Classical limit for Lie algebras

ANNALS OF PHYSICS 19, 187-224 ( 1990)

Classical Limit for Lie Algebras

AUREL BULGAC AND DIMITRI KUSNEZOV

National Superconducting Cyclotron Laboralorv and Deportment qf Physics and Astronomy,

Michigan State lJnic?ersiry, East Lansing, Michigan 48824-1321

Received May 25, 1989; revised October 5, 1989

We present a general method to construct the classical canonical coordinates for any Lie algebra. The symplectic 2-form, which defines the Poisson structure is constructed explicitly. In this way we are able to define the corresponding classical action for a Lie algebra, which

can be subsequently used in recovering the quantum limit, e.g., through a functional integral approach. We pay special attention to the topological structure of the classical phase space, which plays a crucial role in recovering the correct form of the classical equations of motion

and eventually in the requantization procedure. The classical phase space turns out to be “curved” and this induces the appearance of “gauge potentials.” The classical mechanics of such systems is analogous to the Dirac’s constrained dynamics. We detail these constructions

for the case of SU(2) and SU(3) Lie algebras. The main differences for other Lie algebras arise

from a larger number of degrees of freedom. ? 1990 Academic Press. Inc

1. INTRODUCTION

Lie algebras play a significant role in almost any aspect of the quantum description of the nature. The most common example is of course the algebra of spin operators. Other examples range from different types of quantum collective models for nuclei to gauge theories in elementary particles and condensed matter physics and so forth. In spite of such a deep penetration of group theory into quantum mechanics, it remains common opinion that the spin is a pure quantum object with no classical counterpart, and that spin cannot be recovered from a classical object through a more or less standard quantization procedure. Besides the fact that such an attitude is wrong and there is actually a very well defined classical limit for the spin, pursuing such a line of inquiry is not only interesting from a pure intellectual point of view, but can prove to be a very powerful technique in the study of quantum systems. Classical trajectories play an enormous role at almost any level of quantum description. The wave functions are concentrated along the classical trajectories. Often, knowing the classical trajectories and the relatively small quantum fluctuations around them provides not only a simple, but most often a very accurate picture. They also allow a very simple and intuitive classification of the wave functions; and the occurrence of families of different classical trajectories join- ing given initial and final configurations provides a very appealing understanding

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188 BULGAC AND KUSNEZOV

of different interference patterns. It is no wonder that after more then half a century of quantum mechanics classical trajectories and other pure classical concepts are used to conceptualize obvious quantum mechanical situations.

In constructing the classical limit for a Lie algebra we shall follow a route opposite to that used to construct quantum objects from classical objects. The fundamental concept of classical mechanics is the phase space, i.e., the canonical coordinates and momenta, and the Poisson brackets. For a system with N degrees of freedom one needs N pairs of canonical coordinates and momenta, which satisfy the standard Poisson brackets

{q&3 PII = d,, I> k, I= 1, . . . . N, (1.1)

{9k> 42 = {Pk, P/l = 0. (1.2)

For any two function on the phase space one can define the corresponding Poisson brackets as

(1.3)

This defines the symplectic structure of the phase space, or in other words indicates which variables play the role of coordinates and which play the role of momenta. The canonical transformations are those transformations (which are in general non- linear) among canonical variables which preserve the symplectic structure (i.e., both old and new canonical variables satisfy relations (1.1 )-( 1.2)). Given a Hamiltonian, the Hamilton equations of motions read

(1.4)

aPk aH -- at= aqk’

or for an arbitrary observable F(q,, . . . . qN, pl, . . . . pN)

(1.5)

The Hamiltonian equations of motion can be derived from the variation of the classical action

s=J[; ] pkcjk- H dt. &=I

In Feynman’s formulation of quantum mechanics the classical action S plays a

CLASSICAL LIMIT FOR LIE ALGEBRAS 189

central role. The functional integral representing Feynman’s propagator is (we shall use h = 1)

i N

K= 9qQpexp c ~kdk-H dt k=l I >

(1.8)

The traditional route from classical to quatum mechanics consists in defining the canonical variables as operators and the Poisson brackets as commutators and replacing the Kronecker symbols by i times the corresponding Kronecker symbols

[@k* tit] = j6k. /r k, I= 1, . ..) IV, (1.9)

II4k3 @,I = ca/r, $,I= 0. (1.10)

The quantum equations of motion are obtained in a similar way

aik i at= C$k, 81,

i +$= [bk, A],

or for an arbitrary operator F ̂

i g= [F, fi] (1.13)

There is, however, no simple equivalent for the Poisson bracket of two arbitrary functions. In principle the commutator of two arbitrary operators can be calculated only if the operators are known in terms of coordinate and momentum operators, for which the commutation relations are known. Closed formulas can also be obtained if one works in a certain representation of these operators in particular Hilbert spaces. Another practical limitation of the quantum mechanics is the relative poorness of the class of manageable canonical transformations in com- parison with the unlimited richness in the classical case. This fact along with many other known difficulties makes the quantum treatment very complicated and often a less accurate but straightforward classical analysis is preferred, which in many cases provides a fair description of the envisaged phenomena.

In dealing with systems described in terms of generators of a Lie algebra one is left with the impression that no classical analogue exists for the quantum description. The most common example is the spin 4 of fermions for which the commonly used operators form an SU(2) algebra

a 1 ^ cJj, Jk 1 = $klJI. (1.14)

The Einstein summation convention is implied throughout the paper. For an


arbitrary Lie algebra one has similar commutation relations among generators Cl? 21

[AT;., JPk] = icjk12?,, (1.15)

where cikr are the structure constants. It looks like there is no equivalent for coordinate and momentum operators among the generators and consequently a classical limit seems impossible to define, at least in the standard way.

We shall interpret the operators X, as noncanonical variables on the classical phase space and define the symplectic structure through the following Lie-Poisson brackets [ 31

(1.16)

where F, G are some arbitrary functions of X,. If F and G are chosen as the generators Xi, we clearly obtain a classica limit of Eq. (1.15)

{Xi, Xi} = CiikXk. (1.17)

In this way we achieve an obvious symmetry between the classical and quantum description, and the quantum commutation relations have a natural classical counterpart. Our task will consist in identifying the canonical variables as functions of the classical quantities X, and the construction of the classical action corresponding to a given Lie algebra. Once this is done the requantization procedure will be more or less straightforward, either at the level of Bohr-Sommerfeld rules or in the framework of Feynman’s functional integral approach. Even though this program seems to be relatively well defined, there are a number of subtleties and pitfalls, due mainly to the unusual topology of the classical phase space. This nontrivial topology induces gauge fields whose role is crucial in reaching a correct description at the classical level and subsequently in the requantization procedure.

In Section 2 we study the classical limit of SU(2), which proves to be the most simple example, though sufficiently complicated to illustrate the unusual role played by the topology of the phase space. In Section 3 we study the classical limit of the SU(3) Lie algebra, which contains the general features of an arbitrary Lie algebra, while remaining quite tractable due to the relatively small number of generators. In Section 4 we describe the general procedure for identifying the canonical variables and the classical action for any Lie algebra, which simply generalizes the discussion of SU(3) in the previous section. The main conclusions of the present paper are summarized in the Section 5. In Appendix A we derive the Euler relation for SU(3)(e”.“) as an illustration of the techniques developed in this paper.


2. CLASSICAL LIMIT OF THE W(2) LIE ALGEBRA

In order to make the notations simpler we shall denote the three generators of the N(2) algebra

J, = x, Jz = y, J, = -? (2.1)

and use also

J = r = (x, y, z)

J=Jm. v-2)

The SU(2) LieePoisson bracket is then

=VFxVG.J, (2.3)

where F, G are any two functions of x, y, and z. A remarkable property of this LieePoisson bracket is that for an arbitrary function H(x, y, z)

(H,J}=O, (2.4)

which is the obvious equivalent of the well known commutation relations

Consequently, both at the quantum and classical level the quadratic Casimir C2=J~+J;+J~=x2+y2+zz IS automatically conserved, irrespective of the particular form of the Hamiltonian, provided it is only a function of the generators.

As a simple example of using Lie-Poisson brackets we can easily obtain the well known Euler equations of motion for a rigid top [4]. The corresponding Hamiltonian is in such a case

(2.6)

where Fk stand for the principal momenta of inertia and Jk are the components of the angular momentum in the body-fixed frame. In terms of the angular momentum components the Euler equations are

(2.7)


or by introducing the angular velocity Jk = Zkwk one recovers the standard textbook form of these equations

(2.8)

Due to the special form of the SU(2) Lie-Poisson bracket, the classical trajectories of any Hamiltonian will always be confined to the 2-sphere SZ. Consequently, the space of all possible states will always be compact. It is obvious that in this case one can have at most one coordinate and one momentum only, which will satisfy the Lie-Poisson bracket

GA PI = 1. (2.9)

One possible choice of coordinate of momentum is [S-7]

p = z, q = i In x+“y = arctan Y, x - zy X

(2.10)

and therefore

x + iy = Jm exp( ) iq), z = p, (2.11)

or by using spherical coordinates (x, y, z) = v(sin 8 cos 4, sin 8 sin $, cos 0)

4=h p = r cos 8. (2.12)

Another choice is

corresponding to the parametrization

x=p r - 8~’ + q*),

Y=qJqiG7i

z=r-+(p*+q*).

(2.13)

(2.14)


Once we know one set of canonical variables we can construct an infinite set of such variables through standard canonical transformations. In terms of the first set of canonical variables the Lie-Poisson bracket becomes

Now it is obvious why r cannot have any dynamical evolution since there is no term in the Lie-Poisson bracket containing a partial derivative on r, as a consequence the corresponding Hamilton equation of motion for this variable will be exactly i = (r, H} = 0.

Even though the above form of the canonical variables can prove to be quite useful for different applications, they obviously are not universally applicable. For example, the coordinates (2.12) are completely undefined along the z-axis, while (2.13) are undefined along the negative z-axis.

Thus the realization in Eq. (2.12) can be used providing the trajectories never pass through the north or south pole of the S, sphere. However, if one is interested in quantizing the classical motion, one has to consider all imaginable trajectories, not only those which minimize the classical action, and in particular trajectories passing through the poles. One can easily understand that it is impossible to find a global set of canonical variables. The reason is that the state space for the classical SU(2) algebra is compact. In such a case it is always possible to cover the whole W(2) manifold with a finite set of atlases. In the case of S, one needs at least two different atlases, e.g., one which maps the upper hemisphere plus some portion south of the equator and a similar one for the lower hemisphere into some open set of the two-dimensional plane R”. One can define then a set of canonical variables for each such atlas and in the overlap region the two sets will be linked by a standard canonical transformation.

The canonical variables of Eq. (2.12) realize a mapping of the S2 sphere into a cylinder, as illustrated in Fig. 1. The momentum p takes values in the interval (-J, J) while the coordinate q in the interval (0, 27r). In the latter case periodic boundary conditions can be employed to identify the endpoints q(0) = q(2n) and define the circle S,. The north and south poles correspond to the upper and lower edges of the plane in Fig. lb and the cylinder in Fig. lc. Once a classical trajectory crosses the north or the south pole, there is no way one can predict the further dynamical evolution of the system, unless one introduces a new set of canonical variables, which are well defined there. The ambiguity occurring at the poles using canonical variables of Eq. (2.12) is obviously a manifestation of the topological structure of the phase space and not of the dynamics of the system. There will be no ambiguity whatsoever in the language of (x, JJ, z), but only if one desires to define a global set of canonical variables.

The natural question arises, should one go to the trouble and define canonical variables in such a case? The answer is affirmative, if one needs to define the classical action for a Lie algebra. Using the prescription outlined in the Introduction one


(Jx,Jy.J,) Q.P)

P

FIG. 1. The global canonical coordinates of Eq. (2.12) map (a) the sphere to (b) the rectangle on the plane @. Three types of closed trajectories are illustrated. In case (1) there is no ambiguity in the path.

In case (2) the ambiguity in the trajectory can be removed by imposing periodic boundary conditions

for the phase 4, which maps the plane to (c) the cylinder. In case (3) there is no equivalent boundary condition since the boundary p= &-J corresponds to a single point. This problem is pathological to

global definitions of the canonical coordinates.

can define the classical limit for the Hamiltonian, the Poisson bracket, but there is no quantum analogue for the classical action

(2.16)

and one needs to know the canonical variables in order to construct it. The ambiguity of certain trajectories using the canonical coordinates of Eq. (2.12)

can be avoided if we go to a local definition of these coordinates. This type of construction will be shown in following sections to be a general property of all Lie algebras, where we will show that all the canonical structure can be found through


the commutation relations. Consider the SU(2) Lie-Poisson brackets of the generators in a neighborhood of the north pole (0, 0, J,)

(2.17)

{ I $,J; =-$=O(,,.

These Poison brackets assume an almost canonical form near the north pole, where J, - Jy - O(E), where E is arbitrarily small. At this point Jz = J- O(E*). Thus, near the local origin the first Poisson bracket is 1 + O(.s*), and the remaining brackets are O(E). At the point J, = Jy = 0, which we define as the local origin, the structure is canonical. since

(2.18)

So although J, and JI. are zero at the pole, they maintain a nonzero Poisson bracket. Locally on S2 we can rotate the coordinate system (J,, Jy, J,) to (T.X, T.,,, TZ), such that the local i = 2, direction is always normal to the sphere S,. In this way we define local canonical coordinates at every point of the manifold. Explicitly, define p and q as the projections of a vector R onto a local coordinate system on S2 with the local origin at 0. This is illustrated in Fig. 2. Thus

q = R t,,

p=R 44, (2.19)

where R is a point in the neighborhood of the local coordinate system G,, C, and er with the local origin at 0 = (r, 19,#). When R coincides with the local origin we obtain the canonical structure

(4, P)lO=R= 1. (2.20)

595/199/l-14


FIG. 2. Local phase space coordinates on S’. Rotations of the axes in the tangent plane TOS2 correspond to elementary canonical transformations. The norm of the normal vector 2 is related to the Casimir invariant by Z= 6.

In this coordinate system, p and q, defined in this way, are everywhere zero, but have nonzero differentials. The differentials are

dq=dR.h,=rdB,

dp = dg . 6, = sin 0 dd. (2.21)

The rotations of this local coordinate system in the tangent plane correspond to a simple linear canonical transformation. Although we have used spherical coordinates to illustrate this construction, this is not necessary. The reader might be still under the impression that the use of spherical coordinates was somehow crucial. Moreover, Eqs. (2.19) and (2.21) still seem to rely on such coordinates.

We shall now construct the symplectic 2-form for the SU(2) algebra in an obvious coordinate independent manner. If the canonical coordinate and momentum are known, then the symplectic 2-form is defined as

w=dp A dq. (2.22)

The clear advantage in working with such an object is that CO is a geometrical


object, invariant under any canonical transformations, i.e., it is coordinate independent [4, g-101. Using Eq. (2.21) it is easy to show that

o = -r sin 8 dtl A dd = - zdxr\dy+ydz/\dx+xdy/\dz

r’ (2.23)

From this last expression for the 2-form it is evident that o is a scalar, actually the surface element of the S, sphere divided by the radius of this sphere.

In practice the construction of “spherical” coordinates on the manifold of a general Lie algebra is not simple. We now consider an alternate construction of o that is better suited to the general case. Consider the Lie-Poisson brackets for W2)

(2.24)

where

3, = E&~.

In the local coordinate system the canonical structure of $!$ is

(2.25)

(2.26)

where 3, = J is an invariant quantity (i.e., invariant with respect to the Lie-Poisson bracket at that point). In every local coordinate system, the 2-form o has the simple form

(2.27)

where again JZ = & at this point and 59 / s2 = 39 1 s2 = 1. Since det $!? = 0, 3 is not invertible in the strict sense. However, if we consider the geometrical meaning of 3, we will see that we can define this inverse 9.

The SU(2) manifold is defined by the constraint of the quadratic Casimir operator C2 = J*. Hence the gradient of C,, VC,, is everywhere normal to the manifold S,. From the local structure of 9, (cf. Eq. (2.26)) it is clear that VC, is an eigenvector of 9 with zero eigenvalue. Hence $$ acting on an arbitrary vector projects and rotates that vector onto the surface on the manifold. Locally the two nonzero rows of 4 provide a basis for the tangent plane at the origin of each local coordinate system. In this way %q can be used to define basis vectors of the tangents plane at every point of the manifold. Further there will never be a coordinate ambiguity as that encountered in spherical coordinates at the north and south poles.


The trick to inverting ‘S’? then is to add the projection onto the eigenvector corresponding to the null eigenvalue. Since ‘S$ is antisymmetric and the projector is fully symmetric, the antisymmetric component of the inverted matrix can be taken. This defines the inverse RY. This procedure is easily followed for SU(2). In this case the characteristic equation reads

det(Y - 11) = 1(A2 + C,) = 0. (2.28)

The null eigenvalue corresponds to the eigenvector ViC2 = Ji. We’define the matrix ~4 by

M, = .sijk Jk + clJ, J,. (2.29)

This matrix has the inverse

+Jk---& JiJj . 2 1

Thus

qj=; [(A@),- (w’)~J = -$ ~~kJk. 2

The symplectic 2-form for SU(2) then assumes the simple form

w = dp A dq

= ej dJ, A dJ,

= -& EiikdkC2 dJ, A dJ,. 2

(2.30)

(2.31)

(2.32)

In local coordinates (2, y, 2) this reduces to the expected result

dp A dq= -; d.f ,-t dj? (2.33)

There is still another way of “inverting” the singular matrix 9 by defining

P(a)=a+c!J. (2.34)

It is a simple exercise to prove that

%==[P(~)-l-P=(C()-!]lr=O. (2.35)

These prescriptions of “inverting” the antisymmetric matrix 3 work because we add to an antisymmetric matrix a symmetric component, in order to make it nonsingular and afterwards we remove the symmetric part of the inverse matrix. It is


a simple exercise to prove the uniqueness of the final result of such a procedure. When one uses the first of the above tricks, one constructs a matrix M whose symmetric and antisymmetric parts act on orthogonal spaces and consequently the same applies for the inverse matrix.

It is worth while to recapitulate once more the steps we followed in the above construction. First we established that the SU(2) state space is the sphere S,, which is invariant under the action of the SU(2) group (i.e., under the transformations which leave invariant X42) structure). Since no point plays any particular role on this manifold, (i.e., the SU(2) group is transitive, which means that any point on the manifold can be reached from any other point through a group transformation) one can choose an arbitrary point, in our case the north pole in order to find the local canonical coordinate and momentum. Defining the canonical variables at one arbitrary point we actually achieve a global solution and we represent the symplectic 2-form in an obvious coordinate invariant form. This is especially important, since only this object is actually invariant under arbitrary canonical transformations. Whatever one desires to name a coordinate or momentum is completely arbitrary. What we managed to show is that, in the local tangent plane, any oriented system of coordinates is equally suited for the role of canonical variables. When following a given trajectory on the sphere, one can choose to call the parameter which defines the length of the trajectory the coordinate and the local orthogonal coordinate the momentum. The picture which emerges is similar to what one has in Einstein’s general theory of relativity, where the equations of motions are cast in a coordinate independent way. Moreover, one easily realizes that a unique global system of canonical variables is impossible to define and the only suitable solution relies on local definition of these coordinates.

Having now defined the symplectic 2-form one can construct the classical action. In order to do this we still have to establish one more result, namely, that in a certain sense the symplectic form is a closed form. In standard classical mechanics this statement is trivial, since

N

OJ = 1 4, A 4, (2.36) k=l

and obviously dwr0. In the case of a Lie algebra, we are dealing with a phase space that is embedded into a larger space. w will be closed in the phase space, but not in the full space. For SU(2), o is closed on SZ, but not in .G%?~. However, it is possible to construct a closed form for N(2) in ,cA!~. In the present case one can easily check that

dw=-fdxndyhdz, (2.37)

and consequently, it is clear that

,2, dw=O. r (2.38)


In such a case, we can write w = d&, which means that the symplectic 2-form can be represented in a simpler, though nonunique way as

o=rd&, (2.39)

where & is a l-form, or in other words, a vector potential. This vector potential is unique, up to a gauge transformation, i.e., d + & + Vf, where f is an arbitrary function. We will describe in the next section a general procedure for constructing such gauge potentials, based on the converse of the Poincare lemma [9]. One possible representation of this l-form is

d= r(rz~zz) (x~Y--Y~x)=cos~&, (2.40)

where one easily recognizes the vector potential for a Dirac monopole located at the origin and the characteristic singularity of this vector potential along the z-axis. Through appropriate gauge transformations, which in the present case correspond to canonical transformations, this singularity can be moved around. Depending on the gauge, or in other words, on the particular choice of canonical coordinates on SZ, one can have one or several singular points of the gauge potential d. Conse- quently, the symplectic 2-form is nothing else but the intensity of this gauge field, which is manifestly gauge, or in other words, canonically invariant.

The classical action for SU(2) is therefore

S=j [r&-Hdt], (2.41)

where the integral is along the trajectory. If the trajectory is closed, one can use the Stokes theorem and transform the loop integral into a surface integral,

(2.42)

where A,,, represents the two possible choices of the surface enclosed by the closed (periodic) trajectory, the sign depending upon the orientation of the surface. When used in a path integral formulation of the quantum theory, the two different choices of the surfaces must lead to results which differ by an integer multiple of 27r, since the classical action appears as a phase. This amounts to the following restriction upon the allowed spheres at the quantum level

ss co = 4rtr = 2m, (2.43)

s2

or in other words

J=dm=; (2.44)


which is the quantization of the total angular momentum, which can be only half-integer or integer, depending on the radius of the S, sphere. In a geometrical differential language this is equivalent to requiring that [w]/(27c) be an integer cohomology class [S, lo].

We observe that the quantization of the magnetic quantum number m can be obtained by considering periodic trajectories with fixed azimuthal angle 8 = BO. A particular Hamiltonian with such classical trajectories corresponds to a particle in a uniform magnetic field parallel to the z-axis. In this case

4 J.d=JjjA, daf= -JjjAz dazz (mod 271)

= 2nJ( 1 - cos 0,) = 2n( J- J;) = 27~2. (2.45)

This is the usual condition that the projection of J on the z-axis is quantized: J, = -J, . . . . J.

The classical equations of motion which follow from the variation of the classical action Eq. (2.42), assuming that r is constant, read

rBxi=VH, (2.46)

where

B=;, (2.47)

is the “magnetic field” of the monopole. Derived in this way these equations of motion imply in a nonexplicit way the conservation of the quadratic Casimir, i.e., of the radius of the sphere. These equations of motion can also be obtained using the symplectic 2-form as described in Ref. [4]. If one will allow the radius to change in time, obviously Eq. (2.46) becomes contradictory, since if i /Jr// B then B x i = 0 but VH # 0 as a rule. One cannot fail in seeing here a similar situation, which occurs in Dirac’s constrained dynamics [ 111. In the case of Dirac’s constrained dynamics one follows an almost opposite line of argument. One is given a classical action, of a form completely similar to Eq. (2.42), from which one derives the corresponding equations of motion of the type in Eq. (2.46), which imply constraints and one is faced with the problem of defining the corresponding Dirac-Poisson brackets, which are consistent with such dynamics. Our trick, described above for going from the Lie-Poisson brackets to the symplectic 2-form can be followed in the reverse direction to this purpose.

As stated in the introduction, the aim is to find a classical theory for Lie algebras completely equivalent to the quantum theory. It is fortunate to know the correct classical variables that pass to quantum variables (the generators) in the quantum limit. We find the classical action from which one can draw, in a completely parallel way to the quantum theory, all the corresponding relations. The fundamental object that is linked uniquely to the LieePoisson bracket is the symplectic f-form w.

202

S’x R

o= -xdy/\dz - ydzrrdx

x2+ y2

BULGACANDKUSNEZOV

0 q - xdyf,dz + ydmdx + zdmdy

x2+ y2+ z2

J2= x2+ y2 J2 = x2+ y2 + z2

(a) (b)

FIG. 3. The manifolds corresponding to (a) the symplectic 2-form of Eq. (2.48) is compared to that obtained in (b) our local formalism for SU(2). For each manifold we indicate the corresponding symmetry, 2-form and conserved angular momentum.

However, the symplectic 2-form o does not follow from the variation of the classical action. It is rather an independent object which has to be introduced separately. If one naively tries to construct the symplectic 2-form as CO = dp A dq using the canonical variables (2.12) one obtains the wrong result

o’=dp A dq= ydx A dz-xdy A dz

x*+y* . (2.48)

The Lie-Poisson bracket corresponding to this symplectic 2-form has the radius of a cylinder as a conserved quantity (see Fig. 3). An equally incorrect 2-form is obtained when using the canonical coordinates (2.13).

3. CLASSICAL LIMIT OF THE SU(3) LIE ALGEBRA

The “quantum” commutation relations for the SU(3) Lie algebra are

[jj, ],I = ifiklRlr (3.1)

and in complete analogy with the case of the SU(2) algebra we shall define classical Poisson brackets as

{Jjzi, &> =.hA (3.2)


where now the operators Izk should be interpreted as noncanonical variables on the classical phase space. In a similar way we can introduce the Lie-Poisson bracket between two arbitrary functions as

(3.3)

Having a definition of the Poisson brackets one can study the dynamical evolution of any observable, which is expressed as a function of /z’s. At this stage we do not know, however, which are the canonical variables which shall be needed if otie would like to construct the classical action and use it in a functional formulation of the quantum theory.

In analogy with the case of the SU(2) algebra, the Lie-Poisson brackets for the SU(3) algebra have the remarkable property that for an arbitrary function ff(;i, 9 ..., A,), there exists two functionally independent functions, the quadratic and cubic Casimir functions, denoted Cz and C3, whose quantum/cIassicai commutator vanishes identically

The explicit form of these two Casimir functions which we shall use is

(3.4)

For the sake of completeness the fully antisymmetric and fully symmetric tensors flkf, djkl are given in Appendix B.

The existence of the two Casimir functions C, and C, implies that irrespective of the functional form of the Hamiltonian H, a classical trajectory of H will always be confined to a six-dimensional submanifold of .!%*, which we shall denote by &Z6

A(j = (k,, . ..) d, ( C, = const, C, = const}. (3.6)

Since C2 = const defines a 7-sphere S,, A6 represents a six-dimensional submanifold of this sphere. Consequently, any classical SU(3) motion is described by three pairs of canonically conjugate coordinates and momenta. As in the case of the W(2) algebra, the classical phase space for the W(3) algebra is compact and we shall be faced with the same problem as before: it is impossible to find a set of global canonical variables.

Before proceeding any further in constructing the local canonical variables and


the classical action, we shall describe some useful geometrical properties of J$. It is obvious that on any point on J&$ the gradients of the two Casimir functions

PI G, ‘.., v,c,)=(~ )...) $)=(A I,..., /I,),

(V,C, >..., V*Cd=(~ ,..., 23, (3.7)

are orthogonal to the tangent plane TJ&. These two vectors are also eigenvectors of the antisymmetric 8 x 8 matrix

qk = fjkJl>

introduced in Eq. (3.3), with null eigenvalues

(3.8)

qJ, c2 = 0, cQVk cj = 0, (3.9)

in complete analogy with the situation we encountered in the previous section in the case of the SU(2) algebra.

It is straightforward to establish also that these two vectors have the following properties

IVG =&G,

d IVC,I =y c,, (3.10)

vc2 . vc3 = 3c,.

All the eight-dimensional scalar products from here on are understood as in an eight-dimensional Euclidean space with a metric Sjk. Consequently, the lengths of these two vectors and the angle they form are constant on &. These properties are a manifestation of the “isotropic” character of J&, under the action of the W(3) group. No particular point on this manifold plays a special role, in complete analogy with the properties of the S, sphere in the case of the W(2) algebra.

In order to make our following exposure more transparent, we shall introduce the analogue of the north pole for the submanifold Jll,,, defined as

A,=~*=A4=~5=&=A7=o. (3.11)

At the north pole, the expressions for the Casimir functions greatly simplify

c2=; @:+A;,,

Js (3.12)

cj = 12 n,(3n; - n;,.


It is useful to define two new quantities

P=Jz, C7=i p2

y = i arcsin &3 ( > ~312 3 C 3 =$ B’ sin(3y).

2 12

In terms of these variables we have the following simpler relations

IVC2I = B>

3 IVC,I = 4 B2,

a VC2 .VC, = 4 p’ sin 3y,

(3.13)

(3.14)

(3.15)

VC2 = (&, &) = P(cos y, sin y X

a a 2 VC, = 4 (21,&, 2: - 1:) = 4 fl (sin 2y, cos 2y),

and

/I3 = p cos y, 2, = b sin y. (3.16)

Since we evaluated these vectors at the north pole, we used a two-dimensional form for the vectors corresponding to the two-dimensional space normal to TA6. This includes the third and eighth components only, since the remaining components are identically zero at this point. It is evident from the definition that the quantities fi and y are SU(3) invariants. It is convenient to define two new orthonormal vectors normal to .,z?~ in the following way

ii, = vg,

8,, = fl vy,

a,.ii,=o,

ii;., = 1,

(3.17)

which can be simply reexpressed through the gradients of the two Casimir functions and the Casimir functions using Eqs. (3.13)-(3.14). These two vectors have the advantage of being orthonormal, while remaining orthogonal to J&.

At the north pole, (0, 0, A3, 0, 0, 0, 0, A*), the Lie-Poisson brackets for the


variables Jk become extremely simple, as it was the case for the algebra W(2). The only nonvanishing Lie-Poisson brackets are

(3.18)

One has to remember that at the north pole, all the variables entering in the left- hand side of these relations vanish identically, see Eq. (3.1 l), although the corie- sponding Lie-Poisson brackets are different from zero. It is also evident that the right-hand sides of these Lie-Poisson brackets at this point are SU(3) invariants. Actually one can show that they are, up to a factor, the eigenvalues of the antisymmetric matrix 9 defined in Eq. (3.8), which defines the Lie-Poisson bracket everywhere (see Eq. (3.3)). The characteristic equation for 3 is

det($ - K) = K2[K6 + $/?‘K” + &a”K’ + &p” COS2(3y)] = 0,

which has the following eight roots

(3.19)

o,o, *ipcosy, f$cos (3.20)

A few simple manipulations lead to a set of three pairs of canonical conjugate variables at the north pole. One possible choice is

14

q2 = /? cos(y - 43)’ P2 =&9 (3.21)

A,

” =/J cos(y -2x/3) p3=&

Obviously, these canonical variables are defined up to an arbitrary canonical transformation, i.e., up to an arbitrary symplectic transformation. As in the case of the SU(2) algebra, instead of constructing a particular set of canonical variables, we shall construct the symplectic 2-form, which is known to be invariant under an arbitrary canonical transformation. As a final remark, one should note that C, defines the angle y modulo 2rc/3 only, see Eqs. (3.13)-( 3.14). Using other allowed values for y amounts to a mere relabeling of the variables, e.g., if y --, y + 2x/3 then 4, ~2+&, -2, in the case of the north pole, etc. The relation of these coordinates


to the weight diagram is indicated in Fig. 4. The 27c/3 invariance then corresponds simply to the operations of the Weyl symmetry group on the diagram.

Since we have succeeded in identifying the canonical variables at the north pole, we can in a completely analogous way construct them at any point of the manifold &&. From our analysis, it is evident that at the north pole the &-axis and the &-axis point in the directions orthogonal to the tangent plane T.A$. The unit vectors along these two axes at the north pole are obviously

&3 = (0, 0, 1 3 0, 0,&O, 0 1,

&x = (0, 0, 0, 0, 0, 0, 0, 1). (3.22)

Since the north pole is actually an arbitrary point on J.& one can construct these two unit vectors at any point in an invariant way. These two vectors are also eigenvectors, corresponding to the null eigenvalues of the matrix 9. A unique prescription for their definition is obtained by requiring that these two unit orthonormal vectors be expressed in a global way through VC,, 3. The solution is then

* COG? 1 vc, _

e3 = p cos(3y)

4 sin y

J@ cos(3y) vc3y

& = _ sin(2y) vc + 4 cos y X

P COS(3Y) 2 J”;p cos(3y) vc3.

(3.23)

X,=p shv

:!L A,= pcosy

FIG. 4. Correspondence between the SU(3) weight diagram and the Lie-Poisson brackets of

Eq. (3.18) in a local coordinate system (at the north pole). Here, for example, (A,, A,} = A,/2 + a&/2 is plotted at the coordinate (4, d/2). The Weyl symmetry group operations (i.e., reflections across any weight vector axis) correspond to rotations by 2n/3, and to a relabeling of the coordinates and momenta. The existence of this invariance by rotations of 2a/3 is apparent in the delinition of the Casimir functions in Eqs. (3.13k(3.14). Here the rotation of y + y + 2n/3 corresponds to I,, ,?z + -i.,, . . 1.5, &$. i,-r -I,, E., and &,% i.,+i,.dz.


So defined, these two orthonormal vectors represent the “local” normal &-axis and &-axis at the “local north pole.” The other remaining six axes, which are in the tangent plane TAG, will determine the local canonical coordinates completely analoguous to the case of the north pole. For example, the directions corresponding to the eigenvectors of 9 with eigenvalues f ifl cos y, will define the first pair of canonical variables and so on.

From the above analysis it is clear that at the north pole, as well as at any other arbitrary point of &,, the symplectic 2-form has locally the structure

where

p1=Bcosy, p2 = D COS(Y - 71/3), p3 = fi COSCY -h/3), (3.25)

PI-P2+P3=“~ (3.26)

and the corresponding 1, are the “local” variables at the corresponding point, chosen along the eigenvectors of 9 as explained above. (Actually the antisymmetric matrix 9 must be brought to a block diagonal Jordan form, each block being a real antisymmetric 2 x 2 matrix. We hope that this sloppiness of language is not mis- leading the reader.)

As one can see from Eq. (3.24) the symplectic 2-form can be written as

o=*kdAj t, dl,, (3.27)

where the matrix 9 is the inverse of -9 in the space orthogonal to the directions corresponding to the null eigenvectors, i.e., in A6 only. In order to construct this “inverse” one can use the same kind of trick, which was described in the previous section for the case of SU(2) algebra. Namely, one constructs the nonsingular matrix

P=a+8, (3.28)

and then it is easy to show that

9 = +[P(a)-’ - P=(a)-‘I I,=(j; (3.29)

it is exactly the desired matrix. The second alternative is to construct the matrix

M=9+a, li$M$? +u, P,>(fi,l, (3.30)

with arbitrary nonvanishing constants CI,,~ and obtain that

~+[M-‘-(M-‘)q. (3.31)


In the present case, it is possible, however, to obtain a closed form for this matrix

(3.32)

where

N=g;-3C:=[P,P,P,]? (3.33)

Obviously, the symplectic 2-form is SU(3) invariant, being expressed through C,. 3 and the only fundamental SU(3) tensors S,,, &, and d,kl. All other possible SU(3) invariant tensors are simple combinations of these fundamental tensors.

The symplectic 2-form for SU(3) (cf. Eq. (3.32)) deserves further analysis, since its properties are not quite trivial. In standard classical mechanics the symplectic 2-form is closed, i.e., do = 0, which it is not presently the case, as one can check. One can show however, that the interior product of o with the vector fields normal to JZ& vanishes: I

&fl J w = 0, c, J w = 0. (3.34)

Here 6s and kv are the unit vectors along the B and y directions, respectively. Conse- quently, the interior product with any 8-vector normal to J& vanishes. This property can be also understood by recalling that VC2. 3 are null eigenvectors of 9 as well. We encountered a similar situation in the case of the SU(2) algebra. We shall show now how one can express the symplectic 2-form through some l-forms, in a manner completely analogous to the SU(2) case.

It is evident from Eq. (3.24) that at the north pole (which is actually an arbitrary point of J&), the symplectic 2-form has a natural decomposition into three terms. We show now that the symplectic 2-form can be uniquely represented (globally) as a sum of three 2-forms as follows

3

co= c pks,

k-1 Pk

where at the north pole

co2 = L d& A d’x,, Pr

1 03=-

P3

(3.35)

(3.36)


which is also valid at any other point in the corresponding Subsequently, we show that

2

w= c r,dc&,

“local” coordinates.

(3.37) k=l

where dk are l-forms, or in more familiar language, gauge potentials defined on the SU(3) classical phase space. We will see below that since the ri depend only on the coordinates normal to A6 (the phase space), w is closed in the phase space.

The decomposition of w into the sum of three 2-forms can be achieved with the introduction of projection operators. Let us define the matrices

M/l jk = fikl “all Mrjk=fk~ny,, (3.38)

using the orthonormal vectors defined in Eq. (3.17). One can then show that

P,= -Mf-M;, P:=P,, (3.39)

projects vectors onto A$. This can be proved in a variety of ways, the simplest proof, although not the most rigorous, is by simply constructing these matrices explicitly at the north pole and observing that at this point P, is diagonal and the diagonal matrix elements are (1, 1, 0, 1, 1, 1, 1, 0).

Let us define the quantities

8 VI = -2 cd--P2P31,

9P

v2= -+$ cP:+P1P31>

‘13= -$ b-P3P21.

(3.40)

One can then show that for any point on A$,

are projectors ( PT2 = P,, , Pi, = P,,, and P& = P,,) onto the space of eigenvectors of 99 corresponding to the eigenvalues f ip,, f ip,, k ip,, respectively. Equiv-


alently, these matrices project onto the (p,, ql), (p?, q2), and (p3, q3) subspaces. In such a case it is convenient to recast F into a more suggestive global form

8= -[p12P,,+p22P45+p32Ps,]re=~,++g+~~ (3.42)

and therefore one has globally

ok = (Fk),, dl+, A dA,, k = 1, 2, 3. (3.43)

One can check by direct computation that

(3.44)

and consequently, according to the converse of the Poincare lemma [9], there exist three l-forms such that

(3.45)

Thus the symplectic 2-form can be expressed as

3

w= c o,da& k=l

(3.46)

where

ol=Pl+Pz> @z=Pl-P3> “3=P2+P3. (3.47)

In Eq. (3.45) there is a linear dependence between the three l-forms, since

d[&, - ~4~ - ccZ3] = 0, (3.48)

and consequently

d, - d2 - d3 = const, (3.49)

595/199.‘1-15


so that one of the vector potentials can be eliminated. Hence w can be expressed as the sum of two gauge potentials as stated in Eq. (3.37).

We observe that the expressions of the external differentials, entering in Eq. (3.44), have the simple form at the north pole

+ d& A d/l, A d/l, + d& A d& A d&l. (3.50)

The quantity N was defined in Eq. (3.33). They will have exactly the same form at any other arbitrary point if expressed through the local coordinates. As a final remark, one can easily see that our formulas will acquire a more symmetric appearance if one will change the sign of p2 and define it as /I cos(y - 47r/3).

The converse of the Poincart lemma actually provides a procedure for constructing the l-forms from the 2-forms that are closed in 9’. Consider a 2-form @ that is closed in W8

d@-0,

Choose an arbitrary point 1, i, . . . . Los in W8, define the map (trajectory)

Ak(t)=&k+(Ak-AOk)f, k = 1, . . . . 8, 0 < t < 1,

and compute the O-forms (functions)

aj=2(izk-iok) I’ fbkj(d(t)) dt. 0

Then the l-form is simply

d=ajdAj, (3.55)

and the original 2-form can be expressed as

@=dor,r\ dAj=d&, (3.56)

(3.51)

(3.52)

(3.53)

(3.54)

(3.57)

This result holds at least in a neighborhood of the point ;1, , , . . . . A0 *. In general it is not possible to define such a l-form, or a gauge potential, globally. The gauge potential will have singularities, which are gauge dependent, such as the string singularities in the case of the gauge potential for the SU(2) algebra, see Eq. (2.40).


The position of these singularities depends on the way one chooses the point A,, which is arbitrary.

We shall not attempt to compute explicitly here the gauge potentials dk, even though the integrals in Eq. (3.54) seem to be almost elementary. The integrands depend on y and rational functions of 2s. Obviously, if needed they can be computed at least numerically in a straightforward manner.

At this stage the classical action can be evaluated easily, since

s=j [a/&;,-Hdl] (3.58)

and the equations of motion implied by it are obviously

(3.59)

The conservation of the total energy is a simple consequence of the antisymmetric character of the matrix 9, as one can easily see, since in the case of time independent Hamiltonians one has

(3.60)

The fact that it is impossible to define globally a set of canonical variables is to a certain extent disappointing. However, one can always define such variables at least over some relatively large portions of the phase space. Recently, Johnson introduced one such possible set in the following way [7].

1, + i& = exp(iq,) Jm,

A, + iA, = exp h+&qz+qd Jp,+p,A 1 (ql+ q%z - 4 1 ,I%?,&

1, + iA, = exp [’

i (-q1+v’h+qd J%=-ih 1 2 (-41 +,/‘b-qJ 1 Jip,+p,B,

(3.61)

214 BULGACANDKUSNEZOV

where

X 2P+Q ___-

3

X

Here P, Q are linked to the Casimir functions in the following way

Cz=;[P2+Q2+PQ],

C,=&(P-Q)@P+Q)(P+2Q).

(3.62)

(3.63)

This is nothing else but the classical version of the “quantum” representation for the SU(3) operators A, established by Biedenharn more than a quarter century ago [12], which is the exact equivalent of the similar relations for the SU(2) we mentioned in the previous section. As in the case N(2), this set of canonical coordinates is not defined globally, and several coordinate patches are required in order to cover the whole phase space. Here again it must be understood that the correct symplectic 2-form cannot be obtained from w’ = xi dpj A dqi. Instead, the expressions (3.27) and (3.32) must be used, in complete analogy to the case of SU(2) discussed in the previous section, At this point, we practically accomplished our task; we have defined completely the classical limit of the SU(3) algebra and all quantities needed to this end are calculated. We have shown how to find the canonical variables at any point, derived the expression of the classical action, derived from it the equation of motion, and know how to compute the Poisson brackets.

4. CLASSICAL LIMIT OF AN ARBITRARY LIE ALGEBRA

We will now describe the general procedure for recovering the classical limit of any Lie algebra. Let us consider for simplicity a semi-simple Lie algebra of rank r with K generators

[fj, &] = iCjJ!J, j, k, I= 1, . . . . K. (4.1)


The first step in our construction will consist in identifying the Cartan-Weyl basis [ 1, 21, in which the commutation relations have the form

[Fj,, A,] = 0, j, k = 1, . . . . r

[I?,, &J = ccjE’,,

[l?,, k,] = ctjfii, (4.2)

@‘a, &J = N,,k,,, if ct+/?#O,

and where N,,] are constants. According to our rules, the LieePoisson brackets will be

{ xj3 xk > = C,k,Xl,

{Hi, Hk} =O, j, k= 1, . . . . Y

{H,,E,) = -icr,E,,

{E,, Em,} = -ia,Hj,

(E,, E,) = - iN&,+p, if cc+P#O,

or for any two arbitrary functions

aF aG aF aG (F,G}=c,,-- X,=%jkr?xc?x. ax, ax, I k

It is convenient to introduce new variables

C-K + E-xl, P,=;[E&.].

(4.3)

(4.4)

As it is well known, a Lie algebra of rank r is characterized by the existence of r Casimir functions which are polynomials in the generators [ 1, 131. The order of the polynomials for a general Lie algebra is indicated in Table I. For unitary algebras, the r Casimir operators are of polynomial order 2, . . . . r + 1 and can be defined as

c /+I =dj,....,,,+jx,, “.*j,+,, I= 1, . . . . r (4.6)

In this way one introduces into discussion the corresponding fully symmetric tensors dj,, ,.,.,, + , of ranks I = 2, . . . . r + 1, invariant under the action generated by the corresponding unitary group. We tacitly implied that the metric in gK is S,,.

As in the case of SU(2) and SU(3) algebras, one has for an arbitrary Hamilto- nian H

{C,, H} =o, (4.7) 9j*V,C,E0. (4.8)


TABLE I

Number of Canonical Coordinates and Invariants for the Classical Lie Algebras

Lie Number of Polynomial order Number of algebra casimirs of casimirs pairs (4. P)

SW) n-l C?, c,, . ..> c, n(n - 1)/2

SO(2n) n cz, C& . . . . CbZl c, n(n- 1) SO(2n + 1) n cz, cq, . . . . CZ” n2

SiQn) n c,, c,, . ..1 c,, n2

G2 2 cz> c, 6 F4 4 c,, c,, c,, Cl2 24 & 6 cz. C5, Cc, cs, c9. c,, 36 E, I c,, c,. c,, c,o, Cl23 CM. c,, 63 & 8 Cl, c*, Cl,, Cl43 CM> cm, c24, c3l 120

This implies as before, that any trajectory for any Hamiltonian H will be confined to a submanifold & of BK of dimension K-r, where K is the total number of generators. The gradients of the Casimir functions will be orthogonal to &? at any point and at the same time they are eigenvectors with null eigenvalues of Y. In complete analogy with the case of N(2) and W(3) algebras, one can define the analogue of the north pole as the point, where

E,=Ep,=P,=Q,=O, (4.9)

and observe that at this point

{Qa, Pp> = L,.

In this way one finds the local canonical variables at the north pole. Therefore, at the north pole, the symplectic 2-form will be

w=cdP,r\dQ, b:

=; $, dE, A dEp,. J J

(4.11)

At any other point, the symplectic 2-form will have exactly the same structure in local coordinates. Using the gradients of the Casimir functions, one can define globally a set of r orthonormal vectors, normal to the tangent submanifold T./if

nj.nk=Bjk, j, k = 1, . . . . r, (4.12)


and the projectors

P;= Inj><njl,

PjP, = 6,,Pj,

P, = i P,, j= 1

(4.13)

P,,=l-P,.

The projector P, defines the space locally orthogonal to A? and P,, projects onto the local tangent plane T&f’. Obviously, the system of orthonormal vectors nj, j= 1, . . . . r represents also the set of null eigenvectors of 9. The characteristic equation

det(9 - K) = 0 (4.14)

will have exactly r zero roots, corresponding to these null eigenvectors. The remaining K-r roots, {p,, j= 1, . . . . K-r}, will be functions of the Casimirs only and at the north pole will have the form +ict,Hj, which at this point will also be functions of the Casimirs only (compare with the cases of SU(2) and SU(3)). The eigenvectors, corresponding to each such eigenvalue, will define a two-dimensional space in A, corresponding to one pair of canonical variables. One can see that at the north pole 9 has the Jordan form. The 2 x 2 blocks have the off diagonal matrix elements equal to for,H, with all the other matrix elements vanishing identically. In this way one can define projectors P,, similar to P,,, P,,, P,, in the case of SU(3) (see Eq. (3.41)), and define

(K- r)/2 (K-r)/2 _ 1

s*l= c T*= c ~

1= 1 (ajHj)2 “” (4.15)

a=1

which will determine the symplectic 2-form globally as

j,k=l

(K-r)/2

= ,c, O1

(K-r)/2 K

= ,g, 1 (%)jkdxj A dX,. jk = I

(4.16)

The antisymmetric matrix 9 can obviously be computed by “inverting” 9 through the “tricks” we described earlier, see Eqs. (3.28)-(3.31). The fact that the antisymmetric matrices 9 and Y are inverse to one another, if projected onto the manifold A, is a general and well known result in classical mechanics. We only provided here a method of “inverting” $9 in the particular case, when this matrix is defined on a larger space and has null eigenvalues, corresponding to the gradients of the

218 BULGACANDKUSNEZOV

Casimirs. The Casimirs are conserved quantities, irrespective of the form of the Hamiltonian, and consequently they are not dynamic variables. The fact that these quantities are conserved is only a consequence of the symplectic structure of the manifold A, embedded in gK.

We did not check or prove that in the general case one can represent the symplectic 2-form through a set of l-forms, as we did for the cases of the SU(2) and SU(3) algebras. However, it is very likely that as before

1 t,d($O, El= fl,

J

(4.17)

(where the sum is over some limited set of indices only and very likely one can limit the sum to two terms only as it was the case for SU(3)) and consequently one can represent it in a simpler form

co= i a,d&. (4.18) /= 1

The number of independent gauge fields ZZ$ must come out to be equal to the rank of the Lie algebra, which will correspond to the number of independent Casimir functions, whose values have to be fixed by quantization conditions, similar to Eq. (2.43) in the case of SU(2) algebra. Once such a representation for the symplectic 2-form is explicitly established, one can write down the expression for the classical action as we did for the case of SU(3) algebra, see Eq. (3.58). If the trajectory is periodic, or closed, which is usually the case of interest in a path integral formulation of the quantum theory or if one only needs to apply the Bohr- Sommerfeld quantization rules, then the closed loop integral can be transformed into a surface integral, using the Stokes theorem, and the only thing one needs is the symplectic 2-form.

5. CONCLUSIONS

We have presented a general method for recovering the classical limit of a Lie algebra, which is completely analogous but opposite to what one would have done in order to construct the quantum mechanics starting from classical mechanics. There is a very natural parallel between the Poisson brackets for the canonical variables and the corresponding commutators in quantum mechanics, which we used as our starting point in our construction.

We were able to recover not only the canonical variables associated with a Lie algebra, but also all relevant quantities needed in a fully classical approach: the phase space and its symplectic structure, the classical action, Hamilton equations of motion. The most unexpected feature of the classical limit in our opinion was the fact that the classical phase space corresponding to a Lie algebra has a nontrivial topological structure. If the Lie algebra is compact so is the classical phase space associated with it. This fact leads to a series of unusual properties of the classical


counterpart of a Lie algebra. One cannot deline a global set of canonical variables, suited for any type of trajectory. The most one can do it is to define canonical variables over a limited-ven almost entire-phase space, but never in such a way as to exhaust all of it. This leads to the fact that the part of the classical action, corresponding to what one would call f P, dQk has the form of a gauge potential coupled in a nonminimal way, i.e.,

I P, dQk + s q(X) dx,, (5.1)

where a,(X) are functions of the phase space variables X, completely analogous in their properties to a gauge potential. These functions are neither defined globally nor uniquely, they have singularities, whose character and position depends on the particular gauge. In the case of W(2) they are exactly the components of the gauge field of a Dirac monopole. Another unexpected feature is the presence of integrals of motion, whose form and values do not depend on the form of the Hamiltonian, namely, the classical equivalent of the “quantum” Casimirs operators. Namely, due to the existence of these “topological” integrals of motion, the available phase space has the unusual features mentioned above.

The classical limit of Lie algebras has exactly the same kind of characteristics one would find in the case of the Dirac constrained dynamics [ll]. In this respect, the properties of the classical Lie algebras are completely similar to what one of us established recently for the case of collective motion of a many-fermion system, when treated in a time-dependent Hartree-Fock approximation, after integrating out of fermionic degrees of freedom [14]. Namely, the classical collective action has exactly the same structure

S= j [dk dX, - H(X) dt], (5.21

where X, are the collective variables describing the many-fermion system and S!(X) are gauge potentials on the phase space. To a certain extent this is not unexpected now, since the corresponding collective variables X, are nothing else but classical quantities corresponding to quantum generators of the shell model Lie algebra. In the case of pairing in a degenerate shell, the effective collective action has exactly the structure expected for the classical limit of SU(2). This proved to be an essential feature, needed in order to recover the correct quantum solution. In all standard collective models of nuclear motion studied up to the present time, the possibility that the collective space has a nontrivial topological structure has never been considered, which is now manifestly apparent through nontrivial gauge fields on the phase space. In light of our results, it is evident that in order to recover the correct quantum limit one has to explicitly consider the gauge structure of the classical action and the nontrivial topology of the collective manifold.

It is useful also to make another parallel with another class of nuclear models, intensively studied over the past decade, the so called algebraic models. In light of


the present results one can view them as already quantum versions of the classical actions one would derive from an adiabatic time-dependent Hartree-Fock approach. The correct classical action has the structure shown in Eq. (5.2), but we now know that the quantum equivalent of it is nothing else but a Lie algebra, which it is exactly the starting point of algebraic models of collective motion [15]. Of course, this is not a proof of such a statement, but rather an indication of the way one has to follow in order to derive such a algebraic model for the collective motion of nuclei. In view of our results, such a way of reasoning seems very promising.

As a final remark we would like to comment on the path integral formulation of a Lie algebra. Evidently, the main ingredient is the classical action, whose form we established here. When computing the path integral one must pay special care not to integrate over the nondynamic variables, i.e., over the Casimirs. These are conserved quantities, both at the classical and quantum level, and special care must be taken not to consider fluctuations of the Casimirs. However, in order to have a noncontradictory quantum theory, the values of these Casimirs must be fixed in such a way as to be sure that [w]/(27r) represents an integral cohomology class of the considered manifold &!. In the case of the N(2) algebra this amounts to considering only half-integer and integer spins only, see Eq. (2.43).

At this point we would like to acknowledge the fact, that during our work on this paper, we relied on the symbolic manipulator REDUCE [16] for checking many relations.

APPENDIX A

In this appendix we demonstrate the simplicity with which many useful results can be obtained in the classical limit of Lie algebras. As an example we compute the Euler relation (or group elements) for SU(3) [ 171. For SU(2) the results are well known

e ir0.P =cosr+i(a.i)sinr (A.1)

For SU(3) the exponent will contain the term A. x, where the A matrices are complex 3 x 3 matrices (the Gell-Mann matrices). The matrix 1. x has the general


We note that C, = Tr(;l . x)’ and C3 = Tr(J.. x)“. The characteristic equation is

det(J.x-x)= --K’+: C2h+i C3

= -ic3+- fi’f~+~ fi’sin(3y) 1 4 36

which has the three eigenvalues

ti,=-$flsin(?+:)

x2=---/Isin u-? ,:i 3 ( >

ii3 = --!- fl sin y. fi

(A.3)

(A.4)

For SU(3) the obvious way to proceed is in the local frame. In SU(2), a general vector v can be expressed as v = ui. Analogously, for SU(3), a general vector x can be written in the form

x=pcosy$,+/?siny&,

where e3 and G8 are given in Eq. (3.23). In this case

(A.51

e’“.x=e l/i C”S yi P, + $ Sl” .yr’r. o* (A.61

This is a very convenient expression since 1, = A. e3 and 1, = A. S8 correspond to the local forms of these matrices. Hence [I,, I,] = 0, and the exponential of the sum in Eq. (A.6) reduces to the product of two exponentials. In the local frame, these matrices are

(A.7)

Further, for each exponential, the matrix products of 1; and 1: are easily found. For (A. c3) we have

(21. it3)2” = 2 + -!- (22. &*) 3Js

(2R.&,)2”f1=(211.C,). (A.8 1


For (2 .C,) we observe that the recursion relation

* 2 21 ,. (2A.e,) =“-z (2A.e,)

can be rewritten as

( Q fi

(i.e,)+f)2= 1 (A.lO)

Hence a simple redefinition of the 1, exponent leads to a trivial summation of terms in the expansion. Completing these expansions we obtain

e ii~x=(~.~3){eiXI -ei’z}+l (;1.&8){eih-I+eih-2-2eiK3}

,/5

+ i { eihl + eix2 + ei~3 1. (A.ll)

This form is quite remarkable. The coefficients of the exponents in the I, and 1, terms are just the diagonal matrix elements of the corresponding matrix (see (A.7)), and the constant term is simply the sum of the eigenvalues. Further the eigenvalues ~~ can be permuted in the exponents, corresponding to Weyl reflections (y -+ y + 2x/3). It is now clear that this method can be easily extended to a general Lie algebra.

This formula is trivial to use. For any eight-dimensional vector x, the unit vectors h3 and & g, given in Eq. (3.23) and restated here,

w2Y) 4 sin y s3 = p cos(3y) vcz - &j2 cos(3y) vc3y

sin(2y) 4 cos y

68 = - p cos(3y) vcz+ J@ cos(3y) vc3,

simply require the evaluation of the gradients at that point.

APPENDIX B

(A.12)

In this appendix we list the form of the structure constants we have used for SU(3). The fully antisymmetric structure constants filk are

CLASSICAL LIMITFOR LIE ALGEBRAS

The fully symmetric tensor d, has the components

d, ,8 = d,,, = d,,, = i 3

d,ah = d,,, = 456 = 444 = dm = ;,

dzd7 = d,,, = d,,, = - ;,

1 dd4* = d,,, = dee8 = d,,, = - -

23

dRgR= -I. J5

223

U3.2)

The general form of the matrix C!+ = fqkAk defined in Eqs. (3.3) and (3.8) is

c 0 I, -%2 f%, 1% 6 $2, -j& 0

-

-i 0 I., $6 $i, I -2 214 44 0

I AZ -A, 0 $5 +I, -I lJ -7 96 0

?I= ii, 4m5 $2 (&&

an, $4 fA, 1* 2A2 $1 0 (J3/2)L,-fl, gJ3i2)l.,

$4 $5 -$& +, - fi., -(J3/2)&+9., 0 (J312F.6

x0 0 0 (Jmh 4&P, (J5/2h +/5/m, 0

The underscored matrix elements are the only terms that remain nonzero north pole.

,

,

at the

ACKNOWLEDGMENTS

Support for one of us (D.K.) was provided by the National S&m Foundation under Grant No. 87-14432.


REFERENCES

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10. N. WOODHOUSE, “Geometric Quantization,” Clarendon, Oxford, 1980. 11. L. FADDEEV AND R. JACKIW, Phys. Rev. Left. 60 (1988), 1692. 12. L. C. BIEDENHARN, Phys. Leti. 3 (1962), 69; 3 (1962), 254; .I. Math. Phys. 4 (1963), 436; G. E. BAIRD

AND L. C. BIEDENHARN, J. Math. Phys. 4 (1963), 1449. 13. M. UMEZAWA, Nucl. Phys. 48 (1963), 111; 53 (1964), 54; 57 (1964), 65. 14. A. BULGAC, Phys. Rev. C 40 (1989), 2840; Phys. Rev. C 41 (1990), in press. 15. F. IACHELLO AND A. ARIMA, Phys. Reu. Lefr. 35 (1975), 1069. 16. A. C. HEARN, “Reduce User’s Manual, Version 3.3,” Rand Publication CP78, 1987. 17. D. KUSNEZOV ANL) A. BULGAC, “Group Elements for SLI(N),” MSUCL-707, submitted for publi-

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Classical limit for Lie algebras