Volume 69B, number 2 PttYSICS LETTERS 1 August 1977

C O V A R I A N T Q U A N T U M M E C H A N I C S O N A N U L L P L A N E

H. LEUTWYLER lnstitut fffr theoretische Physik ~, der Universitat Bern, Sidlerstrasse 5, 3012 Bern, Switzerland

and

J. STERN Division de Physique Th~orique ~ , Institut de Physique Nucldaire, Universit~ de Paris XI, 91406 Orsay, France

Receivcd 26 May 1977

Lorentz invariance implies that the null planc wave functions factorize into a kinematical part describing the mo- tion of the system as a whole and an inner wave function that involves the specific dynamical properties of the system - in complete correspondence with the nonrelativistic situation. Covarianee is equivalent to an angular condition which admits nontrivial solutions.

The Hilbert space of a Lorentz invariant quantum theory must carry a unitary representation of the Poincar6 group. The generators of this representation determine the dynamical properties of the system; they determine, in particular how a state prepared at a given time evolves as a function of time, i.e. specify the dynamics in Hamiltonian form.

In relativistic quantum field theories the Hamiltonian approach has not received much at tention. These theo- ries are usually specified by a relativistically invariant Lagrangian giving rise to manifestly covariant equations of motion. Bound states are described in terms o f their .Bethe-Salpeter wave functions that obey a covariant four-dimensional integral equation. In view of the lack of calculational methods outside perturbation theory it is of interest to develop the Hamiltonian approach further. In particular, the semiphenomenological des- cription of hadrons as bound states of two or three quarks and the success of simple spin-orbit coupling schemes requires a more detailed analysis of Lorentz invariance for wave functions involving a finite num- ber of degrees of freedom.

Dirac has pointed out [1 ] that a relativistically in- variant Hamiltonian quantum theory can be based on different classes of initial surfaces: spacelike planes, null planes or hyperboloids. The structure of the theory

* Work supported in part by the Swiss National Science Foundation.

** Laboratoire associ~ au C.N.R.S.

is considerably different for these three forms of rela- tivistic dynamics. In the present paper we sketch the main points of an investigation of Lorentz invariant quantum theories based on a null plane as an initial surface [2].

Those generators of the Poincar6 group that leave the initial surface invariant (i.e. generate the stability group of this surface) are kinematical quantities whose representation is independent of the dynamical proper- ties of the system. The remaining generators are the Hamiltonians of the theory; they contain the informa- tion about the specific dynamical characteristics of the system. One reason for the difficulties met in relativis- tic quantum mechanics based on the surface x ° = const. is that the stability group of a spacelike plane is too small: for an observer S who watches the system at time x ° the trivial kinematical part of the Poincar~ group is generated by the momentum P and the angu- lar momentum J. Since the stability group interrelates only states of motion with the same value of P it is a highly nontrivial matter for S to realize that two iden- tical systems which happen to move with different velocities are indeed identical. This problem does not arise for an observer N who watches the system on the null plane x 0 = x 3. His stability group includes the gen- erators

cK = {PI ' P2 ' P+' E l ' E2 ' K3 ) (1)

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as well as the third component J3 of angular momen- tum. [P+ = ~'(Po + P3) ' - 1. • - ± E 1 - 2(K 1 + J 2 ) , E 2 - 2(K2 - J l ) ] . The algebra 9( which we refer to as the kine- matical algebra has the following remarkable property: the three generators E l , E 2 and K 3 that appear in the place of the angular momentum are boosts that, acting on a state at rest, generate a state of arbitrary velocity (i.e. ~ acts transitively on the mass shell of the system). A complete set of states IP, n) of the system is obtained from a complete set of states at rest In) by

[P, n) = exp - i/3rE r exp - i/33K 3 In) (2)

with 3r = pr/p., f13 = ln(2p+/M). For the observer N there is no difficulty to compare systems that happen to move with different velocities - this aspect of Lorentz invariance is a kinematical problem. The price to pay is of course that it is not so easy for N to recog- nize whether two systems at rest that differ only in their orientation are in fact identical - rotations in-

volve the Hamiltonians F 1 = K 1 - J 2 ' F2 = K2 + J l" To analyze the behaviour of the system under rota- tions we introduce the spin operators ~/as follows. We require that on states at rest they coincide with the total angular momentum of the system, ~ In) = JIn) and extend this definition to states in motion by ask- ing ~ to commute with the kinematical algebra oK. Since the spin operators commute with the mass opera- t o r M = [pUpu] 1/2 the algebra

c/) = (M, ~') (3)

has the structure of U(2). We call c/) the dynamical algebra. It contains the essential part of the Hamil- tonians F1, F 2 and P_ -- P0 - P3" The generators of the Poincar6 group that are not contained in 0( may be written in terms of the elements of q( and ,'7) as follows

e_ : (2&) -1 2

F = re+) -1 (e K 3 - s 93 +Mgs))

J3 = (p+)- i (erE 2 _P2Ex } + 93

(4)

where ers = -e~.; e l 2 = 1. This shows that we may re- place the Poincar~ algebra by the direct sum cK ® c-/). A unitary ,representation of the Poincar6 group is the direct product of a unitary representation of the kine-

matical factor c'K with a unitary representation of the dynamical factor c/). The representation of the factor ~X is the same for all physical systems - it is specified in (2). The representation of c/) acts on the inner de- grees of freedom of the system. Particular physical sys- tems correspond to different sets of these inner vari- ables, particular interactions correspond to different rep- resentations of the dynamical algebra on this set.

The facorization of the kinematical and the dynamical properties of the system is analogous to the decomposi- tion of nonrelativistic Galilei-invariant wave functions into a factor that describes the motion of the center of mass and an internal wave function that depends on the specific dynamical properties of the system. For the Galilei group the kinematical algebra is generated by the momentum P and by the Galilei boosts K and the nonrelativistic dynamical algebra contains the Hamil- tonian of inner motion (in place of the mass operator that appears in the relativistic algebra) and the non- relativistic spin operators given by ~ = J + m - t K X P (the total mass m of the nonrelativistic system is a c- number). The commutation rules of the relativistic and nonrelativistic dynamical algebras are the same: The re- quirement of Lorentz or Galilei invariance of a quan- tum mechanical system is equivalent to the condition that the corresponding dynamical algebras constitute a unitary representation of U(2). This shows that Lorentz invariance is not a stronger condition than Galilei invariance: the irreducible unitary representa- tions of U(2) are characterized by a spin quantum num- ber s with 9 2 = s(s + 1) and a mass eigenvalue M. A gen- eral unitary representation of c-/) involves the direct sum of such irreducible representations. Every such representation gives rise to a representation of the Poincar6 group, i.e. describes a relativistically invariant system.

There is an important difference between the rela- tivistic and nonrelativistic cases in the relation of the dynamical algebra to the stability group of the initial surface. In the relativistic case only 93 may be expres- sed in terms of the stability group generators (see (4)), i.e. only 93 is independent of the interaction. In the nonrelativistic dynamical algebra all three generators

are kinematical objects given by the same expression for free and for interacting particles. We will see below that the requirement of covariance that goes beyond pure Lorentz invariance implies an angular condition for the relativistic spin operators. This condition is the

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substitute in the relativistic case for the explicit ex- pression of the nonrelativistic spin operators. Before we discuss the notion of covariance we briefly illus- trate the utility of the factorization of the kinematical variables in connection with the well known no-go theorem of O'Raifeartaigh [3]. It is possible without any difficulty to embed the dynamical algebra q) in the algebra g of a larger group such that the higher symmetry connects eigenstates of the mass operator with different eigenvalues. To obtain a representation of the Poincar4 group it suffices to let cK commute with g . This construction leads to a Lorentz invariant theory with nontrivial mass spectrum without of course contradicting the theorem: the generators of g and those of the Poincar4 group do not close to a finite dimensional Lie algebra. (The commutators of g with P_ give rise to operators of the form ( p + ) - l g . Com- muting two such quantities one gets elements propor- tional to (p÷)-2 etc.). The point we wish to emphasize here is that to have a nontrivial Lorentz invariant higher symmetry it is not necessary to embed the Poincar4 group, it suffices to embed the dynamical algebra - on a null plane the kinematics takes care of itself.

Poincar~ invariance does not imply that there is a covariant description of the system. In particular, it is not necessary for relativistic invariance that the wave functions that describe the states of a composite sys- tem transform in a covariant fashion. In local field theory, covariance is built in. Consider e.g. a theory involving the local scalar fields A(x), B(x). If this theory contains a stable one particle state Ip, n) with the quan- tum numbers of AB we may define a Bethe-Salpeter wave function by

(0lA(x)B(v)lp, rO = e-(i/2)p(x+Y)~on(x - y ) . (5)

(Since we are interested in these wave functions only for points x and y contained in the same null plane we need not distinguish between the product and the time-ordered product of the fields.) If the state IP, n) describes a particle of spin s the wave function is of the form

~On (Z ) = eta * ...t, sz "I ... z us ~b(z 2, pz ) (6)

where eul ~us is the standard polarization tensor. Up to a kinematical factor the 2 s + 1 helicity states are described by the same covariant wave function ¢(z 2,pz).

We claim that the wave functions of the system have this property if and only if the generators of the dy- namical algebra satisfy the angular condition

z lM~l + z 2 M ~ 2 + Z L ~ 3 = 0 . (7)

In the following we briefly sketch a proof of this state- ment. The reader is referred to ref. [2] for a more de- tailed discussion. Invariance under the kinematical Lorentz transformations generated by ~ implies that on the null plane z 0 = z 3 the wave function ¢n(Z) de- pends on the two vectors z and p only through z 1, z 2 and z L = pl zl +p2 z2 + p+z +. Furthermore, invariance with respect to rotations around the third axis which also belong to the stability group requires ~7 3 = - i (z I ~7 2 - - Z 2 ~71 ) where ~71 , ~2 are partial derivatives with res- pect to z 1 , z 2 at f'Lxed z L. The nontrivial part of co- variance concerns those Lorentz transformations that do not leave the null plane invariant. In the rest frame of the particle covariance amounts to the requirement

~,, (Rz) ~,, (z) (s) - -- D,,,,, ( R ) (8)

where R is a rotation and D (s) is the spin s representa- tion of the rotation group. (Lorentz transformations that do not leave the momentum invariant are trivial - the kinematical algebra takes care of these). For the null plane wave function (8) is a restriction only if R has the property that both z and Re are on the null plane. Rotations that fail to have this property merely determine the off-plane values of the wave function. Any rotation with this property may be decomposed into a rotation around the vector (z 1 , z 2, z 3) followed by a rotation around the third axis. Since the latter be- longs to the stability group, the content of the co- variance requirement amounts in the rest frame to the condition that the wave function is invariant with res- pect to the rotations generated by z 1J 1 + z2J2 + z3J 3. It is easy to extend this information to arbitrary mo- menta: the operator Z 1 ~1 +ZI 92 +ZL~3 M-1 com- mutes with ~ and reduces to z lJ 1 + z~J2 + z3J3 in the rest frame. Covariance is therefore equivalent to the condition that the operator specified in (7) an- nihilates all wave functions of the system - this proves our claim.

If the dynamical algebra satisfies the commutation rules of U(2) and obeys the angular condition (7) the null plane wave functions of the system transform co-

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variantly. Note that both the angular condition and the explicit expressions for the Poincar6 generators in terms of q( and Q involve the products M 91, M 92 and M 2 rather than the quantities 9 1 , 9 2 and M - these products are the proper Hamiltonians of null plane quan- tum mechanics. The angular condition (7) is valid for a system of two spinless constituents (in field theory this case corresponds to Bethe-Salpeter wave functions in- volving two scalar fields). For constituents endowed with spin the wave functions have several components - the covariant transformation law mixes these compo- nents. The null plane wave functions are identified with the so-called good components of the covariant wave function. If one eliminates the bad components in the covariant transformation law one arrives at an angular condition involving only the null plane wave functions. In the case of two spin ~ constituents this condition becomes a cubic relation to be satisfied by the dynamical algebra (see ref. [2] for more details). Systems with more than two constituents such as baryons require several internal angular momenta - the corresponding angular conditions and spin coupling rules have yet to be worked out.

One class of solutions to the angular condition is ob- tained in a standard fashion from the perturbation ser- ies o f a local field theory. One writes the equation of motion for the Bethe-Salpeter wave functions in terms of a quasi-potential which, unlike the BS equation, in- volves only the wave functions on a given null plane. The perturbation series for the quasi-potential defines the corresponding mass operator M 2. The same frame- work may also be used to extract the Hamiltonians M 91 and M92. Covariance of the underlying framework guarantees that these operators obey the proper com- mutation rules and satisfy the angular condition. 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.

An entirely different class o f solutions is obtained if one demands that the Hamiltonians M 2, M g l and

M92 are local differential operators in the transverse directions of the null plane. (Locality in the transverse directions intuitively corresponds to causal propagation properties: the operator P_ generates an infinitesimal displacement of the null plane. For a causal propagation one expects that the value of the wave function at a given point A on the shifted plane depends only on the properties o f the wave function on the initial plane at those points that are in the backward cone of A. The intersection of this cone with the initial plane consists of an infinitesimal neighbourhood of the backward null

'ray through A that is contained in the null plane - there is only infinitesimal propagation in the transverse direc- tion.) We have argued in ref. [2] that this class of Hamil- tonians is equivalent to manifestly covariant wave equa- tions. In particular we have shown that a manifestly co- variant relativistic oscillator model [4] is characterized by Hamiltonians that are second order differential opera- tors in the transverse direction and obey the angular con. dition. The eigenstates of this model constitute a com- plete set o f null plane wave functions that satisfy the spectrum condition in the longitudinal momentum. The wave functions are entire functions of the trans- verse momentum - the absence of singularities in the transverse momentum reflects confinement: for a Bethe- Salpeter wave function of the type (5) the spectral rep- resentation shows that intermediate states with the quantum numbers of A or B produce singularities in the transverse momentum whose position is determined by the mass of these nonconfined intermediate states.

References

[1] P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392. [2] H. Leutwyler and J. Stern, Relativistic dynamics on a null

plane, preprint IPNO/TH 77-11. This paper contains fur- ther references to the literature on the subject.

[3] L. O'Raifeartaigh, Phys. Rev. 139 (1965) B1052. [4 ] R.P. Feynman, M. Kislinger and F. Ravndal, Phys. Rev. D3

(1971) 2706.

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