Functional integrals for spin

ANNALS OF PHYSICS 192, 104-118 (1989)

Functional Integrals for Spin*

K. JOHNSON

Center for Theoretical Physics, Laboralory for Nuclear Science,

and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received December 21, 1988

DEDICATED TO HERMAN FESHBACH IN HONOR OF HIS 70TH BIRTHDAY

A method based upon elementary quantum mechanics for constructing a path integral

representation for spin amphtudes is described. A path integral representation obtained earlier for the unitary rotation matrices is rederived. In this system, the appropriate set of classical canonical variables and their quantum counterparts are identitied. It is suggested how this method can be extended to more general systems by using the three-dimensional unitary

group as another example. 0 1989 Academic Press, Inc.

I. INTRoD~JcTI~N

The construction of the three-dimensional angular momentum matrices directly from the commutation rules is elementary and is presented in every book on quantum mechanics. Angular momentum is always carried by a dynamical system but since its properties are independent of the system, it is reasonable to develop them without reference to it. However, it also can be useful to study angular momentum in the context of a specific model. A most elegant way to obtain results about the group W(2) is in terms of the two-dimensional system whose coordinates transform under rotations as a two-component spinor [l]. However, there is an even simpler way to represent angular momentum. This is where the angular momentum is carried by a one-dimensional system described by just a single pair of canonical variables (which shall be called 4 and p). The situation where rotations in a two- dimensional space are related to a dynamical system whose quantum phase space is the surface of an infinite three-dimensional cylinder is a standard example. For some reason the extension from this to that where the quantum phase space is a spherical surface is an example which has eluded being discussed in books on quantum mechanics although it has appeared recently in an article [2] where the quantum theory is set up using functional integrals. It shall first be shown how to reproduce the functional integral using ordinary quantum mechanics. Next this example shall be discussed in a straightforward operator basis. Here everything is

* This work is supported in part by funds provided by the U.S. Department of Energy under Contract DE-AC0276ER03069.

104 0003-4916/89 $7.50 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.

FUNCTIONAL INTEGRALS FOR SPIN 105

well known but in different contexts. Since the quantum phase space of this system is compact, both the coordinate and the conjugate momenta become discretely quantized. Some remarks will be made concerning the “reasonableness” of the infinite range of the classical azimuthal variable in which the functional integration takes place. In this paper it shall be shown that all of the ingredients of the quantum theory for this problem have been available since nearly the beginning of quantum mechanics and indeed the suggestion that it could be used for rotations is implicit in at least one book [3] where treatment of the canonical variables was introduced. The variables of the phase space functional integration are closely related to these quantum operators. The properties of the quantum canonical variables were later elaborated further 141. However, the author has been unable to find a physics reference where they have been used to represent three- dimensional quantum angular momentum. He would not be too surprised if one turned up since the development given here is very simple. Finally, the application to a slightly more complex problem, namely a functional integral representation of the SU(3) unitary matrices, is discussed briefly in order to stress the generality of the methods.

It has been amusing to study this classic problem in quantum mechanics in a different way. I hope that Herman Feshbach will also find this treatment to be entertaining. It has been a pleasure to have had Herman as a colleague, friend, and person with whom it has always been fun to discuss physics. I look forward to this continuing for many years.

II. AN ORDINARY FUNCTIONAL INTEGRAL FOR SPIN

One may call a “spin” a dynamical system where the total angular momentum is not a dynamical variable but a parameter. The goal here will be to derive from the elementary quantum mechanics of the representations of the generators of the rotation group, a functional integral which will yield the matrices which represent a given element of the group,

(ml e’“J Im’), (2.1)

where J’ =j(j + l), and the states are the 2j+ 1 eigenvectors of J, [S]. Of course, these matrix elements are well-known wavefunctions. The aim is to obtain an alternative way of expressing them. Since for all the states in this system i is fixed, this system constitutes a “spin.”

To begin, a set of states based upon a continuous “azimuthal” variable 4 in the range ( -TC, JC) is defined as

(2.2 1

106 K. JOHNSON

The sum is over integers or half integers depending on whether j is an integer or half integer. Here the amplitudes are

The set of vectors (41 is infinitely over-complete and not orthogonal. However, it is still true that

(mIm’)=b,,.= s R 4(ml4>(4lm’>. --x

(2.4)

This equation means that the identity operator is expressed in terms of these vectors in the same way as it is using a complete basis. The overlap of the “4 states” is easily calculated,

(~,~,>=sin((j+t)(l-$‘)) 271 sin(+(d - 4’))

(2.5)

It follows that the discrete subset of vectors <dk( is complete and orthogonal for the set of 2j + 1 values of #k = 2xk/( 2j + 1 ), -j < k <j, integer spacing of k, all of which values lie within the interval (-X,X). They may be regarded as the eigenvectors of an Hermitian operator “canonically conjugate” to Jz. However, to discuss them now would lead in a tangential direction. That direction will be explored later.

A transformation of (2.1) to the 14) states would yield the amplitude to be evaluated as

(q&l eiw.’ I&>= 2 ($flm)(mlei”‘Jlm’)(m’I~,).

By using (2.4) one may obtain the original amplitude from this one. It shall now be shown that one may obtain an expression for (2.6) in terms of a functional integral of the traditional type and thereby determine a form for the “classical” action to be used. There is a standard method of obtaining a functional integral for a unitary operator which may be composed from an infinite sequence of infinitesimal unitary operators if they are expressed in terms of canonical variables. In this case the form of the transformation functions (2.3) suggests that the states 14) and lm) should play the role of the eigenstates of the canonical variables even though the states Id) are not the eigenstates of any Hermitian operator and are infinitely overcomplete. (They are a complete set of eigenfunctions for the Abelian subgroup of rotations about the z axis, when there is no bound on J=.) The Im) states are complete and are the eigenstates of J, which shall be called p. The standard method is to repeatedly switch back and forth between 4 and p states in forming the product. In the case at hand this will not lead immediately to a functional integral because of the discreteness of the p eigenvalues. With the standard method, one will obtain a


path sum and integral rather than a path integral. The finite sums must be transformed into integrals to make the resulting infinite sequence of products correspond to a functional integral. If the rotation operator is regarded as a “Hamiltonian” one would evaluate

A, = (q$l eirw J Id”> = (drl cifH I&l>. (2.7)

Here for convenience, a “time” t (which may be put equal to one at the end) is introduced and the “Hamiltonian,” H = -u . J, has been defined. Now the unitary operator is factored into small time steps At = t/N, and alternately the states 14) and Im) are used to evaluate the products,

e ““~J=limfi(I+iAtdw.J)(l+iA~(l-l~)w~J). (2.8)

Here ,? is a parameter which formally may take any value. It shall be taken as a number between zero and one. It is included so that a comparison between the derivation to be given here and earlier work may be made. It is easy to evaluate the Hamiltonian in the mixed q5, m basis using the standard matrix representation for J, and the transformation (2.2),

and

<mlJ, Id> (tnl16>

=eqj+;)‘-( -~~,1)2)“2=J((~,-m). (2.9)

One should note for future reference that the function rfm is always associated with .I+, whether one has the matrix element (mIJ+ Id) or (411, Im), and similarly forJP,ePi4alwaysappears.0fcourse, (m~J,~~)/(mI~>=(~IJ~Im>/(~Im>=m. One obtains

x exp i 2 Cmk(dk - dk l)+Afo.(J(tik, m,)A+J(d, Ir -m,)(l -A))] , 1

where in (2.10), 4, = qSr. This is the path sum integral which shall now be changed into a path integral. One may transform each intermediate sum over the spectrum

108 K. JOHNSON

of J, =p by using the following sequence of transformations. First, one may write the identity

i F(m)= F’ dPF(p)p(p-m).

Because of the limitation on the range of the p integration, the sum on the right may be extended to be over all positive and negative integers (or half integers). In this way all members of the set of functions (2.3) which make a complete set for the group of axial rotations are formally included. The subset members which are not needed are projected out by the bounds on the range of the p integration. To handle the fact that when j is a half integer the spectrum of p is expressed in terms of half integers, a variable A = 4 which is present only in the case of the half integer spin is introduced. That is, for integer spin, A = 0. The sums on the right may then be replaced by alternative infinite sums by using the Poisson inversion

(2.12)

where on the left the sum takes place on all integers (or half integers) and on the right over all integers. One thus obtains the identity

i

m= -j

F(m) = Jj dpF(p) 1 cPM(p + 4’.

-i M

If this transformation is applied to each of the m sums in (2.10), one finds

$ [Pk(~k-~k--)+AfO.(J(~k,Pk)~+J(~k--, -P,Nl - 2)) 1

+ 2nM,(p, + A) J . (2.14)

The result is an infinite set of path integrals over a compact phase space of a one- dimensional classical system with canonical coordinates qi,p. The phase space may be regarded as a sphere with azimuthal angle 4 + I( and polar angle 0 = cos - ‘(p/j).

However, it is easy to transform the above multiple sum of path sums into a single sum of paths over an extended, classical, phase space where the range of the azimuthal angle is ( - co, co).

To make this transformation, define a new set of summation labels,

M&kfi (2.15)


SO

where Mb = 0. If one puts

M,=M;-M;_,

4; = fpk + 2nA4;

then the expression above is transformed into

.(J(~;,P,),!+J(~;-,.-P,)(~-~))I I

(2.16)

(2.17)

(2.18)

where the 4’ integrations now range over the interval (- co, co ), Here only the last M’ sum is not included in an integral, put ML = M. In this form 4; = $r + 2rrM. To achieve this transformation it was essential that the Hamiltonian be a periodic function of the canonical variable 4. The last M sum may now be interpreted as a sum over “winding numbers” if one associates the C,Z~ path integral with the set of continuous curves which connect the azimuthal angle do to #r+ 2nM. A4 counts the number of times the curves wind about an axis through the poles of the sphere. In the case of half integer j, the even and odd windings differ by an overall sign. Thus, it is imagined that the sphere has the poles removed to make it topologically equivalent to a doughnut.

There is nothing special about the form of the Hamiltonian which has been taken as the rotation operator -w J. The same method could be applied to any function of the operators J,.

The presence of an axis through the poles of the sphere would seem to destroy the attempt to give a geometrical interpretation to the action as being phase space area enclosed by a curve on the surface of a sphere. On the other hand, classical trajectories on a sphere necessarily “wind.” Suppose one were to compute an amplitude with a trace boundary condition where 4, + 2nM = dr = 4, and one also includes an integration over the final azimuthal angle. One would then expect that the action element would be rotationally invariant. The action element which would appear above would be around a closed path including a count for windings about an axis through the poles. The integral around a closed path of the action,

A=$pdtj=$jcos(O)d$,

is proportional to the area on the spherical surface enclosed by the path as can be verified by defining a vector with components (A, = cos (0), A, = 0) and applying

110 K. JOHNSON

Stokes’ theorem to the line integral of A. This transformation fails at the north and south poles of the sphere where the component A, is undefined. Therefore the integral A is rotationally invariant only if the path does not include within it either the north or the south pole. If the north pole is enclosed there is an additional contribution of 2rcj as one easily sees by imagining a path which encircles the north pole and encloses no area so cos (0) = 1. Encirclement of the south pole gives an extra 2rcj. One can remove this singularity at the north pole by subtracting 2rcj, but then the extra contribution at the south pole becomes proportional to -4rcj. If this is a multiple of 27(, it makes no contribution to the amplitude. Thus the geometrical interpretation of the action as an area element on a spherical surface is okay since 2j is an integer. The punctures at the north and south poles do not violate rotational invariance or the interpretation of the action as an element of area on a sphere. In general, up to a canonical transformation, or a harmless multiple of 27c, the quantity

s P d4

is invariant under rotations. The formal continuum limit of the action which appears is

W= ‘(pdc$+wJdr), 5 0

where the angular momentum is

J,=P

and

(2.19)

J~=~fi~C~~j(j+l)-p(p+l)+(l-l)~j(j+l)-p(p~l)]. (2.20)

Note that this is a modified version of the classical angular momentum, which would have, instead of the above elaborate form, Jm between the brackets.*

The representation of (2.1) as a path integral, (2.18), has now been achieved. Of course, the meaningfulness of a formal interchange of the limit as N becomes infinite and a functional integration in (2.18) has been taken for granted. The next qualifications apply in general to path integration in phase space. To be sure whether these remarks are significant one must be cautious in interpreting this limit. For the definite integral (2.19) to exist, it is required that p(t) be continuous and that d+(t)/dt be continuous, whereas the discrete form is at least formally symmetric between p and 4. One can easily shift to a summation procedure where only +5k occurs in each term and the difference pk+ 1 -pk appears. If there is any question about using the continuum form (2.19) it is probably safest to retreat to the discrete form given in (2.18). It is not necessary to assume that the admissible p(t) curves which make (2.19) stationary are more than continuous; it can be proved that the

* See Note added in proo$


ones which make the action stationary are differentiable. Therefore, the continuum form is meaningful, at least in the classical limit. This formula is almost the same as that obtained by Nielsen and Rohrlich from the classical system. They obtained the form where A= 1 so only the first of the expressions for J, in (2.9) appears. One might expect that the more symmetrical version with i = 4 would be more natural since the symmetry between up and down would be present. It might also be that a form with L = 1 and p > 0 and ;1= 0 with p < 0 could result. Clearly a more careful treatment of the continuum limit is needed to determine whether any particular value of 1” is preferable. However, since the procedure here started directly from the ordinary quantum theory, it is certainly correct at least in the discrete form of (2.18) for any value of 1. Nielsen and Rohrlich obtained an ordered form and the need for puncturing the sphere by arguing how to reduce the classical function expression to the proper quantum form so that the rotational group structure would be present. Here we started with the standard quantum theory and have cast it into the form of a classical functional integral. It has also been stressed that from the point of view of classical continuous trajectories on a spherical surface, windings might be expected to be present because the mappings of the points on a sphere produced by continuous trajectories necessarily have singular points.

III. WHAT CLASSICAL SYSTEM CORRESPONDS TO A SPIN?

Consider for a moment the classical system. If there is a system with one degree of freedom with canonical coordinates 4 and p, and the angular momentum is defined to be

J, = cos(d) Jm, J, = sin(d) ,,/m, J;=p (3.1)

then it is easy to verify that the Poisson brackets obey

{J,, J,} =+Jk. (3.2)

This is the form for angular momentum in three dimensions. Conversely, one could have obtained (3.1) by regarding the Poisson bracket equations (3.2) as differential equations in (4,~) for the components of the angular momentum. The solution is (3.1), with J* as a constant of integration. Since the total angular momentum is not a function of the dynamical variables but is a constant, the system is not a top but a classical spin. This is the classical system for spin The classical phase has - J<p < J, - cc > 4 < co. The phase space for d is classically unbounded since a classical motion with 4 increasing (or decreasing) without bond is a solution of the equations of motion for Hamiltonians which are made of functions of J,.

112 K. JOHNSON

IV. WHAT ARE THE QUANTUM CANONICAL VARIABLES FOR A SPIN?

The quantum angular momentum variables for a fixed j may be regarded as the angular momentum of a dynamical system with one degree of freedom with “canonical variables” which are the Hermitian operators 4, whose eigenfunctions were alluded to above, and p = JZ. The Hermitian operator C$ is the one with eigenvalues bk = 2nk/(2j+ l), -j< k <j, integer spacing of k. The unitary transformation which connects the 4 and p representations is

(4.1)

which aside from normalization coincides with (2.3) when evaluated on the spectrum of eigenvalues. The variables 4 and p are canonical at least in the sense that the transformation function from representation with 4 diagonal to that with p diagonal has the canonical form. These are the variables introduced by Weyl in his book on quantum mechanics many years ago. They are not canonical in the sense that [c$, p] = i, as no bounded operators can be, although they share a subset of the transformation functions (2.3) with such variables. One can easily construct the quantum version of the above classical equations which determine the angular momentum as a function of the finite quantum operators 4 and p. The operator commutation rules take the form

PQ = Qpe2”‘/2j + 1, (4.2)

where

p = e2nid2i+ 1 3 Q = e’@. (4.3 1

The unitary operators P and Q play a more central role than the Hermitian variables p and 4. The angular momentum operators are in this basis,

J:=p

(4.4)

It is easy to verify that the operators of this set obey the angular momentum commutation rules. As in the classical case, the total angular momentum J2 =j( j + 1) is not a dynamical variable. The dynamical system with but one degree of freedom is a spin. In contrast to most other dynamical systems, the canonical variables, $ and p, do not transform linearly under rotations. They transform as the azimuthal angle d and j cos(8), where 0 is the polar angle, that is, they transform “non-linearly.” From the commutation rules above one learns what the operator “ordered” version of that transformation would be. In the state space of the canonical variables, the


operators Q and P ’ when acting on the eigenvectors P and Q, respectively, shift the vectors to be that which belongs to the next higher eigenvalue of C$ and p. However, when they act on the state with the largest eigenvalue, it is recycled to the state with the smallest value with a change of sign ifj is a half integer, that is,

(4.5 1

The phase space has a twist in the half integer case. These operators have been studied extensively but, as far as the author is aware, have not been used previously to construct angular momentum.

Now one might ask what would happen if one tried to compute the rotation matrix using these quantum canonical variables in forming the product (2.8). One would again obtain as above a path sum, an infinite sequence of finite sums, rather than a path integral. However, because the angular momentum operators are periodic functions of 4, just as above it would be simple to transform the finite sums over the quantum states of J, into finite p integrals together with infinite discrete sums on the quantum states of 4. If the dependence of J on p were also periodic, by allowing the 4 variable to become continuous by way of the Poisson inversion, one could then extend the range of the p variable. However, because the dependence of the angular momentum on p is not periodic, the result obtained in Section II is equivalent to the statement that all of the extended p sums which would have enforced the discreteness of the azimuthal coordinate interfere destructively, leaving only the p contributions in the central phase strip where -j <p <j. That this magic is true was proved in Section II through the use of the over-complete continuous azimuthal basis.

V. THE PATH INTEGRAL FOR A LARGER GROUP

To illustrate that there is nothing special about the group SU(2) except for its simplicity, a sketch is given of the method used above when it is applied to construct a path integral representation for the unitary matrices which give an irreducible representation for the larger group, SU(3). The extension to other continuous groups will then be obvious.

To obtain a functional integral for the unitary matrices of SU(3) one could proceed as in Section II. One would use the matrices of an arbitrary irreducible representation obtained by standard methods using the commutation rules. Next the transformations used in II to obtain a functional integral would be applied.

Alternatively, one may proceed by first constructing the classical system as was done for the rotation group in Section III by solving the Poisson bracket equations. Then the canonical operator version would be developed as in Section IV. Finally the functional integral can be obtained from this by standard methods. Here the first of these alternative steps will be carried out. In a later paper the remaining steps will be performed.

114 K. JOHNSON

Consider the Poisson brackets of the generators of this group for some dynamical system,

iAm 4) =fobc4 (5.1) For the convenience of the reader the non-vanishing structure constants for this group are given in Table I. In the quantum case the usual operators which are simultaneously diagonalized are A 3, A,, and the SU(2) Casimir operator A: + Ai + Ai. In the classical case, since the Poisson brackets vanish,

{A3,A*}={A3,A:+A:+A:)={A*,A:+A:+A:}=o

one may choose to be the canonical momenta,

A3 =Pl

A* =P2 (5.2)

Now call the corresponding canonical coordinates, di. One may then evaluate the Poisson brackets for the group generators using these as the canonical variables. The result is a set of non-linear first order differential equations for functions A,(q5,, qS2, c,b3,p1,p2,p3). Three of these functions are of course trivial, namely those for a= 3 and a = 8, and the SU(2) Casimir. The totality of generating functions will be the analog for SU(3) of the functions which appear in (3.1) for the group SU(2). With the aid of the structure constants one finds

A3 =P1, 4=p2

A, + i/l, = e’dl{p: -P:}‘/~

A4 + iA, = e(i/2)ch +v’%+hl VfGGiGA

+e(i/2)Ch+&+~h1 JliGKU?

(5.3)

TABLE I

abc f abc

123 1

147 112 156 -l/2

246 112 257 112 345 u-2 367 458

678


where

.;r=’ i(

P-9 P+%

2P, T+p,+? -y-

A( +p3+%

d-1

X (

2P + 4 p3 -pz ‘I2

-- 3 A1 3

B=l- 9-P P + 29

2P, i( -+pj+

3 J --p3+% 3 J-J

2P + 9 ‘;? x- ( 3

+p. -fi 7 A . 3

(5.4)

The “dynamical system” for the group SU(3) and the generators in terms of the dynamical variables have now been determined. These formulas are the analogs to those for the classical spin system used above for the group SU(2). The constants p and q are arbitrary constants where p, q > 0. These correspond in the quantum theory to the number of contravariant and covariant tensor indices in the representation. The phase space consists of the compact domain of p’s where the variables A, are real. This is a volume where a cross section with p, (where -p3 <p, <p3) is the interior of the region shown in Fig. 1.

Of course, quantum versions of these expressions also come up in the standard operator version of the group. One might have anticipated that the solution to these equations would be unique up to constants of integration which correspond to the total angular momentum which appears in (3.1). In the case of SU(3) the constants are equivalent to the two classical Casimir invariants of the group SU(3). Just as above for the spin system, the Casimir invariants are not dynamical variables but are fixed parameters. In terms of p and q the quadratic Casimir is (p” + q2 +pq)/3 and the cubic Casimir is another function of p and q. One could call the resulting dynamical system a classical SU(3) spin.

The closed path integral phase which will result from the extended quantized SU( 3) operators will be

it; PI&~+Pz%+P~~~ (5.5)

FIGUKE 1

116 K. JOHNSON

which is the phase space area enclosed by the path and will be an invariant under the classical canonical transformations generated by L!, which act like gauge transformations acting on the canonical variables, that is,

The integral (5.5) will be singular when the path includes a place where the transformation functions are singular when expressed in terms of these variables. The singularities are determined by the places where the generators above all vanish, which are special places on the boundary of the phase space. Of course, just as for the group SU(2), the “angle” variables, #i, are classically unbounded, even though the quantum phase space for these variables will be compact. One may also note the corner positions where all of the components are zero simultaneously. At each of the corners one of the azimuthal variables becomes undefined so there is a singularity in the coordinate system based upon these canonical variables. This occurs, for example, for the largest values of p1 and p2 when (for p > q)

P+4 PI =P3 =-

P-4 2 ’

p2=2&.

The other positions are indicated in Fig. 1. At the positions of the corners of the phase space, the momenta are fixed at their

extreme values, so a path which encloses no area but which winds about a corner will make the phase space integral finite. In the quantum case, the removal of the singularity will require the removal of the singular points from the phase space. This will produce axes and winding numbers which will be responsible for the discrete quantization of the canonical variables (1,, /is and the SU(2) Casimir, ,4: + nz + .4:. There are three axes and winding numbers since there are three independent “azimuthal” angles. One could further expect a quantization condition on the SU(3) Casimir invariants as a consequence of the requirement that it should not make any difference as to where the winding number axes are located. This will follow from the condition that the contribution of the singular path, that is a path which encircles one end of a winding axis, should produce a contribution which is an integer multiple of 271 and will be required for the SU(3) invariance of the quantized theory. Finally, one anticipates that the windings will be associated with phases connected with the center of the SU(3) group. This is the SU(3) counterpart to the phase ( - 1) needed in the SU(2) case for half integer j in the case where the winding number was odd. The proof of these assertions will not be given here but shall be discussed in more detail in a future paper.

VI. CONCLUSION

It has been shown how to express the unitary matrices which are the members of a definite representation of the rotation group by an ordinary functional integral.


This was accomplished starting with the standard matrices which describe the generators which belong to an arbitrary irreducible representation.

This derivation suggested how quantum three-dimensional angular momentum can be constructed from a one-dimensional system with a single pair of the “compact” canonical variables which were defined by Weyl for quantum mechanics many years ago but not used for this purpose It has been indicated how one might generalize the method to derive a functional integral representation for elements of the group SU(3) in an arbitrary representation in terms of a dynamical system in three dimensions. The classical system is represented by a set of canonical variables where the momenta have a classically bounded range. Ey these two examples, one may see how to determine the appropriate set of classical variables in general by employing the Poisson bracket formalism. It was shown how to determine the functional integral by employing a combination of an extended overcomplete set of states related to the canonical coordinates and the Poisson inversion. There are also interesting possible extensions to the case where discrete quantum operators are coupled to other variables where a path integration is possible. For example, the Dirac Hamiltonian comes to mind.

Unfortunately, the type of functional integral obtained is in phase space rather than in configuration space and therefore so far has been of little use for com- putational purposes. The magic needed for its validity requires delicate phase coherences which are impossible to achieve numerically. Of course, it is the generalization to more complex situations than that of a single spin which one is after. The next step, obtaining a functional integral of the configurational type,

might be the most important contribution.

Note added in proof: After this paper was written, I was made aware of closely related work by A. Alekseev, L. Faddeev. and S. Shatashvilli, “Quantization of the Symplectic Orbits of the Compact Lie Group by Means of the Functional Integral.” to be published in the Journal sf Gromerr~ und Phy.vics;

see also P. B. Weigman. “Multivalued Functionals and Geometrical Approach for Quantization Relativistic Particles and Strings.” MIT preprint, Condensed Matter Theory-Group.

The formula for the path integral obtained by Rohrlich and Nielsen [2] is slightly dillerent from that derived here. It may bc obtained by the small modilications of the above derivation which follow. The

limits in (2.11) are arbitrary except that they include all the states from -j to j in the momentum space interval and exclude those for 1 pl >j. When the states are at the boundary the interval should be understood to be slightly extended to include them. Now the range ofp will be enlarged by one unit over that in (2.18). This will ensure that all of and only the needed states are included. Set the upper limit in

(2. I I ) equal to j + E. 0 <I-: < I. and the lower limit equal to - (.j + c’), 0 < E’ < 1. Any values of e and c’ which fall within the indicated ranges can be assigned. To obtain the integrand of Rohrlich and Nielsen, first put I= I. Next carry out the summations over m which change the range of the azimuthal

integrations to ( ~ X. + q ). Consider the term which involves the integral over pl. where the integrand is

When one performs the pi integration, take the limits on pr to be -( j+ f) to j+ I,‘? (F. = E’ = $), in the

terms which contain 1 and J,. On the term which contains J + , take them to be -(I+ I-O) to j+O (s=O+.~‘=l-O),andforthetermwhichcontainsJ takethemtobe -(j+O)toj+l-O((i:=l-0.

118 K. JOHNSON

E’ = 0+ ). Finally, in the latter terms shift the pr integration variable up or down by f so that the limits for pk become - (j + f) to j + f in all terms. As a consequence of these transformations, the expression for

the angular momentum which appears in the integrand will be

In the continuum limit the integrand will have the classical form of the angular momentum with J* = (j + 4)‘. The path integral for p will take place over the range -(j + 4). j + f. The result now is the same

as that derived by Rohrlich and Nielsen. This expression may be preferable because in the continuum limit J, satisfies the classical Poisson bracket equations. One can see that there are many formally equivalent ways of obtaining an expression for the integrand which appears in the path integral. One

expects that this is a consequence of the over-completeness of the states labeled by the azimuthal angle. As argued above if any of these is best, it must be shown by the mathematically more comprehensive

treatment.

REFERENCES

1. H. WEYL, “Gruppentheorie und Quantenmechanik,” 1929 (Translation, “Theory of Groups and

Quantum Mechanics,” Dutton, New York, 1932); J. SCHWINGER, in “Quantum Theory of Angular Momentum” (L. C. Biedenharn and H. Van Dam, Eds.), Academic Press, New York, 1965.

2. D. ROHRLICH AND H. B. NIELSEN, Nucl. Phys. E 299 (1988), 471, and references therein.

3. H. WEYL, “Gruppentheorie und Quantenmechanik,” 1929 (Translation, “Theory of Groups and Quantum Mechanics,” Dutton, New York, 1932.

4. J. R. KLAUDER AND B. S. SKAGERSTAM. “Coherent States,” World Scientific, Singapore, 1985;

J. SCHWINGER, Proc. 46 (1960), 570. 5. T. KASHIWA, Preprint KYUSHU-88-HE-5 (July 1988). This paper was received after the completion

of this work and also contains a derivation of a path integral for the rotation group matrices starting

from ordinary quantum mechanics. I thank D. Rohrlich for sending me a copy.

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Functional integrals for spin