transitive: every point of the surface Z may be mapped into any other point of 2 by a suitable element of GZ . (In particular, GZ must have at least 3 parameters.) This guarantees that all points of 2 are equivalent. Dirac gave three inequivalent classes of surfaces with this property: (i) the time instant .x0 = 0; (ii) one sheet of a hyperboloid x2 = a2 > 0, x0 > 0; (iii) the null plane x0 = x3. Of course, these representatives stand for the corresponding equivalence class of surfaces that are related by PoincarC transformations. The recently established classification [6] of all subgroups of the PoincarC group makes it easy to complete this list. There are only two further classes of possible initial surfaces with a transitive stability group: one sheet of the hyperboloids (iv) xo2 - xl2 - x22 = uz > 0, x0 > 0 and (v) xo2 - xS2 = a2 > 0, x0 > 0. It is important that the list stops here: the uniqueness of the nonrelativistic Hamiltonian description becomes a fivefold ambiguity in the relativistic case.

The dimension of the stability groups of the surfaces (i)-(v) is 6, 6, 7, 4, 4, respectively. Null planes have the largest possible stability group: only three of the PoincarC generators are Hamiltonians, i.e., depend on the interaction. (Note that if transitivity is dropped, one may also consider the limiting case a = 0 of the classes (ii), (iv), and (v). The stability groups of these cones are of the same dimension as the stability groups of the corresponding hyperboloids.) The stability groups of the surfaces (i)-(v) are not isomorphic. They cannot even be deformed one into another by contraction. Consequently, two Poincart invariant Hamiltonian theories based on inequivalent initial surfaces will in any reference frame differ in the way in which the IO Poincart generators are split into kinematical ones and Hamiltonians.

The question whether this difference is only formal or whether it has physical consequences has remained unanswered since Dirac raised it [I]. In particular, it is not clear whether each theory based on a given surface .Z finds its physically equivalent counterpart among Hamiltonian theories based on an inequivalent surface 2’. Poincarr2 invariance does not require such equivalence. Even if this should turn out to be the case, different forms of Hamiltonian dynamics may emphasize different


properties of a physical system. In the absence of a decisive physical argument in favor of one of the five possible forms, they all deserve attention.

Hamiltonian quantum theories with a finite number of degrees of freedom have been investigated systematically for initial surfaces of the type x0 = constant [3]. In particular, the condition of separability has been solved in the lowest orders of an expansion in powers of a/c [4]. In view of the confinement problem, separability does however not appear to be an obvious requirement. A quite different problem deserves study within this class of theories: covariance. We will comment on the significance of this requirement below. There are a few papers concerning theories based on initial surfaces of type (ii) and (iii)-hyperboloids [7] and null planes [5, 8-101. They are not very systematic as they either concentrate on a particular model or on particular aspects of the problem, such as the nonrelativistic approximation.

In the present paper we systematically investigate case (iii): Hamiltonian dynamics based on a null plane as an initial surface. Our preference for this class of theories derives mainly from field theory: interacting quantum fields restricted to a null plane generate a Fock basis from the physical (Poincare invariant) vacuum [I 11. This (rather unusual) property of fields on a null plane allows us to describe the structure of hadron states in a frame independent manner, which is closely related to the infinite momentum parton picture. Besides this reason (to which we return in Section 8) there are various particular aspects of null plane dynamics, which make it a suitable basis for a description of composite systems with finitely many degrees of freedom. Some of these aspects have already been stressed in the literature [5,8-121.

(1) As mentioned above, null planes are characterized by the largest (seven parametrical) stability group. We will show that this allows a clear separation of the variables that characterize the system globally from the inner variables that describe its structure. It will be shown that these variables actually uncouple as it is the case for nonrelativistic theories.

(2) The stability group of a null piane acts transitively not only on this surface but also on the mass shell p2 = m2, p. > 0. Consequently, the null plane wavefunction of a system is determined if it is known at rest. This property is not shared by theories based on a time instant x0 = constant as an initial surface.

(3) The stability group of the null plane x0 = x3 contains the generator PO + P, of a lightlike translation. The requirement that the spectrum of P, should be contained in the forward cone P2 > 0, P,, > 0 implies significant information even on the kinematical level: the kinematical generator PO + P, must be positive.

(4) It is possible to unambigously specify the position operators in two spacelike directions of the initial null plane in an interaction independent manner. This fact together with covariance (see below) allows us to introduce a consistent spacetime description of the system. The system need not be interpreted as describing a collection of two (or three) point particles interacting at a distance, it may rather appear as an extended object. The typical space-time picture of a composite system we shall meet


throughout this investigation is that of a nondeformable straight line segment of variable length.

Foldy [3] has already concluded that Poincart invariance does not by itself restrict the interaction of an N-particle system very strongly, if one works on the surface x0 = 0. The null plane approach leads to the same conclusion: Poincart invariance alone allows for a large class of interactions. This is illustrated in Section 4, where, as an example, we exhibit a class of Poincare invariant models of confinement. Despite the fact that these models are Poincare invariant, they do not admit a covariant description: APoincarC invariant Hamiltonian theory is not guaranteed to be covariant. Invariance only asks the Poincare group to be an isomorphism of the space of physical states. Covariance requires the wavefunctions that describe physical states to transform under this isomorphism in a particular manner: according to a finite representation of the Lorentz group. Covariance is an additional requirement, which in contrast to Poincart invariance strongly restricts the dynamics. To illustrate this point, it suffices to remember that the problem of constructing a classical Poincart invariant Hamiltonian mechanics of N interacting particles has no solution [13], if one asks for covariance, i.e., requires that the world lines of the particles transform properly. (Although this no-go theorem has been established only for theories of class (i), it likely also holds for the remaining four forms of Hamiltonian dynamics.)

Within a quantum theory of composite systems based on a null plane, covariance appears to be essential. As already pointed out, it allows us to control the local properties of the system. Another aspect of covariance is peculiar to the null plane framework: the spin of the constituents of a composite system is not a kinematical quantity, because the angular momentum is not among the generators of the stability group. The requirement that the constituents carry a definite spin constrains the Hamiltonians of the system. Covariance incorporates this information in an unambiguous fashion. We shall return to the physical motivation for covariance in more details in Section 5.

The exploration of Poincart invariance and covariance within Hamiltonian null plane dynamics is the main contribution of the present paper. We show that these requirements are equivalent to a set of algebraic equations (angular conditions) relating the mass and angular momentum of the system. They are summarized in Sections 5B and C. We show in Section 6 that there exists at least one nontrivial solution of the angular conditions. The solution represents a relativistic oscillator which may equivalently be described by a pair of manifestly covariant wave equations.

2. KINEMATICS AND DYNAMICS

We first review the separation of the Poincart algebra into a kinematical and a dynamical part for Hamiltonian theories based on a null plane as an initial surface. Many general aspects of these theories will already appear at this stage.

To set up notation we specify the null plane

IZ * x = constant


by the lightlike vector w = (3, 0, 0, 4). It is convenient to choose a coordinate system adapted to this plane:

x- = n . x = *(x0 - x”)

x+ = x0 + x3 (2.1) jT = (xl, x2).

The coordinate x- plays the role of time, the coordinates within the plane are collectively denoted by x = (x+, x1, x2). The null plane description specifies the configuration of the system at “time” x-.

A. Poincard Algebra in a Null Plane Basis

In the following we work with the specific null plane

x- = 0.

This plane is mapped onto itself, in particular, by the boosts along the third axis generated by K3 = MO,: Under the transformation exp -i/SK3 the coordinate X+ transforms as

x+ + esx+

and the transverse coordinates x1, x2 remain unchanged. More generally we may classify the components of any Lorentz tensor according to its transformation property with respect to K3: An operator A that obeys

i[K3, A] = yA (2.2)

is referred to as an operator of goodness y. (This word derives from the terminology used in the “pa + co method”: Operators with goodness y = +l, 0, -1 are “good”, “bad”, and “terrible”.) In this language the coordinates x+, x1, x2 have goodness y = - 1, 0, 0, respectively. In particular, this classification may be applied to the generators M,, and P, of the Poincare group. Among these ten generators there are three with y = + 1, four with y = 0 and three with y = - 1. They are listed according to this classification in Table I. (J3 = M,, , Kl = MO,, etc.).

TABLE T

goodness Lorentz generators Translations

y= +1 G = +(K, + Jd, Ez = $(K, - J1) p+ = wo + P,)

y=o Ka, Js Pl , ps

y = -1 Fx = K, - Ja , Fz = Kz + J1 P- = PO - Pa


Goodness is additive. This implies a particularly simple structure of the PoincarC algebra in a null plane basis: the set of all Poincare generators with goodness y generates a subgroup G, of the Poincare group and Ghl are Abelian. Furthermore, there exist two seven-parametrical Poincare subgroups S* with a semidirect product structure

Sk = G,, x G%1. (2.3)

A few comments on the notation may be added. The nonzero elements of the metric tensor are

g+- = g-+ = +l = -g22 = 1. (2.4)

For any vector A,, A, = A- = +(A,, + &) and A- = A+ = A, - A, stand for the y = + 1 and y = -1 components, respectviely. The transverse components AT = (A,, A,) have y = 0. In particular the boosts E, , K3 and Fv defined in Table I may be written as

E, = M+, , K3 = M-, , F, = AL, . (2.5)

(The transverse components 1,2 that correspond to the two spacelike directions in the null plane x- = 0 are labeled by the letters Y, s, t = 1, 2.)

B. Stability Group of the Null Plane x- = 0

The stability group of the plane x- = 0 is generated by the three translations P = (P+ , P, , Pz) and by those Lorentz transformations n that obey

A . “@nv = hu (2.6)

One easily verifies that this group coincides with the group S+ defined above: the stability group is generated by the seven Poincare generators of goodness +l and 0. The nonvanishing commutators among these operators are (E,, = -E,, , + = 1)

VG , &I = --iE, , [&, P+l = -iP+, [J3 , Gl = ids , [J3 , P,l = id, , (2.7) [E, , P,] = -i&P+.

The operator

.A = Ja + $ W’, - Wd (2.8)

commutes with all generators of the stability group. In terms of the Pauli-Lubanski relativistic spin operator


the quantity $a may be written as

W+ wLl+ w3 j3 = p, = PO + P, * (2.9)

The next section reviews the unitary irreducibIe representations (UIR) of the stability group. In this connection y3 plays the role of a Casimir operator. The eigenvalue of x3, referred to as the null plane helicity, uniquely characterizes the UIR of the stability group.

C. Unitary Irreducible Representations of the Stability Group

For any physical system, the momentum P = (P+ , PI, Pz) is among the observables. It is therefore convenient to consider a unitary representation of the stability group in the basis, in which the momentum operator P is diagonal. From the commutation relations (2.7) of the stability group it is clear that the eigenvalue of P,. is not affected by the generators J3 and E, , while the boosts generated by K3 take the eigenvalue p+ into Ap+ , where h is a positive multiplier. This shows that there are three classes of UIR characterized by p+ > 0, p+ = 0 and p+ < 0, respectively. At this point we invoke the spectrum condition, which requires

p+ = &(P, + P3) 3 0. (2.10)

Negative eigenvalues of P+ are therefore ruled out. If we furthermore restrict ourselves to systems that do not contain massless physical particles, then the vacuum is the only state of finite energy for which P+ vanishes. (As usual, states of infinite energy are excluded.) In the following we therefore consider only UIR with p+ > 0.

The action of the stability group is transitive on the half space p+ > 0, - co < p1 , pz < + co: The stability group transformation exp --I’(&E, + p,E,) exp -ib3K3 takes the momentum p = (p+ , p1 , pz) into

P+ ’ = p+& Pi’ = p7 -+ p+Ae*S. (2.11)

We use this property to define the basis vectors of the representation as follows. Consider the collection of states Ifi, n} that belong to the particular momentum @ = (a, 0,O) and detine the states I p, n) by

I P, n> = exp --i&G + rB&) exp --if13& I ii, n> (2.12)

where

P3 = W+/a>, BT = PAP+. (2.13)

This choice of basis vectors fixes the representation of P+ , PI, Pz , El , I&, K,:

P I p, n> = p I p, n>

ET I P, n> = ip+ vp, I P, n> (2.14)

K3 I P, n> = ip+ alap, I P, n>.


To specify the action of J3, it suffices to consider the set of states 1 $, n). Since J3 leaves the standard momentum i, invariant, it may be diagonalized on this set of states. Each one of the eigenvectors of J, gives rise to a separate representation of the stability group. For an irreducible representation we must therefore have a single state for each value of the momentum. The condition that requires the state associated with the momentum 9 to be an eigenstate of J3 may be expressed in terms of the helicity operator 2% introduced in the last section

$3 I P> = h I P>. (2.15)

(In contrast to the eigenvalue equation for J,, this condition applies to states of arbitrary momentum; the helicity y3 is stability group invariant.) The action of J3 therefore reads

J3 I P) = [i (PI & - P2 $-) + h] I p). (2.16)

Unitarity determines the norm up to a constant:

<P’ I P> = N2P+a3@’ - P). (2.17)

This shows that the UIR of the stability group are uniquely characterized by the helicity quantum number h. In the present context we are interested only in those representations of the stability group that arise from unitary representations of the PoincarC group. These are the stability group representations with integer or half- integer helicity.

D. Hamiltonians, Mass, and Spin

The remaining three PoincarC generators that do not belong to the stability group S, form the Abelian Poincare subgroup GW1 . The operator P- translates the null plane X- = 0 into another null plane X- = T. The operators Fl and 103 rotate the null plane x- = 0 around the surface of the light cone x2 = 0. Consequently, in theories which specify the initial data on the null plane x- = 0 and consider X- = T as an evolution parameter, the operators P- and F, describe the dynamics, i.e., the motion away from the null plane x- = 0.

The nonvanishing commutators among the stability group generators and the three Hamiltonians P- and F,. read

I

and [K3, P-1 = iP- , [E, , P-1 = -iP,

K3, F,l = iF,, [J3, FA = kFs [PC, Fsl = &,P- , [P+ , Fcl = if’,. E9 F.J = -@A3 + d3).

(2.18)

(2.19)


We solve these commutation rules as follows. The operator P- is expressed in terms of the mass operator M2 = P2:

P- = (1/2P+)(f142 + PI2 + Pz”)* (2.20)

(The action of the operator l/P+ is well defined on a space that does not contain zero mass particles or states of infinite energy). The commutation rules (2.18) are satisfied if and only if M2 commutes with all generators of the stability group. Similarly, we express the Hamiltonians Fl and F, in terms of the spin operators, which are defined in the following manner. Let / p, n) stand for an arbitrary state of momentum p = (p+ , p1 , p2) with a sharp value m of the mass operator 44. Denoting by 1 n> the corresponding state at rest (p+ = m/2, pr = 0), one has

(2.21)

where the parameters @I , p2 , /$ are determined by the momentum and the mass of the state according to Eqs. (2.13). We define the action of the spin operators $I , yZ, A by

A I P, n> = exp --i(& + &&) exp -iB& Ji I n>. (2.22)

The third component of 3 coincides with the null plane helicity operator #Cs introduced previously. By definition, $ commutes with all generators of the stability group except J3 and satisfies the commutation rules

(2.23)

It is straightforward to verify that the above definition of the transverse components $I and j2 is equivalent to the following explicit operator expression in terms-of the Pauli-Lubanski operator Wu:

M/T = w, - (P,lPJ w, . (2.24),

This relation may be inverted to express the Hamiltonians Fl , F, in terms of the spin and mass operators:

Fr = WP+W& + P-E, - 4PsY-z + KA)l. (2.25)

The commutation rules (2.18), (2.19) between the Hamiltonians P- , F, and the generators of the stability group, as well as the commutativity of any two of the three Hamiltonians are guaranteed if and only if the operators M and 2. obey the algebra (2.23) and commute-with all generators of the stability group except J, .

Note that the generators of the Poincare group involve the products M$; , MJF~ ,


and M2, rather than the operators yl, $ZZ and M themselves. (In particular, the sign of M is of no significance: a simultaneous change of sign in M and in yl, yZ leaves the commutation rules (2.23) as well as the generators of the Poincare group untouched.) This shows that the three dynamical quantities of primary importance are M#r, MfZ, and M2-we will refer to them as reduced Hamiltonians.

The spectrum condition requires P, to be contained in the forward cone P2 > 0, P,, > 0. Since the representation space of the stability group was constructed so that P, > 0, these conditions are satisfied if and only if M2 > 0.

E. Null Plane Position Operators

Let us consider the collection of states j p; h, n) where n stands for the variables that are needed, in addition to the momentum p and the helicity h, in order to completely describe the state of a given (yet unspecified) physical system. We have shown that the kinematical operators P+ , P, , KS, E, , J, act exclusively in the space of the momentum variables p leaving the set of variables (h, n) unaffected. On the other hand, all quantities that commute with these kinematical operators, in particular, the reduced Hamiltonians M2, M,$$ act exclusively in the space of variables (h, n) leaving unaffected the momentum variables p. We now ask to which extent this information suffices in order to define in the space of states 1 p; h, n) a null plane position operator, whose measurement would yield states localized in the surface x- = 0.

The null plane position operators Qr, Qz and Q+ may be defined in complete analogy with the classical investigation of Newton and Wigner [14]. These authors deal with a localization that results from a measurement performed at a given instant of time x0 = 0. The main conditions on the corresponding position operators arise from their transformation properties with respect to the Euclidian group E(3) that leaves invariant the instant x0 = 0. Analogously, we require the null plane position operators Q to exhibit correct transformation properties with respect to the stability group S, of the null plane instant x- = 0. The corresponding algebraic conditions read

I&, Q81 = 0, K3, QTl = 0, V3, Q’l = ksQS

r+,Qv=o, VT, Qsl = & (2.26)

E , Q+l = C, W3, Q+l = iQ+, V3 2 Q+l = 0, P+ , Q+l = i, [P, , Q+l = 0.

(2.27)

In addition, Ql, Q2, and Q+ have to be selfadjoint and it seems reasonable to ask that they commute among themselves:

KY, Q-7 = [Q+, Qrl = 0. (2.28)


On the other hand, one does not expect the position operators to transform simply with respect to those Poincart transformations that do not leave invariant the “instant” x- = 0.

Consider first the transverse position operators Q1 and Qz. An explicit solution of the conditions (2.26) and (2.28) is in this case provided by the generators El , Ez of the stability group:

QT = ET/P, . (2.29)

This result is intuitively reasonable: the generators El and E, boost the transverse momentum, but commute with P+ . The general solution of the constraints listed above reads

(2’ = Wf’, + q’, (2.30)

where q1 and q2 commute with the whole set of kinematical operators P, , P, , KS, ET and therefore act only in the space of variables (h, n). Furthermore, q’ obeys the algebra

[A 3 4Tl = hIS

w, q21 = 0. (2.31)

In other words, y3, ql, and q2 generate a unitary representation of the Euclidian group in two dimensions, which can be nontrivial only if it is infinite dimensional. As long as the space of variables (h, n) is finite dimensional-in particular, for a single particle-one necessarily has q’ = 0 and consequently, the transverse position operators are uniquely given by the formula (2.29). We will actually retain this solution for the transverse position operators even if the space of variables (h, n) is infinite dimensional. The reason for this choice is again in the commutation relations (2.31): According to the first of these relations, a state with a given helicity h can be localized in the transverse directions of the null plane x- = 0 only if q’ = 0.

We call the quantity (2.29) the ‘mean transverse position operator. Its properties are quite analogous to the properties of the center of mass coordinate for nonrelativistic systems. As will be shown later, the main difference between these two concepts is that Q’ weighs the constituents of a composite system with their longitudinal momentum rather than with their mass: Qr measures the position of the center of longitudinal momentum. It should be noticed that the mean transverse position operator is a purely kinematical quantity, as it commutes with the reduced Hamil- tonians M2 and M$r and with the helicity f3. Consequently, Qr describes the transverse localization of a particle with a given mass, spin and helicity, independently of its internal structure.

In this connection it is interesting to observe that the mean transverse position operators do not exhibit Zitterbewegung as do, e.g., the rudimentary x0 = 0 “position operators” x’ of a spin + particle in the Dirac representation. Indeed, the commutator with the Hamiltonian P- is given by

i[P- , Qr] = P,/P.+ . (2.32)


This shows that the transverse velocity operator is given by P,/P+ , which by the way coincides with the correct classical expression for axr/ax-. Equation (2.32) corroborates the analogy between the mean transverse position operator and the nonrelativistic center of mass coordinate: P, plays a role analogous to the mass of the nonrelativistic system.

The situation is quite different for the operator Q+, which measures longitudinal position. In fact, there is no selfadjoint operator satisfying (2.27). Although it is possible to find formal solutions of these commutation rules, it follows from the spectrum condition that the operator exp(--iolQ+), which displaces the eigenvalue of P, by 3, can exist only for positive values of 01. Hence, there is no observable longitudinal position operator.

For any system, one therefore deals with two complementary sets of observables, besides the null plane helicity y8:

and

p, 2 Pl, p, (2.33)

P +7 Q1 = W’, , Q2 = E,/P+ . (2.34)

These two sets are closely related to the two invariant Abelian subgroups of the stability group S+ .

The eigenstates of the set (2.34) are (xr = (x1, x2))

I p+ , g, ; h, n) = -$ s d”pr &‘lxl+ivzxa 1 p; h, n). (2.35)

In particutar

(2.36)

It is of course possible to define a complete set of states labeled by all three coordinates x = (x+, x1, x2) in the null plane x- = 0. A general class of states of this type is (p . x = P+Xf + P& + P2X2)

I xi h, n> = (2& s d3p eiDex I p; h, n>f(p+),

where the integration extends over positive values of p+ . The states ) x; h, n) are eigenstates of the mean transverse position operator-since there is no operator for longitudinal position, the function f(p+) remains at this stage unspecified. Note however that the action of all stability group generators but K3 is independent off:

P 1 x; h, n) = --id I x; h, n}

E,. 1 x; h, n) = -ixra+ 1 x; h, n)

J2 1 x; h, n} = [i(x’a, - x2a,) + h] / x; h, n).

(2.38)


In order to give meaning to the longitudinal coordinate x+, the system has to be further specified. We will see that the requirement of covariance provides us with the necessary additional information to fix the functionf( p,). The basic fact of course remains: the space of physical states contains only positive values of p+-physical wave packets are therefore not localizable in x f. This does not however exclude the existence of localizable observables, like, e.g., a local electromagnetic current: fields that carry both positive and negative values of P, can be localizable in x+.

A measurement performed at a null plane instant x- = 0 can (at most) yield a localization on a lightlike ray parallel to xu = x + II. The coordinate x+ has little to do n with a particle location, while the transverse coordinates x, = (xl, x2) may be used to label the particle position. This conclusion is further corroborated by the locality properties of the Hamiltonians P- , F,. (2.20) and (2.29, which can be read off from their action on the coordinate space basis vectors (2.37). Independently of the function f( p+), this action is seen to be local in the transverse coordinates xr but nonlocal in x+. This result corresponds to what one may expect naively: A causal motion out of the initial surface x- = 0 (viewed from an infinitesimal neighborhood of that surface) has to be local in gT , not necessarily in xf.

F. Summary: Kinematical and Dynamical Algebra

In the previous sections we have shown that the null plane Hamiltonian description of any Poincare invariant quantum mechanical system has an extremely simple structure: The space of physical states spans a representation of the direct sum X @ 9 of the kinematical algebra X’ and the dynamical algebra 9 generated by

(2.39)

respectively. (The commutation rules of X and 9 are given in (2.7) and in (2.23). The elements of X commute with the elements of 9.) The Hilbert space of the system is spanned by basis vectors of the type

I P, n> = I p> 0 I 4, (2.40)

where p = (p+ , p1 , pz) denotes the three kinematical components of the momentum and n labels the inner variables of the system. The algebra X acts on the momentum p,

L 9 acts on the inner degrees of freedom. For any physical system that does not contain massless particles a single irreducible

representation of % is relevant:

p I P> = P I P>

E, I P> = ip+ &

GIP) = ip,g +

I P> (2.41)

I P>.


This representation is unitary in the norm

(P’ I P> = (W &7+~3(P’ - P). (2.42)

All elements of ~6 except K, have a direct physical interpretation in terms of observables: the total momentum and the mean transverse position of the system. Note that the kinematical algebra ~$7 does not coincide with the whole algebra of the stability group-it does not contain the operator J3 .

The dynamical algebra 9 contains the hard core of the Hamiltonians P- , Fl , Fz . The UIR of 53 are of course the standard finite-dimensional representations of U(2)- the mass operator is a c-number on irreducible representations. For a composite system the representation of 53 is reducible, whereas the kinematical algebra X is always represented irreducibly according to (2.41). Every unitary (irreducible) representation of ~33 gives rise to a unitary (irreducible) representation of the Poincare group.

Note that 9 is not a subalgebra of the Poincart algebra, since the elements of SB are nonlinear, Casimir-like functions of the Poincart generators. The Poincart algebra as such is of course not decomposable into a direct sum of two subalgebras. In particular, the Hamiltonians P- , Fl , F, do not generate an invariant subgroup. As we have seen, it is however possible to identify the intrinsically dynamical part of these Hamiltonians: this part is contained in the operators M, $I , jz , which together with y3 close to the algebra U(2). (The helicity y3 is intermediate between a purely kinematical quantity and a Hamiltonian: on the one hand it involves only generators of the stability group, on the other hand f3 is needed to close the dynamical algebra 9.) Explicitly, the representation of the Poincare generators not contained in 3? is given by

J3 I P, n> = [i (P, & - pz -&) + A] I P, n>

P- I P, n> = $- (PT~ + M2) / p, n) + (2.43)

It is clear from these expressions for the Poincare generators in terms of the elements of S and LB that only the products My1 , M$, and M2 rather than $I , j2 and M are involved-they are the proper reduced Hamiltonians.

The fact that the hard core of relativistic invariance resides in four operators M, f1 , $;A , yj generating U(2), is of course a general property of the Poincart group-it is not particular to the null plane description. In fact, the unitary representations of the Poincare group may always be brought to the canonical form given by Wigner; in this form the basis vectors are labeled I 3, n), where p’ denotes three suitable components of the momentum and n stands for the inner variables needed to distinguish the different states at rest. The operators A4 and $ act on inner space in a p’-inde-


pendent manner. What is however peculiar to the null plane description is the fact that the generators of the stability group (except JJ act only on p’ in a manner that is independent of the inner variables n. This feature is related to the transitivity of the null plane stability group in momentum space and allows us to completely separate the dynamical part from the kinematics of the problem. Such a separation is not possible in a Hamiltonian description based on the surface x0 = constant. In this case, the stability group is transitive only in x-space. The corresponding x-space basis vectors are related to the canonical set j 3, n} in an interaction dependent manner.

The remarkably simple structure of Lorentz invariant quantum theory on a null plane perhaps justifies the hope that it is a suitable framework to investigate relativistic interactions. In this framework, the inner variables IZ play the role of the dynamical variables: Particular physical systems correspond to different sets of these inner variables. Particular interactions correspond to different representations of the dynamical algebra 9 on this set. In the following sections we concentrate on one- and two-particle systems.

3. ELEMENTARY SYSTEMS AND COVARIANT WAVEFUNCTIONS

In this section we briefly consider systems that constitute a unitary irreducible representation of the dynamical algebra .9 and consequently, a UIR of the Poincart group. Following Wigner we refer to systems of this type as elementary systems. They are characterized by a single internal variable- the helicity h, which runs from --s to +s (s integer or half integer). The corresponding basis vectors are labeled 1 p, h) = 1 p) @ 1 h). The mass operator takes a fixed value M and the spin operators generate the irreducible matrix representation Ds) of SU(2), 9 = s(.s + 1).

We call the inner product of an arbitrary state 1 y) with the basis vectors 1 p, h)

w&w = <P, h I Y> (3.1)

the momentum space null plane wavefunction. In terms of the wavefunction the norm reads

(3.2)

Note that this norm is independent of the mass m of the particle. The “time” evolution of the system, which is governed by the Hamiltonian P- ,

( v, T) = ewirP- [ y),

implies the Schrbdinger equation

(3.3)


As discussed in Section 2E we may introduce the x-space wavefunction by setting

R&(x, 4 = (x, h I 5% T>, (3.4)

where the x-space basis vectors are given by

0 I x, h) = J (2,rr)3 eYfb+) I P, A). (3.5)

The weight f(p+) is at this stage arbitrary, because there is no position operator conjugate to the observable P+ . In the following we will see that for elementary systems there is nevertheless a unique notion of locality in the longitudinal direction. The crucial point which leads to this notion is the fact that elementary systems may always be described by covariant wavefunctions. For these, there is no intrinsic difference between transverse and longitudinal components of x. We will see that this leads to an essentially unique choice of the weight f(p+). The corresponding x-space wavefunction in fact coincides with the good (i.e., maximal goodness) components of the covariant wavefunction.

To establish these claims we first briefly review the standard transformation of the unitary, noncovariant representations of the Poincare group to a covariant nonunitary basis.

The action of the Lorentz group on the momentum-space wavefunctions may be read off from the transformation properties of the basis vectors, which read

WI) I P, A) = 1 I P’, h’) &JR(4 PI. h

Here, of course, p2 = m2 and p’ stands for the kinematical components of the vector lip. R(A, p) is the Wigner rotation defined in accordance with our choice of the null plane helicity basis:

R(d, p) = Lj’,(IL, . (3.7)

The boost L, is generated by El , E2 , and KS and takes the rest-frame momentum top (see Section 2C).

We reformulate this unitary noncovariant transformation law in terms of a covariant nonunitary representation by imbedding the representation Ds) of the rotation group in some finite-dimensional representation of the Lorentz group. The covariant x-space wavefunction is then defined as

d4p &P+) &p2 - m”) e--iB’z 1 w&, A) qdp), h

(3.8)

where the spinors uoL(p, h) are to be chosen such that the wavefunction Q”(x) transforms covariantly:

@(x) + SpP(A-lx). (3.9)

595/I141-S


We may always choose a basis for the nommitary representation S of the Lorentz group such that the boost K3 is diagonal. In this basis every component of the covariant wavefunction corresponds to a specific eigenvalue of K3, i.e., carries a definite goodness. The components of Qa that have maximal goodness are called good components of the covariant wavefunction. It is shown in Appendix I that there is a canonical choice of the representation S, which contains exactly 2s + 1 good components of goodness s. For this representation, the good components of the covariant wavefunction therefore constitute a multiplet of the same dimension as the null plane wavefunction q&). Furthermore, since the kinematical boosts E, are goodness raising operators, the good components of the spinor u”(p, h) must be annihilated by E,. . Consequently, the good components of u” are independent of p1 and pz and are proportional to (p+)“. For the good components of the covariant wavefunction @, which we denote by v&c) we therefore have

s d”p B(p+) 6(p2 - m”) c~P*~(~P+)~ qua. (3.10)

We thus see that the x-space null plane wavefunction (3.4) may always be chosen to coincide with the good components of the covariant wavefunction provided we put

f(P+) = (2P+F (3.11)

in the definition (3.5) of the x-space basis vectors. The fact that this possibility always exists is particular to elementary systems. It will be shown below that for composite systems, the null plane wavefunction does not in general coincide with some components of a covariant wavefunction. To demand that this be the case is to impose a restriction on the dynamics of the composite system.

The remaining components of the spinor u, the so called bad components are proportional to a power (P+)S-~, n = 1,2,..., 2s, of the longitudinal momentum multiplying a polynomial in the transverse components pT . This clearly reflects locality in the transverse direction and shows that covariance and transverse locality are compatible. We illustrate this discussion by the following two examples.

Spin 0

In this case we have a single momentum space null plane wavefunction y(p). The corresponding canonical covariant wavefunction reads

It coincides with the null plane wavefunction in x-space and obeys

(3.12)

(0 + my @p(x) = 0. (3.13)

RELATIVISTIC DYNAMICS ON A NULL PLANE

In terms of Q(x), the norm takes a manifestly covariant form

The momentum space null plane wavefunction has two components &p) (h The canonical covariant wavefunction has 4 components

The good components of @ are given by

wdx) = (2fi)3 I d3P 2p, e +y2p+)‘l” q+&(p),

113

(3.14)

(3.15)

(3.16)

The redundant bad components x(x) are related to the good ones by (see Appendix I)

x(x) = c3+)-Ym - w4J dx). (3.17)

This relation, together with the wave equation (3.3) can be collected in a single covariant equation

(-iy3, + m) CD(x) = 0. (3.18)

The norm may be written in the covariant form

or, alternatively, in terms of the good components

4. COMPOSITE SYSTEMS

(3.19)

(3.20)

We now turn to systems that are not described by a finite number of irreducible representations of the Poincart group. Clearly, in this case the dynamical algebra 9 is represented on an infinite-dimensional vector space. In the following we focus on systems that intuitively consist of two constituents. The next section characterizes a special class of two-particle systems. The general case is discussed in Section 4C.


A. Systems of Taco Localizable Particles

A natural class of two-particle systems is characterized by the requirement that under the action of the stability group the system transforms like a direct product of two unitary representations corresponding to two elementary systems. This requirement provides us with two momenta p’ and p” and with the corresponding transverse position operators that allow us to talk about momenta and transverse positions of the two constituents of the system.

For a direct product representation of the stability group we have

P = P’ + P”

J, = J,’ + J; (4-l)

and likewise for the boosts E, and K, . The representation acts on basis vectors labeled by 1 p’, h’; p”, h”) according to Eqs. (2.14) and (2.16). The helicities h’ and h” run over the values --s’,..., +s’ and -8” ,..., +s”, respectively. We of course expect that in this case the constituents carry spins s’ and s”. We emphasize however that like the mass of the constituent, its spin is a dynamical quantity. The constituents can be guaranteed to carry spins s’ and s” only by imposing the condition of covariance. This requirement goes beyond simple Poincart invariance-it will be discussed in detail in Section 5.

In order to transform the basis I p’, h’; p”, h”) into the standard form 1 p, n), for which p = p’ + p” represents the kinematical and n the dynamical degrees of freedom, we note that the operators

p = (l/P+)(P:_p+ - P+‘_p’;.)

(4.2) KL = (1/2P+)(P+’ - P;)

are invariant under the action of the kinematical algebra SK. They essentially represent the relative momenta of the two constituents. The desired transformation of the basis therefore reads

1 p’, h’; p”, h”) = I p; K_r, KL ; h’, h”). (4.3)

The following discussion exclusively concerns the dynamical variables _K~ , KL , h’, h”, since the dependence on the kinematical variable p (which describes the free motion of the system as a whole) is trivial. We therefore find it convenient to write (cf. Eq. (2.40))

IP;KT,KL; h’, h”) = I p} @ I ST, KL ; h’, h”) (4.4)

introducing a vector space that carries only the dynamical degrees of freedom. It is on this inner space spanned by the vectors / K_~, ~~ ; h’, h”) that the reduced Hamil- tonians M and $act. In particular, (4.1) implies for the helicity operator $a

A = Ls + s, , (4.5)


where

(9, KL; h',h" 1 L, = -i (K1 & - K 2 &) (9, KL ; h', h" I (4.6)

is the third component of the inner orbital momentum and

(KT, KL; h', h" / s3 = (h' + h')(K_r, KL ; h', h" 1 (4.7)

acts on the constituent helicities. To simplify the notation, we suppress the helicity h’, h”-a compact matrix notation is understood. The operator S, , e.g., becomes a numerical matrix of the type S3’ @ 1 + 1 @ Si , where S,’ and S,” are diagonal.

To further illustrate the connection between the set of basis vectors / p’, p”) and the basis / Q, K~) in inner space, consider an arbitrary state of sharp momentum p, which can be put into the standard form 1 p, v) = [ p) @ j F). For such a state, stability group invariance implies

(p’, p” j p, y,> = (27~)~ 2p+ a3(p' + p" - p) &T, KL), (4.8)

where the inner null plane wavefunction &cT , KL) contains the nontrivial part of the characteristics of the state. We may write the inner wavefunction as a scalar product in inner space:

&T, KL) = (9, KL / T). (4.9)

Note that c&~, L K ) is a matrix in the helicity variables h’, h” of the two constituents. ‘As an immediate consequence of the positivity of the operators P,' and P; the

support of the variable KL is compact:

+<KL<+$. (4.10)

A further consequence of the direct product structure of the stability group representation is that we have two observable transverse position operators for the individual constituents

Q" = E;IP+', Q"' = EJP; . (4.11)

The mean transverse position of the system is related to the positions of the constituents by

Qy = E,/P+ = (+ + KL) Q’? + (; - KL) Q”‘. (4.12)

The factors (4 + KL) and (3 - K~), which represent the longitudinal momentum fraction of the constituents, play the role of the mass fractions in nonrelativistic quantum mechanics.

The relation (4.12) has an interesting qualitative aspect in connection with the spectrum condition (4.10) on the support of KL . The expectation value of the mean


transverse position in a state in which the transverse positions of the two constituents are localized at _x,’ and x; is given by

(&T) = <a + KL) xr'f G - KL) xr"* (4.13)

The mean transverse position of the system will therefore be somewhere on the straight-line segment connecting 8, and & , as long as ~~ satisfies the Constraint

(4.10). The interpretation of the system in terms of two localized constituents clearly fails, if ~~ does not satisfy the spectrum condition.

The stability group representation fixes the norm of the basis vectors:

(P'; ET', KL [ p; ET, KL} = (243 2p+ a3(P' - P&T', K: /ET, KL)

(4.14) &‘, KL’ j Ifr, KL) = 2(2?T)3 (a - KL') 62(s~' - sr) ~(KL' - KL).

In terms of the inner wavefunction (4.9), this norm reads

(4.15)

Poincare invariance of such a two-particle system is equivalent to the requirement that there exists a set of three operators M, $I , and f2 that together with the helicity f3 generate a unitary representation of U(2) on the inner space spanned by the vectors / or, L K ). The generators must be selfadjoint in the norm (4.14). These requirements clearly do not restrict the dynamics of the system very strongly.

B. Two Noninteracting Particles

The special class of two-particle systems defined above contains the trivial case of two noninteracting particles. In that case, the vectors 1 p’, h’; p”, h”) span a basis of a direct product of two UIR of the Poincare group with the masses m’, m” and the spins s’, s”: In addition to (4.1), we have

P- = P-’ + P” (4.16)

FT = F,.’ + F,“.

The operators P-‘, F,.’ and P! , F: act irreducibly on the momentum and helicity variables p’, h’ and p”, h” as described before.

We put the free Hamiltonians (4.16) into the canonical form (2.43). This yields the following expressions for the reduced Hamiltonians M2, M$; and My2 in the space of inner wavefunctions 9)h,h+T, KJ:

RELATMSTIC DYNAMICS ON A NULL PLANE 117

(4.17) + Kr2 + m’12

i - KL ZG m”(K)

1 +++KL

~ [‘G(&’ 0 1) + m’(&’ 0 1)l

&’ and S: stand for the matrix UIR of SU(2) that correspond to the spins s’ and s” and act on the helicity variables h’ and h”, respectively. The reduced Hamiltonians (4.17) and (4.18) are hermitian in the norm (4.15); M2 is positive, provided ~~ satisfies the spectrum condition (4.10).

The formulas (4.17) and (4.18) provide us with a definition in terms of inner variables of what is meant by interaction: A representation of the dynamical algebra 93 describes two interacting constituents, provided it cannot be brought into the form (4.17), (4.18) by a canonical transformation of the dynamical variables.

Canonical transformations may be used to simplify the above expression for the free reduced Hamiltonians. In particular, the spin orbit coupling terms which appear in (4.18) may be transformed away by means of the following unitary transformation v [151, (KT = 1 KT 1~ Gr = ‘%/KT)

V = exp 2k,&[G(l @ S:) - cL(Ss’ 0 I)]

a’ = arctg (a + KL);(K) + m’ an = arctg (3 _ KL) “m;K) + m” *

The transformation V, which commutes with the mass operator M2 and with the helicity f13 , brings the angular momentum into the standard form

V$iV+=Li+Si* (4.20)

The spin operator Si reads

si = (Si’ @ 1) + (1 @ sa. (4.21)

The third component L3 of the orbital momentum operator is given by Eq. (4.6) and the transverse components L, are

L, = k,, C - -&& + (; - "P) Fs]. (4.22)


Consequently, the angular momentum operator yi can be rewritten as

where the operators

A = =% + z , (4.23)

Pi = v+Liv, Yi = v+sgv (4.24)

generate the group W(2), @ O(3), , commute with the mass operator M2 and therefore represent an exact symmetry of the system of two noninteracting particles.

In the case of equal mass particles, m’ = m” = m, the orbital angular momentum operator Zsimply generates rotations of the vector

6 = (K1 , K2 2 m(K) KL)

E = -iR X 9,.

Furthermore, in terms of E the mass operator may be written as

(4.25)

M2 = 4(m02 + R2) (4.26)

and the invariant measure becomes

d2tcT dKL d3k 2($ - KL2) = (mo2 f R2)1/2 ’

(4.27)

This clearly exhibits the close analogy between nonrelativistic quantum mechanics and the null plane framework. The mapping of the momenta x -+ c leaves the transverse components untouched, whereas the relation between the longitudinal momentum K~ and the Cartesian variable k, involves the mass of the system.

C. A General Class of Composite Systems

The key ingredient in the characterization of two-particle systems introduced above was the requirement that they constitute a direct product of two unitary representations of the stability group. This is not strictly necessary-it suffices to require that the degrees of freedom of the system are those of a direct product, such that (i) we have a complete set of states labeled by / p’, h’; p”, A”), (ii) the stability group acts on these states as before (cf. Eq. (4.1)) i.e., it transforms the variables of each one of the two constituents in the standard fashion, and (iii) the stability group representation so obtained is unitary, i.e., it conserves a norm.

There are two points contained in this generalization we wish to emphasize here. The first concerns the norm, the second concerns the spectrum of the inner variable KL .

(1) The norm need not be the direct product of single particle norms. In contrast to the situation for elementary systems, the norm is not determined uniquely by the


transformation properties of the basis vectors under the stability group. Unitarity of the stability group representation only requires

<p’; _Kr’, KLI I Pi Kr 7 KL) = (27~)~ $+ a3(p’ - P)(lf;, KL) / _KT, KL). (4.28)

The norm in the inner space is arbitrary, except for the condition that the helicity operator y3 , whose action on the states 1 !r , K~) is given by Eqs. (4.5)-(4.7) must be selfadjoint.

The possible need for such a generalization may be illustrated by the following examples: Suppose that the separation of the two constituents in the transverse direction never exceeds a certain value d. (This constraint is Lorentz invariant, as it requires that the Minkowski distance between any two points on the world lines of the constituents satisfies z2 = zo2 - zr2 - zZ2 - z3 2 > -d2.) In this case we would , expect that the difference of the eigenvalues of the two transverse position operators of the constituents never exceeds d. Clearly, for the class of systems considered in Section 4A this does not happen: the space of states contains vectors with arbitrarily large transverse separations of the two constituents. An alternative possibility is that the variable K~ of transverse inner momentum has compact support (“confinement in momentum space”). We may envisage for example, that the system is described by wavefunctions that vanish outside the region ~~~ < p2($ - Key), where p is some mass parameter. In this case, the set of vectors 1 K* , KL) restricted to this compact region should be complete-the norm (JC~‘, K~’ / K_~, KJ should vanish outside this region.

We therefore generalize the definition of a two particle system by admitting an arbitrary norm in the inner space 1 +, K~). This clearly restricts the list of observables of such composite systems. The operators or , K~ (4.2) will not in general be selfadjoint and the same therefore applies to the operators P’ and P” or &’ and &F that describe the momenta and transverse positions of the individual constituents. (In other words, this generalization admits that the two factor representations of the stability group that occur in the direct product are not unitary-the ,product must of course be unitary). Needless to say, that the total momentum P, the helicity f3 and the mean transverse position QT are still among the observables of the system.

(2) The support of the inner oariable K~ need not be contained in the interval 1 K~ ] < t. The spectrum condition only requires P, > 0; if the variables p+‘ and p; that label the basis vectors I p’, p”) do not represent eigenvalues of observable longitudinal momentum operators P+‘, PJ for the individual constituents, there is no obvious reason to ask them to be separately positive. The question whether or not the K,-distribution in a physical state may contain a tail with j K~ / > 4 can therefore not be decided on kinematical grounds. In particular, K~ need not coincide with the scaling variable E, that denotes the longitudinal momentum fraction of the degrees of freedom coupled locally to the photon. It is conceivable that both .$ and the longitudinal momentum fraction carried by the neutral gluon-like degrees of freedom collectively contribute to the K,-distribution in a physical state. Although the condition j K~ j < 4 is a natural requirement within our framework, the physical content


of this constraint is not obvious. We return to this problem in Section 8 where it will be shown that the spectral representation for the wavefunctions associated with local fields imply I KL [ < & whereas the norm of the wavefunction does not in general reduce to the special form (4.15).

We now turn to the properties of the norm (4.28) in configuration space. As already pointed out, the transverse position operators &r’ and &F of the two constituents are in general not expected to be selfadjoint in this norm. Although the basis vectors obtained by Fourier transformation in the transverse momentum variables

(4.29)

are eigenstates of&r’ and $I; with eigenvalues xr’ and of , they cannot, in general, be interpreted as describing two point particles localized at xr’ and CX;I. : the inner product of two states (4.29) may be different from zero even if the two states are labeled by different transverse coordinates.

The states (4.29) are however eigenstates of the mean transverse position operator &r with the eigenvalue

XT = (4 + KL) XT’ + (i - KL) _X; (4.30)

and this operator is selfadjoint in any norm that guarantees unitarity of the stability group representation (cf. (4.28)). The scalar product of two states (4.29) therefore vanishes, unless the corresponding mean transverse positions (4.30) are the same. Since the mean traverse position is located on the straight line connecting x,’ and R we have the result that any two states of the type

s d’+’ d’Xp+‘, P;) I P+‘, XT’; P; , x;> (4.31)

are orthogonal if the corresponding straight lines do not intersect. This suggests that the qualitative picture to be associated with the general class of

composite systems considered here is that of an object extended along a straight line in the transverse variables rather than that of two localized point particles. The actual extension of the system along this straight line is governed by the distribution of the longitudinal momentum variable K= . In particular, if the support of K~ is contained in the interval 1 K~ 1 < 4 (i.e., if only the states (4.31) with p+’ > 0 and pl; > 0 are allowed) the picture becomes that of a straightline segment connecting the two points x,’ and XJ . If, on the other hand, the values I K~ [ > ) are allowed, the extension of the system in the transverse direction exceeds the points x,’ and d .

An infinitesimal evolution of the system preserves the cotiguration space support properties of the norm (4.28) described above. This follows from the locality of the Hamiltonians P- and F, in the mean transverse position Qr (cf. Section 2E.): The matrix elements of any polynomial in the Hamiltonians taken between two states of the type (4.29) vanishes, unless the corresponding mean transverse positions coincide.

Between the general case of an arbitrary norm (I+‘, K~’ I ~fr , ~3, for which only


the total momentum, the total helicity and the mean transverse position are observables, and the special norm (4.14), for which the momenta, helicities and transverse positions of each one of the two constituents are observable, there are a number of intermediary situations. They may be characterized by the corresponding set of observables, or equivalently, by the corresponding restriction on the norm in inner space, as summarized in Table IT.

TABLE II

Momentum observables Transverse position

observables (ST’, KL’ i ET , ~3

fT,P+',P[:

pr’, p; , P+‘, p:

pT,p,

QT %KL' - KL)N(_KT', ST, KL)

QT aa&- - _KT)8(KL' - KLM~, KL)

v QT$QT -QT - N(_KT' - ET, KL', KL>

In configuration space, the special cases of the norm listed in Table 11 exhibit support properties, that are stronger than the general “straight-line segment locality” described above. (In particular, the last two cases of Table II yield a scalar product of two states (4.31) that is local in the transverse positions of both constituents.) It is however clear that such particular configuration space properties of the norm will not be preserved during an infinitesimal evolution of the system, unless the reduced Hamiltonians W, My1 , and My. satisfy certain locality constraints. For example, for systems whose norm is local in the transverse’ separation z, = x,’ - & of the two constituents (cf. last three cases in Table II.), it is natural to require the reduced Hamiltonians to be finite order differential operators in the inner variable z, , which is conjugate to the relative momentum + .

D. Lorentz Invariant Models of Confinement

To illustrate the content of the requirement that the dynamical operators M and 2 must generate a representation of U(2), we consider two types of models. Models of the first type were suggested by Altarelli et al. 191. These authors make use of the following device to take care of the spectrum condition that may restrict the support of the variable ~~ . They replace the variable K~ by a variable K~ in such a fashion that the interval -4 < tcL < 4 is mapped into -cc < K~ < + co. (More specificaly, these authors consider the mapping K~ = a In@ - K~)/(+ + K~).) Then define the configuration space wavefunction as the Fourier transform with respect to ri = (‘Q 3 K2 2 ‘G)

q(Z) = (27r)e3 s d3c e&%p(K, , K~ , K~) (4.32)


Next, define the operators of inner angular momentum by (we take spin 0 constituents)

j&(Z) = --ii x a,@). (4.33)

The third component is of the required form (4.6). Furthermore, any mass operator of the form

A4 = -(1/2m) A, + V(l i I) (4.34)

commutes with 2. If the potential tends to infinity as j Z 1 becomes large, the two

constituents will be confined. The mass operator M and the angular momentum 2 are selfadjoint in the norm

(y I y) = J d3C I 949 3 K31”. (4.35)

They have a discrete spectrum of eigenstates of arbitrarily high mass and spin. Alternatively, we may take the potential Y such that it becomes infinite at some fixed value R of I Z 1, in which case we have confinement within a bag of radius R.

We emphasize that systems of this type are fully PoincarC invariant, as they explicitly solve all the conditions that we have imposed. For the particular mapping KL -+ ~~ suggested by Altarelli et a/. [cf. ~~ = a ln($ - KL)/($ + K~)], the norm (4.35) even reduces to the special norm (4.15).

An alternative class of models may be constructed as follows. Take again spin 0 constituents and unite the three inner momentum variables K~, K~, KL in a single three-vector

k = (4 3 K2 , k“k) (4.36)

where p is some constant with the dimension of mass. It is easy to satisfy the algebra of the angular momentum operators: put

j = -ilt X Vk. (4.37)

Clearly, the third component of this object generates rotations in the plane K1 , K2

as is required by (4.6). All three generators leave the square g2 = K12 + K2’ + p2KL2

invariant. The spectrum condition may therefore be satisfied if we demand I%~ < &.L”. A class of mass operators that commute with the angular momentum defined above

is easily found. Consider, e.g., differential operators of the form

M = --a& + V(l E I). (4.38)

It suffices to take a “potential” V that becomes infinite at I z I = 4~ to garantee that the eigenstates of M are confined to the region

ii” = Kr2 + /.L’KL~ < &L2. (4.39)


The operator M and the angular momentum are hermitian in the norm

The spectrum consists of an infinite set of discrete states-there are again bound states of arbitrarily high mass and spin.

These simple models make it clear that PoincarC invariance is in no way essentially stronger than Galilean invariance: If one restricts oneself to Hamiltonians that are local in -u-space, then the freedom left in the dynamics in both cases qualitatively reduces to a potential that depends on a single variable. The reason for this is that the reduced Hamiltonian M must commute with the inner angular momentum. In the relativistic case the additional restriction on the support of the inner momentum variable K~ does not represent any particular difficulty. It can always be solved by brute force, if one makes use of the device described above.

5. COVARIANCE AND ANGULAR CONDITIONS

Poincare invariance does not imply that there is a covariant description of the system. In particular, it is not necessary for relativistic invariance that the wavefunctions that describe the states of a composite system transform in a covariant fashion. The examples given in Section 4D illustrate this statement: the eigenstates of the mass operators given there do not transform according to a covariant representation of the Lorentz group. Covariance is an additional constraint on the dynamics.

Before working out the explicit form of this constraint, let us mention two arguments that justify the requirement of covariance. One point clearly emerges from our analysis of elementary systems. Covariance in that case allowed us to introduce a unique notion of locality in the longitudinal direction-invariance alone only led to a concept of locality in the two transverse directions. For composite systems the problem of localization of the two constituents is even more acute. In this case even transverse locality is lost in the general case; the general picture to be associated with PoincarC invariant two-particle systems in terms of the null plane observables is that of a straight-line segment, rather than that of two point particles. We may expect that covariance has important implications concerning localization also for two-particle systems.

Why should we put emphasis on locality when we are attempting to describe extended systems ? One basis reason is that the system must be capable to interact with local fields such as the photon field. To describe this interaction the observables associated with the system presumably include a current operator that transforms like a vector field under the Poincart group. The existence of such a covariant observable is a stringent condition on the dynamics of the system, since the transformation law involves the reduced Hamiltonians M2 and M$r . A simple way to satisfy this condition is to require that the wavefunctions describing the system transform covariantly.


The second point is relevant in connection with the quark model and concerns the spin of the constituents. The spin of an elementary system is uniquely determined by its stability group representation. This is closely related to the fact that the null plane wavefunction of an elementary system is the restriction of a covariant wavefunction, which is unambigously determined by the null plane wavefunction and satisfies a covariant wave equation. On the other hand, two interacting constituents that belong, say, to the stability group representation with zero helicity and whose PoincarC invariant interaction is described according to Section 4, do not necessarily carry spin 0. Poincare invariance alone allows for an interaction that “excites” the spin of the constituents. This (somewhat unusual) feature of the null plane dynamics should not surprise one more than the (less unusual) fact that two interacting constituents cannot be characterized by a given mass: The angular momentum is not in the stability group and consequently, it plays a similar dynamical role as the mass operator. Covariance solves this problem, too, as it associates with a given spin of the constituents a definite transformation law of the wavefunctions that describe a composite system.

A. Covariance for Spin 0 Constituents

Spin 0 constituents may be characterized by the following covariance assumption. For every state ] F) of the composite system there is a covariant wavefunction @(x’, x”) that under the action of the Poincart group 1 v) + U-l(A, a) 1 v) transforms like

@(x’, x#) + @(Ax’ + a, Ax” + a).

Equivalently, we can assume that the Hilbert space of the composite system contains an overcomplete set of covariant vectors 1 x’, x”), , such that

@(XI, x”) = c(x’, x” 1 qJ) (5.2)

and

e(x’, XI 1 P, = i(3,’ + a@ c(x’, X” 1 (5.3)

e<x’, x” I n/i,, = i[x,‘a,’ - X”?,’ + xp: - xp;] c(x’, xn I.

In order to incorporate this assumption into the null plane Hamiltonian description of composite systems, we first note that every system whose states can be described by a set of covariant vectors I x’, x”), belongs to the class of two particle systems defined in Section 4C. To establish this claim, we restrict the covariant vectors 1 x’, x”)~ to the null plane x’- = x”- = 0 and introduce the null plane basis vectors as

1 X’, x”> = 1 x’, X”)c ]zc’-=z’--o. (5.4)


The corresponding momentum space basis vectors 1 p’, p”) defined by

transform like the direct product of two stability group representations with h’ = h” = 0. Consequently, Eqs. (5.4) and (5.5) establish the connection between the covariant vectors 1 x’, x”)~ and the momentum space basis vectors of the composite system Hilbert space that have been introduced in Section 4.

We now invert the argument and consider Eq. (5.5) as a definition of the basis vectors in coordinate space that is suggested by covariance. As in the case of elementary systems, covariance provides us with additional information on the longitudinal coordinate x+, which does otherwise not have an unambigous meaning because there is no longitudinal position operator, Covariance alone does however not imply the special form of the norm in inner space given in Section 4A, nor does it require the spectrum condition on the variable K~ . We therefore deal with the more general class of composite systems discussed in Section 4C.

It is useful to separate in Eq. (5.5) the total momentum p = p’ + p” and the inner variables K~ , ‘cT defined in (4.2). One gets

where z = x’ - x”. This shows that the transverse relative coordinates -zT = (zl, z”) and the dimensionless variable

z, = p*z (5.7)

are the variables conjugate to gT and K~ , respectively. We may introduce the corresponding coordinate basis in inner space as

1 1 1 ZT, 2~) = 3 c2,,.j3

I

@ET dKL eiKr.IT+i+cLzr i - Kt

1 K-T, ‘CL). (5.8)

This leads to the following expression for the coordinate space null plane wavefunction of an arbitrary state 1 p, 91) = 1 p} @ 1 y) with sharp momentum p:

(x’, X” 1 p, q) = e+p*(x’+x”)(gT , ZL 1 fJ.3). (5.9)

We refer to the quantity

&T , %) = &T 3 =L 1 d (5.10)


as the inner wavefunction in coordinate space. According to Eq. (5.8) it is the Fourier transform of the momentum space wavefunction (rr, K~ j ~JJ) introduced in Section 4. The factor ($ - Key)-’ that appears in this transformation has its origin in the weight (p+‘,pl;)-l which occurs in the definition (5.5) of the coordinate space basis vectors. The norm of inner space basis vectors 1 _K~, K~) is not, in general, related to this factor.

To appreciate the content of the covariance condition, consider an eigenstate of the mass operator M = m with spin 0 and sharp momentum p. In this case covariance implies

Ax’, xv / p, qJ> = e- i/2Pw+;c”)~((p . z, z2), (5.11)

where z = x’ - X” and p stands for the four vector on the mass shell p2 = m2, whose kinematical components are p. The inner wavefunction &+, zL) that corresponds to the state / p, v) may be read off if one restricts x’ and x” to the null plane x- = 0. This restriction implies z- = 0 and therefore p . z = p . z, z2 = -zT2. The quantity F coincides with the inner null plane wavefunction in coordinate space and covariance therefore only states that the wavefunction must be independent of the direction of z, . This is however not a new piece of information: for constituents of helicity zero the inner momentum space wavefunction of any state of helicity zero is independent of the direction of “T . Covariance is therefore automatically satisfied for states of spin zero. Furthermore, the null plane wavefunction determines the covariant wavefunction completely. (This is not the case in the x0 = 0 description of a state at rest where the degree of freedom contained in the variable p . z disappears.)

Covariance does however imply nontrivial information for states of nonzero spin. Let / p, h) be a state with spinj > 0, helicity h and mass m. Its covariant wavefunction is given by

c(x’, XI / p, A) = e-i’2v*(z’+r”)zul ... z%;l...uj(p) F(p * z, z2), (5.12)

where l :,...~j(p) is the standard polarization tensor. The 2j + 1 helicity states are described by a single covariant wavefunction F(p . z, z2), apart from an explicit kinematical factor. Consequently, covariance relates the null plane wavefunctions of different helicity. These relations are not guaranteed by the Poincare invariant Hamil- tonian description of the system, unless one imposes additional constraints on the mass and angular momentum operators M and J? We are now going to derive the operator form of these constraints.

B. Angular Condition for Spin 0 Constituents

Let us consider the covariant wavefunction (5.12) in the rest frame and rewrite it as

e(x’, x” I jj, /)) = e(-i/2)p^.(s’+z”)F~(z),

where f = (m, 0, 0,O). Covariance reduces to the requirement

F*(R . z) = F,,(z) D:,,(R-l), (5.13)


where R is a rotation. (Lorentz transformations that do not leave invariant the momentum are trivial, the stability group takes care of these.) Thes corresponding inner null plane wavefunction is obtained by restricting F,(z) to the plane z” = z3. For the null plane wavefunction, Eq. (5.13) represents a nontrivial condition only if the rotation R is such that both z and Rz are on the null plane z” = 9. Any rotation with this property may be decomposed into a rotation around the vector (zl, z2, 9) followed by a rotation around the third axis. Since the latter belongs to the stability group, the content of the covariance requirement amounts in the rest frame to the condition that the wavefunction is invariant with respect to the rotations generated by zlJ, + z2J2 + z3J3 .

In order to derive the angular condition in an operator form, it suffices to find the component of the Pauli-Lubanski operator

that reduces at rest to the rotation zlJ, + z2J2 + 2J3 . The relevant operator that has this property is the projection zL1 W, . Due to the transformation law (5.3) of the covariant vectors 1 x’, x”)~ , one in fact has (z = x’ - x”)

c(x’, X” I Z’L w, = 0. (5.14)

We again restrict this equation to the null plane x’- = xc- = 0. Using further the expression (2.24) of the Pauli-Lubanski operator in terms of Hamiltonians, one finds

Z” w, Is-=* = zlM$, + z”Mf2 + z * Pf3 . (5.15)

Consequently, the covariant Eq. (5.14) becomes an operator condition in the space of inner wavefunctions v(gT, zL):

zQfy1 + z”M$2 + z&g3 = 0. (5.16)

When applied to the wavefunction of a state with mass m and spin j at rest, z, = +mz+ = mz3 and Eq. (5.16) reduces to m(zlJ, + z2J2 + z3J3) = 0 as expected.

We show in Appendix II that the angular condition (5.16) is not only necessary but also sufficient to guarantee a covariant description of the system of two spin 0 constituents: Any wavefunction describing an eigenstate of the operators M2, 22 and $; that satisfy (5.16), is the null plane restriction of a covariant wavefunction (5.12). The angular condition shows once more that the spin Hamiltonian relevant for the null plane dynamics is the product M$; rather than the operator $; itself.

The problem of constructing a covariant Hamiltonian null plane dynamics that involves two spin 0 constituents can now be summarized as follows: Consider a Hilbert space of functions ~(zl, z2, ZJ and define the operator y3 by

y3 = -i (zl & - z2 &). (5.17)

595/IIz/I-9


In this space, find three reduced Hamiltonians M2 > 0, M&;l and M$, that in addition to the relations

LA, MAI = k$fA , [A > M21 = 0

WA, WA1 = iM2$,, Pf2, KP-rl = 0 (5.18)

required by Poincart invariance, satisfy the angular condition (5.16). Find a suitable norm of this space such that the three reduced Hamiltonians together with fZ are selfadjoint.

C. Spin 4 Constituents

We are now going to analyze the more complicated case of spin -$ constituents. In view of possible applications to the meson spectroscopy, we restrict ourselves to systems that consist of a spin 4 particle and its antiparticle. These systems are characterized by the following covariance condition: For any state 1 q~) there exists a covariant wavefunction (Pars-(x’, x”), (a’, a” = I,..., 4) that under the Lorentz transformation 1 y) -+ U-+l, a) 1 y) transforms as the product YI,(x’) ‘ya-(x”) of two Dirac wavefunctions, i.e.,

@(x’, x”) + S(A-1) @(Ax’ + a, fix” + 0) S(A).

Equivalently, the Hilbert space of the system should contain an overcomplete set of covariant vectors / x’, x”; a’, a”), , (a’, a” = l,..., 4) such that

e(x’, x”; a’, a” 1 P, = i(a,’ + 8;) c(x’, x”; a’, a” 1

c(X’, X”; II’, au I MU, = i[(x,‘a,’ - x,‘a,‘) + (x:az - x;$)] e(x’, x”; a’, ax /

+ (~;;~‘?i@“~” - Sa’b’u;;““) e(~‘, 2’; b’, b” 1, (5.19)

where a,’ = a/axU’, a: = a/&?* and CT,,, = (i/4)[~, , ~“1. In order to establish the connection with the null plane Hamiltonian description of

composite systems, it is convenient to consider the covariant vectors [ x’, x”; a’, a”)c in the canonical basis, in which K3’ and Ki are diagonal so that every component of the covariant vectors carries a definite goodness y = y’ + y”. (See Section 3 and Appendix I.) We recall that in this basis, the four component Dirac spinor ul, becomes

where h = *& is the helicity and v, x respectively stand for the good (goodness y = +&) and bad (goodness y = -4) components of Y. Furthermore, the y- matrices take the form given in Appendix I. Consequently, like the product Y&x’) ‘p,$x”), the covariant vectors may be represented in an extended notation, in


which the four-valued indices a’(a”) are replaced by the pairs of two-valued indices h’(h”) and y’(y”), (h’, h”, y’y” = &+):

(CW, x” I YO)a’a” = e(X’, x”; h’, h”; y’, y” I. (5.21)

This notation explicitly shows the goodness content of the individual components of the covariant vectors.

The complete set of basis vectors of the composite system Hilbert space can now be identified with the null plane restriction of the good-good (i.e., y’ = y” = +$) components of the covariant vectors (5.21):

J x’, x”; h’, h”) = / x’, x”; h’, h”; + i ) + f)c )2--es-=o . (5.22)

Using the covariance assumption (5.19), the momentum space basis vectors defined according to

I x’, x”; h’, h”) = (2~)~~ 1% -$ e ‘D’.x’+“P”.x”(~,+‘P:>~‘~ 1 P,, Pfj; hr, h~j) (5.23) f

are seen to transform as the direct product of two stability group representations with helicities h’ and --h”. (The factor (4p+‘pl;)li2 in Eq. (5.23) is required by covariance. It has the same origin as the factor (2~+)l’~ that occurs in the definition of the x-space wavefunction of an elementary spin 4 system; see Section 3.) Consequently, every system, whose states transform according to (5.19) belongs to the class of two-particle systems described in Section 4C.

It is worth mentioning that the third component of the constituent’s spin operator S, , which occurs in the decomposition

of the total helicity J$ of the system, now satisfies

(x’, x”; h’, h” / s, = (h’ - h”)(x’, x”; h’, h” 1. (5.24)

The sign flip of h” is a consequence of covariance and is related to the fact that the corresponding constituent is an antiparticle. In this connection one may recall that under charge conjugate U, the basis vectors (5.22) transform as

(x’, x”; h’, h” I U,-’ = &%;“““(x”, x’; A”, A’ 1, (5.25)

where the sign accounts for the statistics of the constituents. The operator S, (5.24) is invariant under U, .

It is again useful to separate in Eq. (5.23) the total momentum p = p’ + p” from the inner variables K~ , sT , h’, h” writing, as usual,

1 p’, p”; h’, h”) = 1 p) @ 1 TT, KL ; h’, h”).


One may introduce the corresponding coordinate basis vectors in inner space

IZTY~L; h’, h”) = (2n)-3 ; 1 (kd~;$;,2 ,@“-ZrfiKLa / ET, uL ; h’, h”) (5.26)

and rewrite Eq. (5.23) as

/ x’, x”; h’, h”) = (2~)-~ s d3p e(i’2)D’(x’+x”) 1 p) @ [ gT, p - z; h’, h”), (5.27)

where, as before, z = x’ - x” and z, = p . z. The coordinate space null plane wavefunction of an arbitrary state 1 p, v) = j p) @ 1 v> with sharp momentum p then becomes

(x’, x”; h’, h” 1 p, y) = 2p+e-‘i/2’p’X’+““‘~*,~“(~~, zL) (5.28)

where

w&~ 3 zL) = (zz- , ZL ; h’, h” I v) (5.29)

stands for the inner wavefunction in coordinate space. It is the Fourier transform of the momentum space wavefunction (E T, K~ 1 y) introduced in Section 4 with the weight factor (+ - K~‘)-‘/~. (Note that this weight factor is not the same as in the case of spin 0 constituents; cf. Eq. (5.8).)

We now turn to the implications of the covariance condition (5.19) for the dynamics of a system of two spin 4 constituents. To appreciate the content of this condition, let us consider the 2j + 1 (2 x 2 component) null plane wavefunctions (5.28) that describe 2j + 1 helicity states of spin j, given mass and delitrite parity. Stability group and parity invariance allow to express this set of wavefunctions in terms of 4j + 2 y3-invariant functions of the variables zT2 and z, . On the other hand, covariance requires the null plane wavefunction of the particle state 1 p, h) with helicity X to be the restriction to the surface x’- = x”- = 0 of the good-good components of the (4 x 4 component) covariant wavefunction

c(x, x”. a’ a” /p A) = e(-ilzbw+r) h 3 , > 3 @a’a”k PI. (5.30)

The set of covariant wavefunctions @i,,” can be decomposed in the standard way into Lorentz scalar amplitudes that depend on the variables p . z and z2. It can be shown (see e.g. Ref. [18] and Appendix II) that for any j > 1 only six (two if j = 0) such scalar amplitudes contribute to the good-good components of Qp,,,- restricted to the null plane z- = 0. Consequently, covariance starts to be operative for states of spin j > 2: It imposes 4j - 4 relations among 4j + 2 kinematically independent (y3-invariant) components of the null plane wavefunctions that describe a composite particle of spin j. We are now going to translate these relations into an operator angular condition for the reduced Hamiltonians M2 and MfV .


As before, the nontrivial part of the covariance condition may be expressed in terms of the rotation zlJ1 + zzJ2 + z3J3 restricted to the null plane z- = 0:

J(z) = (z’Jr + z2Jz + z3J3),-c, = zlJl + z2J2 + $z+J3. (5.31)

Without losing generality we can consider the covariant vectors 1 x’, x”; a’, a”), at x’ + X” = 0 and z- = (x’ - x”)- = 0. The covariance assumption (5.19) then implies

c(+z, -&z; a’, a” 1 J(z) = [Pb’(z) Saab’ - 8”‘b’~b”~“(z)] &, -4~; b’, 6” 1

(5.32) where

Z(z) = zlu23 + z2u,, + *z+u12 (5.33)

is the spin part of the rotation J(z). In order to get an operator constraint on the Hamikonians, one has to convert relation (5.32) into an equation that involves the basis vectors ($z, -&z; h’, h” I (5.22), rather than the whole set of unknown bad components of the covariant vectors. The redundant components of the covariant vectors can be eliminated, if one iterates Eq. (5.32) twice: Using the matrix representation (5.21) of covariant vectors and the canonical form of the y-matrices given in Appendix I, one easily finds

(tz, -&z; h’, h” I [J(z)]~ = [zT2 + &(z+)“]($z, -$z; h’, h” 1 J(z). (5.34)

This relation may be rewritten as an operator equation jn the space of inner wavefunctions glh&z , zL): It suffices to use the decomposition (5.27) of the basis vectors and to express the transverse components J, of the angular momentum in terms of the reduced Hamiltonians. The resulting angular condition reads

:(zrM$, + z&3)3: = :(zr2M2 + zL2>(zpMyT + ZL93)3, (5.35)

where the dots indicate that all operators -z, and z, stand to the left of the operators M and $$. Applying Eq. (5.35) to the inner wavefunction (5.29) that describes a state of mass m at rest, the variable zL becomes &mz+ and one recovers the relation (5.34).

We show in Appendix II that the angular condition (5.35) is not only a necessary but also a sufficient condition of covariance: Any null plane wavefunction that is an eigenfunction of the operators M2, j2, and JK~ satisfying Eq. (5.35), is the null plane restriction of the good-good components of a covariant wavefunction (5.30).

The problem of constructing a covariant Hamiltonian null plane dynamics involving two spin 4 constituents can now be summarized as follows: Consider a Hilbert space of functions ~~,~“(zl, z2, zL) (h’, h” = *$) with a norm in which the operator

95 = --i (zl& - z2 &) + s3 (5.36)


is selfadjoint. In this space find three hermitian reduced Hamiltonians A@, M&1 , and My2 that, in addition to the relations (5.18) required by Poincare invariance, satisfy the angular condition (5.35).

The nonlinear character of the angular condition is the price to pay for spin 4 constituents. There exists however a particular class of systems, for which the cubic angular condition (5.35) effectively reduces to the linear form (5.16) that we encountered in the case of spinless constituents. This class is discussed in the next section.

D. SU(2), @ O(3), Symmetry

We now consider a particular class of systems, of two spin Q constituents, for which the angular momentum operator yi can be separated into orbital and spin parts:

$i = -% + x., i = 1, 2, 3. (5.37)

Such systems might be relevant in connection with the quark model [lo, 15, 161. The operators Zi and Yi act in the space of inner wavefunctions ~)~,&r , zL). They

are supposed to be selfadjoint and to generate the group SU(2), @ O(3), . The Hilbert space should contain all representations of O(3), with integer I, but only the s = 0 and s = 1 representations of SU(2), , since we consider two constituents of spin 4. If, in addition, L$ and Yd separately commute with the mass operator M2, the system exhibits an exact SU(2), @ O(3), symmetry. The SU(2), generators Yi need not coincide with the canonical spin operators

Si = Si’ + Si) Si’ = &q @ 1, s; = 1 @ &Jiul (5.38)

(a1 accounts for charge conjugation-cf. (5.25)). The example of two noninteracting particles illustrates this fact [15]: In this case the generators 9$ and Yi of the SU(2), @ O(3), symmetry are rather complicated nonlocal operators given by Eq. (4.24).

In general, covariance does not guarantee a separation of the angular momentum into orbital and spin parts with the above properties. One can, however define a particular class of systems that allow such a separation, for which the angular condition (5.35) effectively reduces to a linear problem. In this context, the decomposition of the angular momentum into orbital and spin parts appears to be a useful ansatz rather than a general property of relativistic systems of two spin 4 constituents.

We first note that the condition, according to which only the singlet and triplet representations of SU(2) should be involved, may be expressed in operator form as

Yn” = Yn ) (5.39)

where 9” is the projection of 3 onto an arbitrary direction n’. Next, we observe that if the orbital momentum L? satisfies the same linear angular condition as the angular momentum j& belonging to a system of two spin 0 constituents, i.e.,

zlqkf + z2&i!f + ZL-EP, = 0, (5.W


the total angular momentum 3 = L% + 3 solves the angular condition (5.35). (This statement actually holds even if 4 does not commute with the mass operator-in this case the order of operators in Eq. (5.40) is crucial.)

To establish the above statement, we consider an arbitrary inner wavefunction F~,~“(+, zL) describing a state of mass WI and apply the operator that stands on left-hand side of the angular condition (5.35). Due to the constraint (5.40) on the orbital momentum, one gets

:(z’$,M + z&J3: CJI = :(mzTyI, + z~CY~)~: y. (5.41)

The property (5.39) of the spin operator reduces the right-hand side of this equation to

(m2zT2 + zL2)(mzTY: + zLY3)p) = :(zr2M2 + ~L~)(.z’$TM + zJ&p, (5.42)

the latter equality following again from (5.40). The angular condition (5.35) is thus verified for any eigenfunction of the mass operator and therefore holds in the whole space of inner wavefunctions.

The constraint (5.40) on the orbital momentum L? is however not a necessary consequence of the angular condition (5.35): The angular momentum 2 that admits the decomposition (5.37) with the operator 3 satisfying (5.39) may belong to a covariant system, even if the constraint (5.40) is not satisfied. For example, the reduced Hamiltonians (4.17) and (4.18) of the system of two noninteracting spin 4 particles satisfy the angular condition (5.35), the corresponding spin part 9 of the angular momentum exhibits the property (5.39) and the corresponding operator 2 fails to satisfy the condition (5.40). Consequently, separating the angular condition (5.35) into the “spin condition” (5.39) and the “orbital condition” (5.40), one loses generality. On the other hand, the fact that the spin and orbital conditions (5.39) and (5.40) assure the validity of the angular condition (5.35) turns out to be a powerful simpli- fication of the search of particular covariant systems of spin + constituents: As illustrated below, the spin condition (5.39) may be satisfied by hand and one remains with the linear orbital condition (5.40).

One can for instance use the fact that the helicity f3 always splits into two commuting operators L3 and S, (5.24) and identify the third components of the orbital momentum L& and of the spin spi with those operators:

2x3 = L, = -i(z’ a/a22 - 22 a/as)

9, = s, = $[(u3 @ 1) - (1 @ a&]. (5.43)

For this choice, the spin condition (5.39) is satisfied identically. Furthermore, commuting the constraint (5.40) with g3 = L, , one finds

[L, , M] = 0. (5.44)


Since, however, [yi , M] = 0, this implies

[2g ) M] = [Z ) M] = 0, i= 1,2,3, (5.45)

and one therefore deals with systems that exhibit an exact SU(2), @ O(3)Z symmetry. In order to construct the corresponding Poincare invariant and covariant dynamics,

it suffices to find in the space of functions 9~~,& r , ZJ a solution of the problem involving two spinless constituents (summarized at the end of Section 5B.) that exhibits an SU(2) symmetry generated by hermitian operators Yi with Ya = S, . The solution of this problem yields the mass operator and the transverse components _Ep, of the orbital momentum. The spin operators Yi are identified with the generators of the SU(2) symmetry of this solution.

It should be noticed that the requirement of SU(2) symmetry is not very restrictive. A particular solution of the above problem that exhibits such a symmetry can always be constructed as follows. Consider any solution M(,) , 2~~) of the spin 0 constituent problem and take the direct product of the corresponding basis vectors ( z, , zJ with a four-dimensional vector space carrying the helicities h’, h” = &j:

1 -zT > zL ; h’, h”) = 1 -z, , ZL) @ 1 h’, h”). (5.46)

Identify the mass operator MZ with M& @ 1, the orbital momentum 2 with e = 2(O) @ 1 and the spin operators L? with 1 @ S, where S are the matrices (5.38) acting in the helicity space. The reduced Hamiltonians MZ and 1 = L? + L? so constructed satisfy all the constraints listed in Section 5C.

This rather trivial construction provides an example of a class of systems that consist of a spin + constituent q and its antiparticle g such that (i) it exhibits an exact symmetry SU(2)$ @ SU(2)$ 0 O(3), , (ii) the spin part of this symmetry is generated by the canonical spin operators S’ and 3” (5.38), (iii) it is Poincare invariant and covariant. We have shown that there are at least as many systems of this class as there are solutions of the spin 0 constituent problem. Note however that a noninteracting system of two spin Q particles does not belong to this very particular class.

We finally mention a more general category of systems for which the angular condition (5.35) is satisfied via the spin and orbital conditions (5.39) and (5.40). For these systems the orbital momentum J& and the spin operator Yi still commute with the mass operator, but gs and Ys are not identified with the canonical operators Ls and S, (5.43). The spin-orbit coupling schemes that have been considered in connection with the Melosh transformation [lo, 15, 161 between the constituent and current quark basis [IO, 15-171 are of this type.

We may solve the spin condition (5.39) by the following ansatz [15]:

Yi = v+s,v, i= 1,2,3, (5.47)


where Si is the canonical spin operator (5.38) and V is a unitary transformation that commutes with the helicity operator

As in the previous case, mass and orbital momentum are taken to obey the commutation rules of the dynamical algebra B and commute with the operators Yi (5.47). The angular condition is then replaced by the orbital condition (5.40), which now involves the transformation V. The resulting theory exhibits an exact SU(2), @ O(3), symmetry, the generators Yi of SU(2)y however do not coincide with the canonical spin operators Si (5.38), i.e., the system exhibits spin-orbit mixing. There is com- pelling phenomenological indication that theories of this type might provide a good framework for meson spectroscopy.

6. MANIFESTLY COVARIANT WAVE EQUATIONS

Since the early attempts by Schrbdinger [19] many authors have considered covariant wave equations as a common denominator of quantum mechanics and relativity. Indeed, covariant wave equations provide a natural framework that incorporates relativistic invariance in a manifestly covariant manner.

In the context of null plane Hamiltonian theories the requirement that the dynamics of the system may be described by a manifestly covariant wave equation is neither necessary nor sufficient. The dynamical properties of the system are contained in the reduced Hamiltonians M2, Mjl , M$, and the wavefunctions of the system transform covariantly if and only if these operators obey the angular condition; it is not necessary that the time evolution generated by the Hamiltonians is equivalent to a covariant wave equation. Even if the system does obey a covariant wave equation this does not by itself guarantee that it belongs to the class of systems investigated here. For this to be the case the wave equation in question must admit null planes as Cauchy surfaces such that the knowledge of the wavefunction on the null plane determines the state of the system and its time evolution completely: the wave equation must be equivalent to a null plane Schrodinger equation with a suitable Hamiltonian.

Despite these reservations we expect that covariant wave equations lead to an interesting special class of covariant Hamiltonian systems, i.e., give rise to particular solutions of the angular conditions. We first consider a special example, the relativistic harmonic oscillator model proposed by Feynman, Kislinger, and Ravndal [20]. This example solves all the constraints we have imposed on a system of two spinless constituents. Next, we discuss covariant wave equations from a somewhat more general point of view. In particular, we emphasize that a covariant description of a Hamiltonian system involves a pair of covariant wave equations (a proper wave equation and a constraint that eliminates the excitations in the relative time variable) and illustrate the amazing strength of the requirement that these two equations are consistent with one another.

136

A. Relativistic Oscillator

LEUTWYLER AND STERN

We consider a system of two spinless constituents characterized by the covariant wave equation

[U’ + 0” - 2w4(x’ - x”)2 + 2m,2] @(lx’, x”) = 0.

States with a sharp value p of the four-momentum

@(X’, X”) = ew2hJw+z”kqX’ _ -J)

obey the wave equation, provided the covariant inner wavefunction Q(z), z satisfies

[O - adz2 + mo”] Q?(z) = $m”@(z),

(6.1)

(6.2)

x’ - x”,

(6.3)

with ma = p2. A hyperbolic differential equation of course admits solutions for any value of m2. To turn it into an eigenvalue equation for m2, we use a standard method: we suppress the excitations of the system in the relative time variable by a constraint

pqa, - dz,) G(z) = 0, (6.4)

such that in the rest frame the dependence on z” is fixed: Q(z) N exp &(wzO)“. Note that the sign in the constraint (6.4) is that of Feynman et al. The opposite sign leads to the wavefunctions considered in Ref. [21] that behave as exp ---$(wz~)~. We comment on this difference below.

The ground state of the system is described by the wavefunction

Qo(z) = e*wezn = exp $w2(z,2 - z12 - zs2 - 22)

and, in the rest frame, the excited states are given by

(6.5)

Qi ,,,,,,&) = K&4 fL,(wz2) fL,(~z31 f+@. (6.6)

The corresponding mass spectrum amounts to a sequence of linear Regge trajectories:

Mn2 = 8w2(nl + n2 + n3 + 2) + 4m,“.

(We denote the sum of the three quantum numbers n, + n2 + n, by n.)

(6.7)

The covariant wave equation (6.3) supplemented by the constraint (6.4) admits a null plane Hamiltonian description. This can be shown as follows. In accordance with our general prescription we define the inner null plane wavefunction ~(3~ , zL) as the restriction of the covariant wavefunction Q(z) to the plane z- = 0. Both the wave equation (6.3) and the constraint (6.4) involve a derivative with respect to z-, i.e., do not restrict the null plane wavefunction, but determine its dependence on the relative “time” z-. There does however result a constraint on the null plane wave-


function from the requirement that these two equations lead to the same dependence on z-. The constraint may be found by eliminating the derivative a/&- between (6.3) and (6.4) with the result

(a + VL2) m29, = [-VT2 + w4zT2 + 2w2V,zL + mo2]T, (6.8)

where V L , V, = (V, , V,) denote the partial derivatives with respect to zL = p a z, z1 and :2, respectively.

This shows that the reduced Hamiltonian M2 is given by

M2 = (1 + V,2)-1[-V,2 + w4zT2 + 2w2V,z, + mo2]. (6.9)

The null plane eigenstates of M2 are easily found from their representation in the rest frame given above:

This representation is valid in any frame. To find the remaining two reduced Hamiltonians My1 and My2 we note that the generators of the Lorentz group are given by

M,, = +,'a, - x,/auf + ga: - ga;). (6.11)

The Pauli-Lubanski operator therefore becomes

(6.12)

The reduced Hamiltonians MyI , My2 and the helicity xa may be worked out from this expression by using (2.9), (2.24) and again eliminating a/az- with the constraint (6.4). The result is

Mfr = ic,,{zLVs - zSVLM2 + 02zszL}

f2 = -i(zlV, - z2V,). (6.13)

It is easy to verify that these expressions indeed obey the angular condition (5.16) and that the operators M2 (6.9) and M$1, My2 , Ya (6.13) satisfy the proper commutation rules (5.18) of the dynamical algebra 9. The system considered here therefore constitutes at least a formal solution of the algebraic problem posed by covariant null plane dynamics for two spin 0 constituents. (Note that the Hamiltonians reduce to those of two free spinless particles if w is put equal to zero.)

What remains to be discussed is the requirement that the Hamiltonians are selfadjoint in a suitable norm. This problem is closely related to the question of whether or not the null plane eigenstates (6.10) constitute a complete set of functions. The norm must of course have the property that the eigenfunctions that belong to

different values of M2, I2 and $j are orthogonal. There is a natural norm with


this property: the integral taken in the rest frame of the system over the variables zl, z2 and z3 = zL/M, with the weight e-w2z~2. (Note that this prescription does not allow us yet to calculate the norm of an arbitrary null plane wavefunction, but only determines the norm of the eigenstates of the mass operator M2.) In this norm we have

A natural space of functions is therefore given by those superpositions of the eigenstates

for which the norm

is finite. It is remarkable that this space contains a very large class of wave packets defined by

j&L(z) = exp[--ikLzL - ST * zr - +w22r2]. (6.17)

These wavefunctions describe Gaussian wave packets in the transverse directions with an arbitrary transverse momentum; in the longitudinal direction they are plane waves. We claim that these wavefunctions can be expanded in terms of the eigenstates of M2 if and only if the longitudinal momentum variable k, , is contained in the interval -4 < kL < 4, i.e., satisfies the spectrum condition.

The expansion with respect to the transverse variables is easily obtained using the generating function of the Hermite polynomials:

e-iax = jTo A (s)” e-ua/4w2Hn(wx).

The expansion of the longitudinal plane wave is more subtle, because of the mass dependence of the argument appearing in the corresponding Hermite polynomial. The nontrivial part of the claim made above is contained in the following mathematical property of the Hermite polynomials:

(6.19)

valid if 1 01 1 < 8, pn = (8 w2n + p)l/“, /3 > 0 arbitrary. (The expansion converges for any fixed value of x.) The validity of this expansion is demonstrated in Appendix


III. Using these results we get an expansion of the wave packets (6.17) in terms of the eigenstates of the mass operator with expansion coefficients given by

x exp - 1

kr2 ~ 4w2

+ kr2Mn2 i 46J* !

(6.20)

where the mass M, again involves the sum n = n, + n2 + n3 . With the techniques described in Appendix III, it is straightforward to compute the norm of the wave packets h.,,,,,,(z) with the result

Il&,,lc, 11’ = (1 - 4k,‘). (6.21)

We emphasize that we have no right to require that every function of z1 , z, and zL may be written as a superposition of the eigenfunctions of M2-even for free particles only those functions, whose Fourier transform with respect to z, vanishes outside the interval 1 K~ 1 < 4 are contained in the space of inner wavefunctions. The fact that this turned out to be the only restriction also within the present model strongly suggests that the space spanned by the eigenfunctions (6.10) of the operators M*, MyI , MB;, , and yS si a complete set of physical states. (Note that the eigenfunctions themselves have their support in K~ concentrated on the point K~ = 0.)

In order to physically interpret the model in detail, we have to consider observables such as the electromagnetic current and to work out the corresponding form factors, structure functions and the like. We intend to do this elsewhere.

At first sight, the covariant wavefunctions given above which consist of a polynomial multiplying the factor exp &2z2 may appear to be physically unacceptable since they give exponentially increasing weight to large timelike separations of the two constituents. Within our framework the behavior of the covariant wavefunctions is however relevant only for spacelike or lightlike separations. In a Hamiltonian theory based on a null plane it must be possible to express the observables of the system in terms of operators associated with the null plane z- = 0, for which we always have z2 < 0. It is the null plane wavefunctions rather than the covariant wavefunctions in Minkowski space that carry a similar physical interpretation as nonrelativistic wavefunctions.

We could instead have considered the model for which the sign in the constraint (6.4) is reversed. In this case, the covariant wavefunctions decrease in all directions of the relative spacetime variable z [21]. Indeed, the corresponding eigenstatse of M2 differ from those of the above model only through the factor exp[--w2(p . .z)~/M~~], which on the null plane reduces to exp(-W2z,2/M,2). It is evident that the support of these eigenfunctions in the inner longitudinal momentum variable is not contained in the interval 1 K~ / < +. This does not exclude that the model admits a decent description within null plane dynamics. At any rate it is clear that in this case the completeness problem is not solved by simply imposing the spectrum condition.


B. What else?

The harmonic oscillator model is a nontrivial representative of the class of systems that admit a covariant null plane Hamiltonian description and whose evolution is governed by a manifestly covariant wave equation. We now add a few general comments concerning this class of systems restricting ourselves to the case of spinless constituents.

The specific property that characterizes covariant wave equations concerns locality in the transverse inner coordinates 9, z2: The reduced Hamiltonians M2, MyI , and My2 that correspond to a covariant wave equation are differential operators with respect to the transverse variables. We conjecture that the converse is true: If the reduced Hamiltonians are differential operators in the transverse inner coordinates of at most second order, then the angular condition and the commutation rules for the reduced Hamiltonians imply that the system may be described in terms of a covariant wave equation supplemented by a covariant constraint [30]. Transverse locality thus appears to be the bridge between global covariance that requires the wavefunctions of the system to transform in a covariant manner and manifest covariance that requires the evolution law to be expressible by means of a local covariant wave equation.

It may appear that the relativistic oscillator model amounts to a very special choice of the force between the two constituents and that there might be a wealth of alternative covariant models that all lead to Hamiltonians that

(i) satisfy the dynamical algebra and the angular condition,

(ii) are local differential operators in the transverse variables.

This is not the case. To illustrate the strength of the requirement that the Hamiltonians are local in the transverse direction we first emphasize the importance of the constraint equation involving the quantity

(6.22)

For the relativistic oscillator the constraint involving D is crucial in the transition from the manifestly covariant framework to the Hamiltonian description. This is not peculiar to the relativistic oscillator. For spinless constituents the Pauli-Lubanski operator is always given by (6.12). It is easy to generalize the calculation of the reduced Hamiltonians given in the last section and to establish the general representation

My,. = iE,.,{zLVs - zsVLM2 + z”D}

of M$, in terms of M2 and D. Note that this representation automatically satisfies the angular condition.

The representation (6.23) makes it clear that to specify the reduced Hamiltonians M2 and MfT means, in particular, to specify D. Since D contains the derivative a/az-,


the dependence of the covariant wavefunction on the relative “time” z- is fixed once the Hamiltonians are given- there are no degrees of freedom in the dependence of the wavefunction on z-.

A Hamiltonian system that admits a manifestly covariant representation therefore involves a pair of covariant wave equations. Consider, e.g., the pair

(0 + ,‘(g - -y)) e-(i/2)u.(r’+s”)~(.~’ _ y) = 0

(0” + qx _ X”)] e-(i/2w’+z”kqx’ _ f) = 0, (6.24)

where the potentials V’(z) and V”(z) are for the moment arbitrary functions of zz and of pz. This pair is equivalent to

{cl + U(z)} Q(z) = @P/4) Q(z)

p”a,@(z) = V(z) G(z). (6.25)

(U and V are linear combinations of V’ and V”). The oscillator amounts to the special case

U(z) = -co423 + mo2

V(z) = ClPpz. (6.26)

It is straightforward to calculate the reduced Hamiltonians in the general case:

M2 = (4 + VL2)-l {-VT2 + U(-zr2, zL) + 2VLV(-zT2, zL)}

Myr = kTs(zLV, - zSVLM2 + zsV(-zT2, zL)f. (6.27)

(It is important that the potentials U and Vare independent of p2; otherwise M2 is not in general a local differential operator in the transverse direction.)

The crucial point we wish to emphasize here is that the pair of covariant wave equations (6.25) admits solutions only for a very restricted class of potentials U and V. If the potentials do not belong to this class the two differential equations are incon- sistent. The differential operators 0 + U - km2 and p a - I’ must obey an integrability condition that requires their commutator to vanish weakly in the sense of Dirac. This integrability condition is very strong. It requires

o=o

.

U’ + V” = 0

vv’+pzO=o

(6.28)

where . and ’ denote derivatives with respect to z2 and pz, respectively. (A systematic way to work out the integrability conditions precisely amounts to a calculation of the commutators between M2, M$l and M$, . The conditions (6.28) are necessary and sufficient to guarantee that these operators obey the dynamical algebra.) It is


straightforward to determine the general solution of the differential equations (6.28). This solution involves only three constants a, b, c and a sign:

V = &{a(pz)” + b}l12

U = -az2 - V’ + c. (6.29)

The relativistic oscillator corresponds to b = 0. The plus sign in the expression for V leads to the model of Feynman et al. [20], the minus sign to the model whose wavefunctions are considered in Ref. [21]. For b + 0 the motion in the transverse directions is still harmonic-the anharmonicity in the longitudinal motion is measured by b. Furthermore, transverse and longitudinal motions remain uncoupled. This illustrates the strength of the consistency condition for a pair of covariant wave equations. We intend to analyze the general class of Hamiltonians that are local in the transverse directions (differential operators of at most second order) in a forthcoming publication.

7. NONRELATIVISTIC INTERNAL MOTION

We now consider a two-particle system whose internal motion is slow. One way to discuss this limiting case is to study the limit c + co of the Poincart algebra itself (contraction of the Poincare group to the Galilei group). In this case one restricts all velocities including the velocity of the system as a whole, to the nonrelativistic regime. We will instead only suppose that the internal motion is nonrelativistic and leave the kinematical algebra untouched.

A. Free Particles

To characterize nonrelativistic internal motion, we first consider two free particles of mass and spin m’, s’, and m”, s”, respectively. The mass operator M2 is then given by (4.17)

&f2 = (i - ~,e)-l [--& KT2 f d2 (k - KL) + id2 (i + KL)]. (7.1)

If the relative motion is slow the mass of the system exceeds the sum of the masses of its constituents only by terms of order v2/c2. The difference

M2 - (m’ + m”)2 = c-q - KL2)-1{K~2 + [m”c(+ + KL) - m’c(+ - KL)]~}

(7.2) is of order v2/c2. This implies that the square bracket in (7.2) remains finite as c + co: The transverse momentum is finite, whereas the longitudinal momentum tendsfo a fixed value that depends on the masses of the constituents-for constituents of equal mass KL tends to zero. In general

KL = 5 m, + m,t * m' - m" + 0 p,. (7.3)


The deviation of K~ from this limiting value is of order v/c and is proportional to the third component of the nonrelativistic internal momentum which we define as

f? = {K1, Kg, m”C(+ + KL) - WZ’C(+ - KL)}. (7.4)

1n terms of z, the mass operator can be rewritten as

where the reduced mass p = m’m”/(m’ + m”) comes from the limiting value of the factor * - Key:

1 KL~ + P --

4 m’ + mn .

The nonrelativistic limit does not affect the helicity f3,

whereas the operators My1 , My2 become

MA- = (m’ + m”) 2fR + (~(~-1). (7.7)

The limit c + 00 of the free reduced Hamiltonians Mj, (4.18) coincides with the standard form of a nonrelativistic inner angular momentum:

@Rx -i,$ X -&+ [St@1 + 1 OS”].

The limit of the position variables z, and z, is obtained from their expression ZT = i a/aK, and ZL ‘v i a/aK,. . the null plane transverse position is identified with the transverse nonrelativistic relative coordinate, whereas the longitudinal position must be resealed:

?= zJ,z2 t ’ (m’ l&m’) c f (7.9)

This behavior of the coordinates in the nonrelativistic limit is easy to understand in the rest frame of the system. In this frame we have P, -+ +(m’ + m”)c, PI = P2 = 0. The coordinate 9 introduced above therefore coincides with the third component of the covariant position variable zU in the rest frame of the system. Note that since we are working on the null plane z 0 = z3 the relative time t = 9/c tends to zero as c -+ cx, as long as z3 or, equivalently, 3 stay finite.

595/l x2/1-10


The nonrelativistic limit of the inner wavefunction is easily worked out for free particles. We define the nonrelativistic approximation to the wavefunction as

(7.10)

where KL is to be resealed according to (7.4). The renormalization factor in (7.10) has been chosen such that the nonrelativistic limit of the norm becomes

Note that the restriction on the support of K~ disappears in the nonrelativistic limit as the points K~ = &+ are mapped into k, = f co. The nonrelativistic limit of the coordinate space wavefunction is the Fourier transform

The expansion of the null plane wave function for c -+ cc reads

(+41/2 e(i/2)(m'-m")/(m'+m")"Lp)(~T, zL)

( m’ 8’

> ( mn

1

8’ =

m’ f mu m’ f m” P)NR(?) + qc-1).

(7.12)

(7.13)

B. Interaction

In the case of two interacting constituents we may define the nonrelativistic limit as follows. The system has a nonrelativistic limit if the mass operator admits an expansion of the type

ilk2 = (m’ + m”) c2 + H + O(c-2), (7.14)

where m’ and m” are c-numbers. The algebra of the operators 2 requires that the corresponding nonrelativistic expansion has a nontrivial zero-order term

2 = $R + O(c-1). (7.15)

The nonrelativistic dynamical algebra, generated by jNR, H has the same structure as the relativistic one: the operators jNR generate SU(2) and H commutes with them.

To discuss the nonrelativistic limit of the internal coordinates and momenta for an interacting system we recall that in general neither internal coordinates nor internal momenta are represented by hermitian operators-they are not necessarily observable in the usual sense of this word. The significance of the internal variables is nevertheless unambiguous once we require that the system is covariant. In this case the internal


coordinates zL , zT are essentially the restrictions of the covariant relative position zU to the null plane, more precisely zL = pz, z, = (zl, zz). We define the nonrelativistic relative coordinate 7 in an analogous fashion: we identify i: with the three space components zl, z2, z3 of the covariant coordinate in the rest frame of the system. (Note that we are always working on the null plane z” = z3, we do not put z” equal to zero). It is straightforward to express the variable i; in a frame-independent manner in terms of z, , zr . The result is given in (7.9).

Let us for simplicity restrict ourselves to spin zero constituents. In this case covariance is equivalent to the angular condition (5.16), which in the limit c -j co reduces to

fZ . $:NR = 0. (7.16)

Furthermore, the third component of 2 retains its form in the nonrelativistic limit

yfR = -i (rl& - 12 &). (7.17)

The standard expression for the orbital angular momentum

$ = -2 x 9, -;VR (7.18)

clearly satisfies both the angular condition and the constraint (7.17). In order for this solution of the angular condition to be acceptable it must of course be hermitian. This is the case only if the nonrelativistic limit of the norm is invariant with respect to rotations of the vector i;. We have stressed that covariance by itself does not restrict the norm in inner space very strongly. It is therefore conceivable that there are covariant two-particle systems for which the operator of internal angular momentum does not reduce to (7.18) in the nonrelativistic limit.

The relativistic oscillator illustrates this discussion. The nonrelativistic limit of the corresponding wavefunctions is

y;zn3(F) = Hnl(wrl) Hn,(wr2) Hn3(bw3) exp - &.02[(r1>” + (r2)21 (7.19)

and the norm becomes

The nonrelativistic Hamiltonian is easily worked out from the explicit expression (6.9) for the mass operator:

m’ = m” = m,

H= $ {-VT2 - (V, - w2r3)2 + w4F2 + co”}. (7.21)


The transverse components of the angular momentum may be read off from (6.13):

9:” = iE,,{r3Vs - zs(V3 - w2r3)). (7.22)

This shows that the transformation:

YNR(F) = 03/3n-3/4e-(1/3)w2(,3)e9)NR(~) (7.23)

of the nonrelativistic wavefunctions reduces the norm to the standard Galilei- invariant form. On the renormalized wavefunctions the angular momentum operator is indeed given by (7.18). The nonrelativistic Hamiltonian is Galilei-invariant and describes two particles of equal mass m, that oscillate around one another with the frequency Q = 2w2/m, .

8. FIELDS

As pointed out in the introduction, the framework developed here is not a complete theory of the strong interactions. Its scope is limited to properties of single hadron states for which a finite number of degrees of freedom is relevant. In the present section we briefly discuss the relation between these collective degrees of freedom and those of an underlying local field theory. We first consider a conventional scalar field theory defined in terms of its perturbation series and comment on the connection between our framework and the standard Bethe-Salpeter approach. Needless to say the purpose of this paper is not to reformulate the perturbation expansion of the BS equation in terms of null plane variables. It is however instructive to investigate the spectrum condition on K~ , the norm of the wavefunction and the significance of the angular condition in the context of a given field theory. We return to more general questions concerning the relation between wavefunctions and local fields at the end of this section.

It is clear that any local field theory gives rise to covariant amplitudes whose null plane restrictions obey the angular conditions discussed in the previous sections. Consider, e.g., a theory involving the local scalar fields A($), B(Y). If this theory contains a stable one-particle state with the quantum numbers of AB we may define a Bethe-Salpeter wavefunction by

(0 j A($) B(y) 1 p) zz ~4/2)~(~‘+mqX’ - x”). (f-4.1)

(Since we are interested in these wavefunctions only for spacelike or lightlike separations we need not distinguish between the product and the time-ordered product of the fields). The amplitude 0(z) transforms covariantly under the Lorentz group. Its restriction to the null plane z- = 0 therefore satisfies the spin zero angular condition. Furthermore, inserting intermediate states in (8.1) and using the positivity of the total longitudinal momentum one easily establishes that the support in the variable ICY conjugate to p * z is contained in the interval -4 < rcL < 4.


As pointed out by Yabuki [22] the dependence of the null plane wavefunction on the transverse variables is strongly restricted by locality. To see this consider the DGS representation 1231 of the wavefunction which incorporates both locality and spectrum condition (we take the state 1 p> to describe a spin zero particle for simplicity):

@p(z) = f Iom ds J’:l dc p(s, 5) e-icpzLl(+)(z, s). -l/2

In this representation d (+) is the positive frequency part of the Pauli-Jordan function for a free particle of mass s1i2. Locality and spectrum condition by themselves do not forbid @ to be singular on the light cone z 2 = 0. We note however that the Bethe- Salpeter amplitude is expected to be less singular near z2 = 0 than the vacuum expectation value. Following Yabuki we assume that at least the zeroth moment of p vanishes

s - ds p(s, 5) = 0 (8.3) 0

such that on the light cone Q(z) is less singular than (z”)-l. (Note that the zeroth moment of p has the dimension of a length and may therefore be expected to vanish order by order in renormalized perturbation theory. In the ladder approximation, (8.3) holds for the full B-S amplitude [24].) This allows us to introduce a new spectral function a(~, 5) as

P(SY 5) = f 4% 5) (8.4)

and to integrate (8.2) by parts. For spacelike values of z we have the explicit representation

g Lv+yz, s) = - & Ko((-sz”)‘/“) (8.5)

where K, is the Bessel function of order zero. For spacelike values of z we therefore get the following spectral representation of the covariant wavefunction:

The restriction of this representation to the null plane is easily obtained: the Minkowski distance -z2 becomes zr2 = (z’)~ + (z”)” and p . z reduces to zL . The null plane wavefunction is therefore given by

s I w ds I” dt o(s, 0 e-ifzUCo((sgT2)l/2). 0 -112

(8.7)


The corresponding momentum space null plane wavefunction is essentially the Fourier transform with respect to g, , z,:

w3)

This representation shows that if the null plane wavefunction is the projection of a Bethe-Salpeter amplitude associated with local fields then (i) it satisfies the spectrum condition on the longitudinal momentum, --$ < KL < 4. (ii) it is an analytic function in the transverse momentum K$ whose singularities are located on the negative real axis. The second property of course reflects causality. The cut for negative real values of KT2 is related to intermediate states for which the matrix elements (0 1 A / n)(n 1 B I p) or (0 1 B / n)(n j A 1 p) are different from zero. The stability condition for the state 1 p) requires that the cut does not start at s = 0, but at s = M2($ + ] K~ 1)” where A4 is the mass of the state 1 p). The representation (8.8) may therefore alternatively be written in terms of the variable p2 = s - M2~,2 as [22]

q’(!?r , ‘CL) = (; - KL~) jm dp2 c(P2, KL) p2 + Kr;+ M2K 2 - (8.9) 0 L

If the local operators A(x’) and B(x”) may be restricted to the null plane x’- = x”- = 0 we may decompose them into positive and negative frequency parts with respect to p+:

A(X)I, = (27~ Jp,>o 2 +-+~a(~) + PV(~)). (8.10)

The spectrum condition then implies that a(p) annihilates the vacuum and the null plane wavefunction in momentum space therefore describes the projection of the state 1 p) onto the Fock space sector a+b+ j 0):

(0 I ah’) W’) 1 P> = (27~)~ ZP+ a’(~’ 4 P” - P) d&, KL) (8.11)

with tcT = ( pT’pt - p;p+‘)/p+ , ~~ = (p+’ - p3/2p+ . If, furthermore, the restrictions of A and B obey the canonical null plane commutation rules appropriate for scalar fields we have

Md, a+Wl = (W 2p+Wp - P’) (8.12)

and similarly for b(p). The projection operator on the Fock space sector a+b+ IO) then takes the form

P = (2Tr)-6 $ g$ g a+(~‘) b+(p”) I OX0 I 4~‘) 0”). (8.13)


The probability that an observation of the system at the fixed “time” X- = 0 reveals exactly two bare particles is therefore given by

s

This probability coincides with the special norm of the wavefunction discussed in Section 4A. If the local fields are canonical on the null plane we may therefore interpret the Bethe-Salpeter amplitude in terms of the observables P’, P” and Q,‘, QG as described in that section. In particular the square of the momentum space wavefunction determines the probability to find two constituents with relative momentum Kr , KL .

In local field theory the probabilityp, is of course expected to be less than one-there is a nonvanishing probability to find the system in the higher Fock space sectors. In fact, in perturbation theory where the renormalization constants formally vanish, the fields have to be multiplied with infinite factors in order to arrive at a nonvanishing Bethe-Salpeter amplitude. It is not clear whether in that case the norm (8.14) is of any significance for the renormalized amplitudes-in any case there is no reason to expect it to remain bounded by one.

In field theory the dynamical information on wavefunctions of the type (8.1) is usually expressed in terms of the Bethe-Salpeter equation. Approximate B-S wavefunctions are obtained by suitably approximating the kernel of this equation. The B-S equation is a global integral equation that involves the behavior of the wavefunction at all times-it does not answer the question how an arbitrary state specified at a given time evolves as a function of time. Our formulation of Hamiltonian dynamics for two-particle wavefunctions replaces the global B-S integral equation by a differential equation in the time variable. In the literature the reformulation of the B-S equation as a differential equation with respect to time is known as the quasipotential approach. This approach was developed in a series of papers both at fixed time [2.5] and on a null plane [26].

In contrast to the B-S equation the quasipotential equation is not manifestly covariant. If one starts from a given local field theory and works out the corresponding quasipotential equation for the B-S amplitude in a power series of the coupling constant without any approximations, covariance of the wavefunctions is of course guaranteed by the fact that one has only rewritten a manifestly covariant dynamical system in a noncovariant form. The perturbation series of a local field theory therefore defines a set of Hamiltonians M2, My1 , M$, that satisfy all the conditions formulated in Section 5. This class of solutions of the angular condition of course involves a power series in the coupling constant with the free particle Hamiltonians as zero order terms.

If one approximates the quasipotential equation, e.g., by including only certain graphs and omitting others, the resulting wavefunctions will in general not transform covariantly. To insure covariance of the approxmiation scheme one needs, in addition to the quasipotential - which amounts to an implicit expression for the operator M2-a


representation for the two other Hamiltonians My1 and Mj, . The approximate wavefunctions are covariant if and only if these quantities obey the proper commutation rules and satisfy the angular condition.

The angular condition by itself does not specify the qualitative dynamical properties of the system. One class of solutions (perturbation theory of local fields) corresponds to perturbations around free particles, another class (manifestly covariant wave equations) is characterized by interactions that are local in the transverse directions- there presumably are further, qualitatively different classes of solutions. The question then arises whether it is possible to interpret these alternative solutions in terms of an underlying local field theory. For the relativistic oscillator model described in Section 6 the wavefunctions can certainly not be written as (0 [ A(x’) B(x”) 1 p} where A and B are Wightman fields: the oscillator eigenstates are entire functions of KT2 and therefore fail to satisfy the representation (8.8). (The absence of a cut for negative real values of KT2 indicates the absence of nonconfined intermediate states.) In this connection it is interesting to note that if the strong interactions are described by gauge fields coupled to quark color, a suitable representation for meson wavefunctions is given by

<0 1 4(x’) TA exp ig /“” dh z@BJ(L[(B + A) x’ + (4 - A) x”] q(x”) / p> -l/2

= e-(i/2)Pw+z”qj(z)~ (8.15)

The gauge factor has to be included to arrive at a gauge invariant wavefunction-there is no gauge in which this factor is 1 for all points of a given null plane. (Note that in order for the wavefunction to transform covariantly the path from x’ to x” must be a straight line). The wavefunction (8.15) presumably fails to satisfy the spectral representation (8.8). Furthermore, the gauge factor has important consequences for the physical interpretation of the wavefunction. Since the gluon field carries momentum, the quantities sr,

. . KL cannot be identified with the relative momentum

of the current quarks; they rather represent the relative momentum of two constituents whose total momentum is the momentum of the meson. The gauge factor also affects the norm, even if the fields q(x) and B,(x) are canonical on the null plane, in which case the support of the variable KL can still be shown to be contained in / K~ ( < i.

Quite generally, consider an operator p(x’, x”) that describes any configuration of fields localized in some set 0(x’, x”) of finite extension and let

(0 1 p(x’, xy / P) = e-(i12)Pld+qj(x’ - xy (8.16)

be the corresponding wavefunction. Equations (8.1) and (8.15) represent two particular examples of such field configurations. In the case (8.1) the set 0(x’, x”) consists of two points x’ and x”. In the case (8.15) the set 0(x’, x”) involves the whole straight-line segment connecting the points x’ and x”. It is interesting to remark that this is almost the most general situation: the spacetime properties of the local field configuration that appears in Eq. (8.16) are to a large extent independent of any dynamical model.


The reason is that the operator p(x’, x”) must transform covariantly under the Poincart group:

U(A, a) p(x’, x”) u+pl, a) = S’pl-1) @ s”(fl-1) &lx + a, fix” + a). (8.17)

The set of points 0(x’, x”) which defines the spacetime extension of the field configuration p(x’, x”) should therefore be determined in a Poincare invariant manner by the two points x’ and x”. For spacelike or lightlike separations x’ - x”, the only sets 0(x’, x”) of finite extension with this property are the subsets of the straight line through x’ and x”. (This can be seen as follows. The set 0(x’, x”) must be invariant under those Poincart transformations that do not affect the pair of points x’, x”. If the pair has spacelike separation these transformations form the group O(2, 1). Let now y E 0(x’, x”) be a point not contained on the straight line through x’ and x”. Under the group 0(2, 1) the orbits of this point constitute an infinitely extended hyperboloid-if the set 0(x’, x”) contains y it must contain the entire hyperboloid in contradiction with the requirement that 0(x’, x”) is of finite extension. A similar argument applies for lightlike separation. In the timelike case the statement is not true; this case does not concern us here because the null plane wavefunctions involve only spacelike and lightlike separations.)

The operator p(x’, x”) therefore describes a configuration of fields localized at the points

x(A) = (4 f A) x’ + (a - A) XI, (8.18)

where the parameter X is contained in some finite interval D that may at most depend on the invariant distance of x’ and x”. (For the particular case (8.15), D = (-a, a), independently of (x’ - x”)“.) The discussion of the transverse position operators in Section 4 suggests that the interval D is closely related to the support of the inner longitudinal momentum variable K L: for finitely extended field configurations p(x’, x”), one may expect that fcL has compact support. (This may indeed be verified, if p(x’, x”) admits an expansion in terms of polynomials in canonical local fields taken at the points (8.18).) This leads us to the conjecture that for wavefunctions that admit a representation of the general type (8.16) such that p(x’, x”) can be expressed in a covariant manner in terms of fields localized in a finite spacetime region, the support of KL is

KL ED(-+'), (8.19)

where D(-zr2) is compact. The spectrum condition ] ~~ / < $ is of course a special case of this general situation.

Operators with the above properties satisfy the following local commutativity:

provided [/4x’, x7, PW, Y”)l = 0, (8.20)

cm - Yh)12 < 0


for all h E D[(x’ - x”)~] and 7 E D[( y’ - JY)“]. This property of the operator p(x’, x”) provides a manifestly covariant formulation of the “straight-line segment locality” that we encountered in Section 4C in connection with the spacetime properties of the norm: it further corroborates the interpretation of a composite system as an object extended over a segment of a straight line.

It is conceivable that bilocal fields of the type p(x’, x”) may even be used as basic field variables. A suitable field theoretic framework for the class of solutions to the angular conditions that are local in the transverse directions may involve a covariant bilocal field p(x’, x”) that obeys a pair of covariant wave equations of the type discussed in Section 6 and satisfies a commutation rule of the type

Mx’, x7, PW, Jo

= I

dz’ dz” A(x’, x”; y’, y”; z’, z”) p(z), z”) + B(x’, x”; y’, y”). (8.21)

The functions A and B are subject to the equations of motion and to the locality requirement (8.20).

9. SUMMARY AND CONCLUDING REMARKS

We have analyzed the structure of relativistic quantum theories for systems with a finite number of degrees of freedom. For Hamiltonian quantum theories the surface in spacetime on which the state of the system is described initially plays a central role; we have shown that null planes constitute a particularly suitable class of initial surfaces. One of the main virtues of the null plane framework is the fact that the kinematical features of Lorentz invariance may be separated very neatly from its dynamical content: the generators of the Poincart group are replaced by the generators of two commuting algebras s and 9. The kinematical algebra X = {P+ , P, , P, , El , E, , &} is generated by three components of the momentum and three boosts, (Z essentially generates the stability group of the null plane), whereas the elements of the dynamical algebra 9 = {M, I} are the mass and spin operators of the system generating the group U(2). (The algebra 9 contains the nontrivial part of the Hamiltonians of the theory, i.e., of those generators of the Poincart group that do not leave the initial null plane invariant.) Different physical systems amount to different representations of Q-the representation of Z is always the same, as long as we exclude physical states of zero mass.

The factorization of the kinematical and the dynamical properties of the system is very similar to the decomposition of nonrelativistic Galilei covariant wavefunctions into a factor that describes the motion of the center of mass and an internal wavefunction that depends on the specific dynamical properties of the system. We describe the connection between Lorentz invariant quantum mechanics on a null plane and Galilei invariant nonrelativistic quantum mechanics in more detail below.

RELATIVISTIC DYNAMICS ON A NULL PLANE 1.53

The requirement of relativistic invariance is a rather weak condition on the theory- every unitary representation of ~2 gives rise to a unitary representation of the Poincare group and thus describes a relativistically invariant system. For elementary systems, the representation of 9 and hence of the Poincare group is irreducible, for composite systems it is reducible. We have investigated in detail only the simplest class of composite systems, those that are described by the degrees of freedom of two particles. The relativistic description of three-particle systems such as baryons is beyond the scope of the present paper.

The central theme of the paper is comriance. For elementary systems relativistic invariance implies that the wavefunctions describing the system transform covariantly under the Poincare group. This is not the case for composite systems-to demand that the wavefunctions transform in a covariant manner is to impose strong restrictions on the dynamical properties of the system, i.e., on the representations of the algebra 9. We have shown that for systems composed of two constituents the covariance requirement is equivalent to an algebraic condition to be satisfied by the generators of g-the angular condition. In the case of spin zero constituents this condition reads

zlMA + z2M$2 + zJ3 = 0, (9.1)

whereas for constituents of spin 4 the angular condition is nonlinear. The problem of finding two-particle interactions that are not only relativistically invariant but are such that the wavefunctions transform covariantly is equivalent to the algebraic problem of finding simultaneous solutions to the commutation rules of the dynamical algebra 9 and to the angular condition.

Standard local field theories defined by their perturbation expansion give rise to a particular class of solutions of the angular condition. The wavefunctions are in this case identified with the Bethe-Salpeter amplitudes of the theory and the Hamiltonian M2 is closely related to the quasipotential: perturbation theory specifies M2 and 2 as a power series in the coupling constant with the free particle Hamiltonians as zero order terms.

We have described an alternative class of solutions to the angular condition: two-particle interactions that are local in the transverse directions of the null plane. This class of solutions is closely related to a particular type of local manifestly covariant wave equations and might be relevant for a description of systems that consist of confined constituents. In particular, we have shown that the relativistic oscillator model of Feynman et al. satisfies the angular condition. The corresponding eigenstates of M2 constitute a complete set of wavefunctions on the null plane.

Finally, it is interesting to specify in more detail the analogy and the differences between the theory developed in this paper and Galilei invariant quantum mechanics. The Galilei group, too, may be replaced by the direct sum & @ 9G of a kinematical and a dynamical algebra. The kinematical algebra X, = {P, I?} of the Galilei group is generated by the total momentum p and by the Galileian boosts I?. Every non-


relativistic system of total mass m carries the same projective representation of the group generated by the operators XG . In this representation only the commutators

[Ki , Pj] = -ima,, (9.2)

do not vanish. The dynamical algebra C3%c = {H, &‘> of the Galilei group contains the nonrelativistic inner Hamiltonian

H = PO - (1/2m) pz (9.3)

(which replaces the operator M of the relativistic case) and the nonrelativistic spin operator given by

2==+(l/m)R x P. (9.4)

The commutation rules of the relativistic and nonrelativistic dynamical algebras are the same: Lorentz or Galilei invariance of a quantum mechanical system are guaranteed, if the corresponding dynamical algebras constitute a unitary representation of U(2). The general algebraic structure of both theories thus appears to be rather similar. We now turn to the differences.

One difference between the null plane and Galileian quantum mechanics concerns the interpretation of the boosts that occur in the corresponding kinematical algebras. The origin of this difference is the fact that one of the three relativistic momentum components, viz. P, is positive. As an immediate consequence of this property, there is no position operator in the longitudinal direction, whereas the two transverse relativistic position operators ET/P+ are analogous to the corresponding components of the Galileian center-of-mass position operators z/m.

The two theories further differ in the relation of their dynamical algebras to the stability groups of the corresponding initial surfaces. In the relativistic case, only $a belongs to the stability group of the null plane X- = 0 and is therefore independent of the interaction. In the nonrelativistic case, all three components of 2 are contained in the 9-parametrical Galileian stability group of the instant x0 = 0; they are kinematical quantities, i.e., are given by the same expression for free and for interacting constituents.

The different character of the relativistic and nonrelativistic spin operator $ has an important corollary concerning the requirement of covariance. In the case of the Galilei group, this requirement does not affect the dynamics: the angular condition for spin 0 constituents for instance takes the form

(9.5)

and is automatically satisfied by the kinematical expression for $. The angular


condition (9.1) for the relativistic case on the other hand contains important information on the Hamiltonians of the theory. In a sense, this information is the substitute for the explicit expression of the nonrelativistic spin operator.

The last important difference between null plane and nonrelativistic quantum mechanics we would like to stress is related to the norm of the two particle inner space. In the nonrelativistic case, the two-particle Hilbert space may always be identified with the representation space of the direct product of two unitary representations of the Galileian stability group of the instant x0 = 0. The norm therefore coincides with the product of two single-particle norms and the system may always be interpreted as consisting of two-point particles. This is closely related to the fact that in the nonrelativistic case, covariance-cf. the angular condition (9.5)-does not affect the dynamics. The norm is independent of the interaction. In the relativistic case-as illustrated by the oscillator model-one has to admit a more general, interaction dependent norm. A relativistic two-particle system in general appears as an object extended along a straight-line segment, rather than as a collection of two- point particles.

It is clear that the present paper leaves untouched various important questions such as: the complete classification of solutions of the angular condition that are local in the transverse coordinates [30],the construction of local observables such as the electromagnetic current, stability of solutions of the angular condition with respect to small perturbations, inclusion of quark flavor, extension to baryons and last but not least, the construction of a realistic model for hadron spectroscopy 1311. We intend to return to these and similar questions elsewhere*. Here, we would like to add one more remark that concerns the nontrivial inclusion of internal symmetries and is relevant in connection with the well-known no-go theorem of O’Raifeartaigh [27].

The fact that the Poincare algebra may be replaced by the direct sum ~6 @ 9 makes it possible to embed the dynamical algebra &@ in the algebra of a larger group 9 such that the higher symmetry connects eigenstates of the mass and spin operators with different eigenvalues. In order for the higher symmetry 9 to be relativistically invariant, it suffices to let it commute with X, i.e., to let it act exclusively in the space of inner degrees of freedom. This construction leads to a nontrivial mass spectrum, without of course contradicting the theorem: the generators of FZ and those of the Poincart group do not close on a finite-dimensional Lie algebra. (This can be seen, e.g., by observing that the commutators of 3 with the generator P- give rise to operators of the form P;l%‘. Commuting two such quantities, one gets elements proportional to P;‘, etc.) The fact that the theorem can be avoided if one allows in a rather ambiguous way for infinite-dimensional algebraic structures is of course well known [28]. The point we wish to emphasize here (see also Ref. [29]) is that the null plane framework provides a natural way of uniting spacetime and internal symmetries: to have a nontrivial Lorentz invariant higher symmetry it is not necessary to embed

* Footnote added in proox Since this paper was submitted for publication some of these questions have been solved [30, 311. A brief account of the work reported here has appeared [32].


the Poincare group in a larger group, it suffices to embed the dynamical algebra-on a null plane, the kinematics takes care of itself. The same remark of course applies to relativistic spin symmetries, as illustrated in Section 5D.

APPENDIX I: CANONICAL COVARIANT BASIS FOR ELEMENTARY SYSTEMS

The covariant wavefunction (3.8) can be expressed as inner product of the state 1 y> with the covariant vectors

d4p &P+) %P’ - m2) e-iv2 C u=@, h> <p, h /. (1-l) h

At x = 0, these vectors span a finite-dimensional representation S of the homo- geneous Lorentz group, which is constructed as follows.

The operators J3 and K3 may be diagonalized on the set of states 1 a) = j x, 01)~ j2=,,:

(a I J3 = ALa I, (a I K2 = iYa<~ I, (I.3

where h, and 3/a stand for the helicity and for the goodness of the state (a I. It is convenient to introduce the operators

NY’ = +(Ji & Xi) (1.3)

which generate the complex SU(2)+ @ SU(2)- basis of SL(2, C). In particular, NJ*’ are diagonal on (a 1 with eigenvalues

(1.4)

In the basis (I.3), the kinematical boosts E1 , E, may be expressed as

i(E, + iE2) = NF’ = If,‘+’ + iN,‘+’

i( --El + iE2) = NL-’ = N,‘-’ - iNz(-) (I-5)

and the dynamical boosts Fl , F2 become

-& (-Fl + iF,) = N?) = N,‘+’ - iN,‘+’

4 (Fl + iF2) = Ni? = IV,‘-’ + iNz(-‘. (I-6)

The operators (1.5) and (1.6) respectively raise and lower the goodness y by unity. A finite-dimensional representation therefore contains a subset of vectors <h, yrnax 1 that belong to the maximal goodness y = ymax , i.e., satisfy

(A, ymax I N>) = (A, ymax I N:-’ = 0. (1.7)

An irreducible representation of the Lorentz group contains only one state of maximal goodness-the state with minimal eigenvalue --n+ of N.j+’ and maximal eigenvalue IZ- of Ni-‘. The corresponding irreducible representation D(n+, n-) is uniquely character-


ized by the two positive integers or half-integers n+ and n-. Alternatively, the representation may be specified by ymax = rz+ + n- and by the helicity of the state with maximal goodness, h = n- - nf.

The canonical covariant basis involves a reducible representation S of the Lorentz group which may be characterized by the following property: S is the smallest representation of the Lorentz group for which the subspace of good vectors (vectors with maximal goodness) contains all helicities h = -s,..., s. For this representation we have

and Y max = s m

(1.9) .4=-S

The covariant wavefunction (3.8) that transforms according to the canonical representation (1.9) of the Lorentz group is determined by the spinor amplitude ua(p, h), which may now be constructed in the following manner. The good components 0, ymax / of the vector (a I are obtained from their stability group transformations properties, i.e., from Eqs. (1.2) and (1.7). Using the action (2.14) and (2.16) of the stability group on the basis vectors (p, h 1, Eqs. (1.2) and (r.7) are seen to determine the states (X, yrnax 1 up to a constant:

(1.10)

The good components of the amplitude u”(p, h) therefore read

uGnax) (P, A) = (2P+)s SAh, (I.1 1)

as claimed in Section 3. The bad components of the spinor amplitude u” are obtained, applying repeatedly to the states (1.10) the goodness lowering operators Fr , F, , whose action on the basis vectors (p, h I is given by Eq. (2.43). (The (2s + 1)-fold application of F, gives zero: the minimal goodness contained in the representation S (1.9) is ymin = -s.)

Let us consider a particle of spin Q. The corresponding canonical representation S coincides with the standard Dirac representation D(0, 4) @ o(+, 0). (Note that all vectors that belong to this representation may uniquely be labeled by the helicity X = && and by the goodness y = -&.) The general prescription described above gives in this case

zP*+qp, h) = (2p+)+ SAh

ucA*-+)@, h) = (2p+)+ (Pp,aSh + imPh).

The expression (3.17) for the bad components

(1.12)

I d4p O(p+) S(p2 - mz) e--ip*z C u(++)(p, h) v*(p) (1.13) h


is a straightforward consequence of the formula (1.12). Equations (1.12) and p2 = m2 imply that the covariant amplitude

ucA,+f) U= ( ) u(A,-3) (1.14)

satisfies the Dirac equation

(P,Y~ - 4 u(P, h) = 0. (1.15)

The canonical null plane basis corresponds to the following representation of the y-matrices:

0 -i Y”=i 0’ i ) y3 = (Ji ii);

(1.16)

y1 = ( “2 y2 = ( -7

With this representation, Eqs. (1.12) and (I. 15) are equivalent.

APPENDIX II: EQUIVALENCE OF ANGULAR CONDITIONS AND GLOBAL COVARIANCE

In the case of spin 0 constituents, the null plane restriction of the covariant wavefunction (5.12) that describes a state of mass m, spin j, and helicity h reads

$(gT, 23 = F(zT2, 23 lim zyl .** ~%~~...~~(p). z--b0

(11.1)

It is convenient to introduce the vector

F = (zl, z2, zJm) (11.2)

and the corresponding spherical coordinates

z1 = r cos a sin p, z2 = r sin 01 sin #I, zJm = r cos /I. (11.3)

The limit standing on the right-hand side of Eq. (II. 1) depends on z and p only through the vector F. It can therefore be determined going to the rest frame, in which 7 = (zl, 22, 23):

lim zU1 ... ~~j&...~~(p) = riYj,&3, a). z-+0

(11.4)

Let now $(zr, L z ) be an eigenfunction of the mass and angular momentum operators:

My” = mq?, $;$ = c &*A, (11.5) A’


where Ji are the matrices that generate the (2j + I)-dimensional UIR of the rotation group and J3 is diagonal. We are going to prove that provided the operators M and 2 satisfy the angular condition (5.16), the wavefunction @ is of the form (11.1).

The angular condition (5.16) implies

,P4 = 0, (11.6)

where a matrix notation in the helicity space is understood. The matrix ?j may be written as

Fj = rDf(a, p, 0) J,[Dj(a, /3,0)]+, (11.7)

where Dj(a, p, y) = e-i*Jw isJze-i?‘Js is the standard rotation matrix. The angular condition therefore states that the rotated wavefunction

‘PA = 1 ‘P~‘D$&, /3, 0) (11.8) A’

vanishes, unless X = 0. This implies

~A(~~ = y’(r, kND:,(~, A WI*.

The formula (11.1) follows from this representation, if one uses the identity

(11.9)

and Eq. (11.4).

[Dio(a, 8, 011” = (&)“’ rid/% 4 (11.10)

In the case of spin & constituents, the covariant wavefunction (5.30) that describes a state of mass m, spin j, helicity h and normal parity can be decomposed as follows

WP, 4 = hJ”(~lP, + v,a,7 + r”r5&%@;

+ U~T~Z,P, + r,z,a,z + T3pua:) + -1 ~“1 - ~~+...,i(p). (11.11)

The functions I’, A, and T depend on p and z through z2 and p . z. The dots stand for terms that do not contribute to the good-good components of the covariant wavefunction @ at z- = 0. (Note that among the 16 y-matrices only the following four matrices actually contribute to the good-good components: y+, r+r5, and u+‘.) In the case of a state of abnormal parity, the right-hand side of Eq. (11.11) gets multiplied by y5.

We now restrict the good-good components of the covariant wavefunction (11.11) to the null plane z- = 0. The resulting 2 x 2 matrix &thv(r) reads

q?(F) = [Ul + u,v, + a,u(zlV, - ZT,) + w, - z2q)(f1 + f,V,) + f,(qV, - fJ2v31 f-jYj*,@, a), (11.12)

where 9 = a/a?. The six functions u, a, and t coincide (apart from trivial numerical

595/I 12/I-I I


factors) with the six scalar amplitudes V, A and T taken at z- = 0: they depend on the variables zr2 and z, . Note that all terms in the square bracket in Eq. (11.12) are $a invariant and even under the action of the null plane parity 8, . (Let us recall that under 9, , z1 + zl, z2 + -z2, Z~ --f z, and crl + -ol , u2 ---f u2 , ug -+ --us .) In the case of a state of abnormal parity, the right hand side of Eq. (11.12) gets multiplied by 03 . The derivatives a in Eq. (11.12) may be reexpressed in terms of the generators of rotations of the vector ?

Et= -ii- X 9. (11.13)

The formula (11.12) then becomes

cp”(F) = [F’l’(~) + P(i.)(zlL, - 2X,) + u3F’yq L3] Yj,,(/3, a), (11.14)

where P are f3 invariant, 8,-even matrices in the space of the constituent’s helicities:

F;;“(F) = fi)2ihy + &Ji)(z1u2 - z2u&y , (11.15)

The six functions fu) and gci) are $,-invariant combinations of the amplitudes v, a and t.

Consider now the null plane wavefunction &la& , zL) that satisfies the eigenvalue equations (11.5) and corresponds to a state of normal parity. We are going to prove that provided the mass and the angular momentum operators M and ,$ obey the angular condition (5.35), this wavefunction is of the form (11.14), i.e., it coincides with the null plane restriction of the covariant wavefunction (II. 11).

As before, the angular condition (5.35) may be expressed in terms of the matrix ?jj:

rp&“(F)[(d)3 - r”(Q)] = 0. (11.16)

Using the identity (11.7), this equation becomes

(A3 - A) Y;,,“(F) = 0, (11.17)

where ?Pi,,, is related to the null plane wavefunction P);,~- by the formula (11.8). Consequently, ?Pi h , . vanishes for X # 0, &1 and one has the representation

(11.18)

All three matrices Y$,. , (A = 0, *I) must be j3 invariant. Furthermore, for a normal parity state, Y” and Y+l - Y-l must be even, whereas Y+l + Y-l must be odd under the action of the null plane parity P= . The three matrices

ylo(~~, u+yi;> - Y-l(F), u3[Yl+l(jt) + Y-l@)] (11.19)


are therefore of the form (11.15). This fact, the formula (11.10) and the identity

P:,*l(% B, WI*

[ 47-r 1

l/2 1 = j(j + I)(3 + 1)

r L--r& It i(zlL, - z2L,)1 Y&z 01) (11.20)

make it clear that the representation (11.18) of the null plane wevefimction indeed coincides with Eq. (11.14).

In the case of an abnormal parity state, the right-hand side of Eq. (11.15) should be multiplied by c3 . The same modification however applies to the form of the matrices (11.19): they are yS-invariant and odd under PL and therefore take the form fu3 + g(zlu, + 2 g2) as claimed above.

APPENDIX III: AN EXPANSION THEOREM FOR HERMITE POLYNOMIALS

To prove the expansion (6.19) of plane waves into distorted Hermite polynomials we first define an analytic function g(y) as follows. Consider the mapping z ---f y

y = zeu/2)22 (111.1)

which maps the disc 1 z 1 < 1 into a region D, that contains the disc 1 y 1 < e-1/2. Since the derivative dy/dz vanishes only at the points z = *i, y = &ie-li2 the mapping z + y has an analytic inverse z = g(y) which maps the disc 1 y / < e-112 into a region D, that is contained in 1 z 1 < 1. This defines the function g(y) as an analytic function on the disc 1 y 1 < e-lj2.

Next, we consider the function

f(z) = (1 + ,2)-l esz-(1/2)tz2 (111.2)

analytic except for poles at z = +i. The quantity f[ g( y)] is analytic on the disc I y I < e-lj2 and thus admits a Taylor expansion

.fMY)l = f Y”fn (111.3) n=o

that converges for 1 y / < e-li2. The expansion coefficients may be calculated with Cauchy’s formula:

Transforming the variable of integration from y to z = g(y) this integral becomes

1 d” =- n. I I( 1 z

esz-u/2Ht+n)e* 1 * 2-O

(111.5)


The derivative may easily be evaluated by using the generating function for Hermite polynomials:

&z2+bz - -

with the result

fn = ; [; (n + q2 Hn ( [.+ ; t)]1'2 )* (111.7)

Inserting this result in (111.3) we conclude that the representation

es2 = (1 + z2) gO 2 [f (IZ + ‘)lniz e(1/2)(t+n)z2Hn ( L2cn ” tj,l,2 )

converges provided z is contained in the region D, . This region contains the interval z = -2ia 9 -4 < 01 < & of the imaginary axis as claimed in Section 6 (see Fig. I).

1 Re z

FIG. 1. The convergence domain D, of the expansion of en2 in terms of distorted Hermite polynomials.

ACKNOWLEDGMENTS

The authors are indebted to Professors M. Ida, R. Nataf and to Dr. J. Gasser for discussions, to Drs. L. Oliverland A. L.e-Yaouanc for a useful remark concerning the oscillator model and finally to Professor N. H. Fuchs for reading the manuscript.

REFERENCES

1. P. A. M. DIRAC, Rev. Mod. Phys. 21 (1949), 392. 2. E. P. WIGNER, Ann. of Math. 40 (1939), 149; Nuovo Cim. 3 (1956), 517; V. BARGMANN AND E. P.

WIGNER, Proc. Nat. Acud. Sci. 34 (1948), 211. 3. L. L. FOLDY, Phys. Rev. 122 (1961), 275; B. BAKAMIIAN AND L. H. THOMAS, Phys. Rev. 92 (1953),

RELATWISTIC DYNAMICS ON A NULL PLANE 163

1300; F. COESTER, I-I&. Pfiys. Acta 38 (1965), 7; R. FONG AND J. SUCHER, J. Math. Phys. 5 (1964), 456; T. F. JORDAN, A. J. MACFARLANE, AND E. C. G. SUDARSHAN, Phys. Rev. B 133

(1964), 487; L. B. REDEI, 1. Math. Phys. 6 (1965), 487; E. H. KERNER (Ed.), “The Theory of

Action-at-a-Distance in Particle Dynamics,” Gordon and Breach, New York, 1973; S. N.

Sokolov, Theor. Mat. Fis. 24 (1975), 236; L. L. FOLDY AND R. A. KRAJCIK, Phys. Rev. D 10 (1974), 1777.

4. Yu. M. SHIROKOV, JETP 9 (1959), 330; F. A. ZHNOPL~T~EV, A. M. PEROLOMOV, AND Yu. M.

SHIROKOV, JETP 9 (1959), 333; L. L. FOLDY AND R. A. KRAJCIK, Phys. Rev. D 12 (1975), 1700; S. N. SOKOLOV, Theor. Mat. Fis. 23 (1975), 355.

5. S. N. SOKOLOV, Doklad. Acad. Nauk SSSR 221 (1975), 809; V. B. BERESTETSKY AND M. V.

TERENT’EV, ITEP preprint, Moscow, 1976.

6. H. BACRY, P. COMBE, AND P. SORBA, Rep. Mafh. Phys. 5 (1974), 145; 5 (1974), 361; J. PATERA, P. WINTERNITZ, AND H. ZASSENHAUS, J. Math. Phys. 16 (1975), 1597; J. PATERA, R. SHARP, P. WINTERNITZ, AND H. ZASSENHAUS, J. Mark Phys. 17 (1976), 977.

7. S. FIJBINI, A. J. HANSON, AND R. JACKIW, Phys. Rev. D 7 (1973), 1732; C. M. SOMMERFIELD, Ann. 0fPhy.s. 84 (1974), 285; D. GROMES, H. J. ROTHE, AND B. STECH, Nucl. Phys. B75 (1974),

313. 8. L. C. BIEDENHARN AND H. VAN DAM, Phys. Rev. D 9 (1974), 471; L. P. STAUNTON, Phys. Rev.

D 8 (1973), 2446; 13 (1976), 3269. 9. G. ALTARELLI, N. CABIBBO, L. MAIANI, AND R. PETRONZIO, Nucl. Phys. B69 (1974), 531.

10. J. S. BELL, Acta Phys. Ausrriaca Suppl. 13 (1974), 395; J. S. BELL AND H. RUEGG, Nucl, Phys. 893 (1975), 12; 104 (1976), 245.

11. J. R. KLAUDER, H. LEUTWYLER, AND L. STREIT, Nuovo Cimento A 66 (1970), 536.

12. F. ROHRLICH, Schladming Lectures (1971); PH. MEYER, Cargese Lectures (1972); F. G~.?RSEY AND S. ORFANW~, Nuovo Cimento A 11 (1972), 225; G. DOMOKOS, Comm. Math. Phys. 26 (1973), 15; J. K~CXJT AND L. SUSSKIND, Phys. Reports CS (1973), 75; J. S. BELL, Nucl. Phys. B66 (1973), 293; H. LEUTWYLER, Nucl. Phys. B76 (1974), 431; H. SAZDJIAN AND J. STERN,

Nucl. Phys. B 94 (1975), 163; M. IDA, Prog. Theor. Phys. 54 (1975), 1519,1775; V. A. KARMANOV, Z. Eksper. Theoret. Fiz. 71 (1976), 399; JETP Lett. 23 (1976), 54; M. HUSZAR, Nuovo Cimento A 31 (1976), 237.

13. D. G. CURRIE, T. F. JORDAN, AND E. C. G. SUDARSHAN, Rev. Mod. Phys. 35 (1963), 350; D. G. CURRIE, J. Math. Phys. 4 (1963), 1470; J. T. CANNON AND T. F. JORDAN, J. Math. Phys. 5 (1964),

299; H. LEUTWYLER, Nuovo Cimento 37 (1965), 556. 14. T. D. NEWTON AND E. P. WIGNER, Rev. Mod. Phys. 21 (1949), 400. 15. H. J. MELOSH, Phys. Rev. D 9 (1974), 1095.

16. H. OSBORN, Nucl. Phys. B 80 (1974), 90; R. CARLITZ AND WU-KI TUNG, Phys. Rev. D 13 (1976), 3446.

17. J. WEYERS, Lectures given at Louvain 1973, CERN preprint TH-1743; J. L. ROSNER, talk at the

XVII International Conference on High Energy Physics, London 1974; S. P. De ALWIS AND J. STERN, Nucl. Phys. B77 (1974), 509; CERN preprint TH-1783, 1973, unpublished; A. J. G. Hey, P. J. LITCHFIELD, AND R. J. CASHMORE, Nucl. Phys. B 95 (1975), 516; Rutherford Laboratory

preprint RL-76-111. 18. M. IDA AND H. YABUKI, Prog. Theor. Phys. 55 (1976), 1606; 56 (1976), 297; H. LEUT~~LEX,

Nucl. Phys. B 76 (1974), 431. 19. P. A. M. DIRAC, “The Development of Quantum Theory,” Gordon and Breach, New York,

1971. 20. R. P. FEYNMAN, M. KISLINGER, AND F. RAVNDAL, Phys. Rev. D 3 (1971), 2706. 21. K. FUJIMURA, T. KOBAYASHI, AND M. NAMIKI, Prog. Theor. Phys. 43 (1970), 73; A. L. LICHT

AND A. PAGNAMENTA, Phys. Rev. D 2 (1970), 1150, 1156; 4 (1971), 2810. 22. H. YABUKI, Research Institute for Mathematical Sciences, Kyoto University preprint 1976.

23. S. DESER, W. GILBERT, AND E. C. G. SIJDARSHAN, Phys. Rev. 115 (1959), 731. 24. M. IDA, private communication.


25. M. LEVY, h’zys. Rev. 88 (1952), 72; 88 (1952), 725; A. KLEIN, Whys. Rev. 90 (19.521, 1101; 94 (1954), 1052; W. MACKE, Z. Nuturforsch 89 (1953), 599; 89 (1953), 615; A. A. LCIGUNOV AND

A. N. TAVKHELIDZE, Nuovo Cimento 29 (1962), 380; R. BLANKENBECLER AND R. SUGAR, Phys.

Rev. 142 (1966), 1051; V. G. KADYSHEVSKY, NucI. Phys. B6 (1968), 125; C. ITZYKSON, V. G. KADYSHEVSKY, AND I. T. TODOROV, Phys. Rev. D 1 (1971), 2823; I. T. TODOROV, Phys. Rev. D 3 (1971), 2351; A. KLEIN AND T. S. H. LEE, Phys. Rev. D 10 (1976), 4308.

26. S. WEINBERG, Phys. Rev. 150 (1966), 1313; G. FELDMANN, T. FULTON, AND J. TOWNSEND, Phys.

Rev. D 7 (1973), 1814; J. F. GUNION, S. BRODSKY, AND R. BLANKENBECLER, Phys. Rev. D 8 (1973), 287; R. N. FAUSTOV, V. R. GARSEVANISHVILI, A. N. KVINIKHIDZE, V. A. MATVEEV, AND

A. N. TAVKHELIDZE, Theor. Mat. Fiz. (USSR) 23 (1975), 310; 24 (1975), 3; 25 (1975), 37;

S. P. KULESHOV, A. N. KVINIKHIDZE, V. A. MATVEEV, A. N. SISSAKIAN, AND L. A. SLEPCHENKO,

Dubna preprint (1974); N. M. ATAKISHIYEV, R. M. MIR-KASIMOV, AND SH. M. NAGIYEV, Dubna

preprint (1976); V. R. GARSEVANISHVILI, JETP Lett. 23 (1976), 51. 27. L. ~‘RAIFEARTAIGH, Phys. Rev. 139 (1965), B1052.

28. J. FORMANEK, Czech. J. Phys. B16 (1966), 1; Nuovo Cimento A 43 (1966), 741; Nucl. Phys. 79 (1966), 645.

29. J. JERSAK AND J. STERN, Nucl. Phys. B 7 (1968), 413. 30. H. LEUTWYLER AND J. STERN, Nucl. Phys. 1978, to appear.

31. H. LEUTWYLER AND J. STERN, Phys. Lett. 73B (1978), 75. 32. H. LEUIW~ER AND J. STERN, Phys. Lett. 69B (1977), 207.

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Relativistic dynamics on a null plane