The relativistic spherical top

ANNALS

OF PHYSICS

87, 498-566 (1974)

The Relativistic

Spherical

Top

A. J. HANSON* Institute for Advanced Study, Princeton, New Jersey 08540 and Laboratory of Nuclear Studies, Cornell University, Ithaca, New York 14850

T. REGGE Institute

for

Advanced

Study,

Princeton,

New Jersey

08540

Received February 11, 1974

The classical theory of the free relativistic spherical top is first developed from a Lagrangian viewpoint. Our method allows the invariant mass to be an arbitrary function of the intrinsic spin. A canonical formalism is established following the approach suggested by Dirac for constrained Hamiltonian systems. There is a second arbitrary function in the theory, in addition to the usual one due to reparametrization invariance. The usual Newton-Wigner variables are supplemented by the Euler angles. The quantum theory of the free top is discussed. The classical theory is generalized to include charged tops with magnetic moments.

INTRODUCTION

In this paper, we attempt to treat the relativistic spherical top and its electromagnetic interactions from as general a viewpoint as possible. We begin with the free classical system and progress systematically from the Lagrangian and Hamiltonian descriptions to a possible relativistic wave equation for the free top. We then introduce electromagnetic interactions and give a Lagrangian derivation of the Bargmann-Michel-Telegdi equation [l] for a spinning particle with a magnetic moment. The Dirac formalism [2-51 for constrained Hamiltonian systemsis then applied to find the interacting Poisson brackets for a spherical top with no magnetic moment. Much of our presentation consistsof the restatement and * Research supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant 70-1866C, and in part by the National Science Foundation. 498 Copyright All rights

0 1974 by Academic Press, Inc. of reproduction in any form reserved.

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synthesis of well-known results into a unified conceptual framework. Some of our results appear to be new. Our description of a spherical top consists of a point on a world-line in Minkowski space and a rotating frame attached to that point [6, 71. More precisely, we begin by considering a full element of the PoincarC group and then restrict the degrees of freedom so that the remaining independent variables accurately describe a relativistic spherical top. The canonical variables in the resulting system include the spatial four-momentum, its conjugate coordinate variable, the spin momentum, and the angular coordinates conjugate to the spin momentum; these may be combined to give an extended Poincart group algebra, including the algebraic properties of the often-omitted angular coordinates. In our treatment, the classical and quantum mechanical mass spectra are given by a trajectory function of the angular momentum. The quantum-mechanical spectrum may be interpreted either as a family of related particles, or as a family of bound states for which the binding mechanism need not be specified. Our quantum-mechanical states all have the extra (2j + I)-fold degeneracy of the nonrelativistic spherical top. The first section of the paper is devoted to the development of a classical Lagrangian formalism for the relativistic free spherical top. In order to have the correct nonrelativistic limit, we impose constraints upon the Lagrangian in the form of a set of differential equations. We argue that the invariant mass of the top is a function of the spin determined by the form of the Lagrangian. However, we find one unusual solution of the differential equations the Lagrangian must satisfy in which the mass and spin are separately fixed. The next section outlines a general procedure which we adopt from Dirac for setting up consistent Poisson brackets in a constrained Hamiltonian system; these Dirac brackets are potentially the basis of a quantum theory. The relativistic spinless particle is treated in detail as an example. Section 3 deals with setting up the Dirac brackets and the Hamiltonian formalism for the free classical top. After fixing all possible constraints, we postulate a quantum theory of the relativistic spherical top which includes the angles as canonical variables. Section 4 introduces electromagnetic interactions using Lagrangian equations of motion. We are able to derive the Bargmann-Michel-Telegdi equations from the action principle by introducing the magnetic moment in an unusual nonlinear way. In Section 5, we use the fact that there is a certain freedom in the canonical system to iix the constraint equations of the interacting top in a convenient way. We then derive the Dirac brackets and the Hamiltonian system for the charged spherical top (with no magnetic moment) interacting with an electromagnetic field. The final section is devoted to a discussion of various questions and problems suggested by our treatment.

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Appendices give useful identities, summarize the different quantum-mechanical operators which have been used to describe spinning particles, and show how to find a classical Lagrangian corresponding to a given trajectory function relating the invariant mass and the spin.

1. LAGRANGIAN APPROACH TO THE FREE CLASSICAL RELATIVISTIC SPHERICAL

TOP

There are many treatments of the classical relativistic top in the literature. One of the earliest Lagrangian approaches seems to be that of Frenkel 181. Related approaches are given by Thomas [9], Mathisson [lo], Bhabha and Corben [ll], Weyssenhoff and Raabe [12], and Shanmugadhasan [13]. Recent treatments include those of Hughes [7], Itzykson and Voros [6], and Rafanelli [14]. Additional references may be found in Corben [15]. We have not seen a treatment exactly like the one we give here, although equivalent approaches must surely exist. In any event, we shall attempt here to clarify and generalize the Lagrangian approach to the relativistic spherical top. We will represent a spinning top as a point on a world-line to which a rotating frame has been attached [6, 71. Our first step is to denote by 7 an arbitrary parameter specifying the position of the particle along its world line. Then at each value of T, we give an element g = (P(T), Ally(~)) of th e P oincare group. The Lagrangian coordinates are taken tentatively to be the spacetime position X“(T) of the particle and the Lorentz matrix f@“(T) obeying (l.la)

(l.lb) The velocities appearing in the Lagrangian would then be (l-2)

Now we must restrict the way in which A@, and AU, appear in the Lagrangian so that only independent variables consistent with Eq. (1.1) appear in the Lagrangian. (Later, we will impose additional conditions so that only the three

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rotational degrees of freedom enter the dynamics.) Observing that the derivative of Eq. (l.la) is &YP + AA4~ = 0, we deduce that the quantity grr” = flAW/@ = -*w

(1.3)

has only six independent components. aUv has just the right properties to be the analog of the usual angular velocity ~9 of a rotating frame, which gives the velocity of a point relative to the origin as dxi/dt = xjwji We will take our free Lagrangian to be a jhction

(1.4) of ouuyand uw alone.

Next we must indicate how the variation of the Lagrangian is to be performed. We have found it convenient to express the variation of aULyin terms of a composite variable defined as 6&u = flf SAA’, = -8&u

(1.5)

so that &Y = ij&

-+

aiLA

6BAu- 8BuAaAy.

0.6)

Equations (1.5) and (1.6) will be seen to give a sound Lagrangian basis for the spin equations of motion and the Poisson brackets. A. PoincarBInvariant Lagrangian To cons’truct the most general possible Lagrangian, we wish to consider all the independent Poincare-invariant quantities which we can -form from our variables. It is therefore essential to define Poincart-invariance precisely. There are in fact two types of invariance under a group: right invariance and left invariance. We define an element (a, M) of the Poincare group to act as follows on a generic group element g = (x, A): Right transformation: Left transformation:

(x’, II’)=

(x, A) . (a, M) = (M-lx

+ a, AM),

(x’, A’) = (a, M) . (x, A) = (A-la

+ x, MA).

(1.7a) (1.7b)

A convenient way to represent this action is to write (a, M) as the 5 x 5 matrix (1.8a) with

(a,M)-l = (-yG-l y).

(1.8b)

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Since a dr = A-l dA and u dr = dx, the right-invariant & . 8-l = (uA-~, /la&l> and the left-invariant

form is &

form is g-l . dg = (u + ax, CJ)d7,

where our notation

(1.9a)

(1.9b)

for dg may be represented as i) = (u, Aa) d7.

dg = (zf

(1.10)

We now examine the quantities ukl = Au, Aod-r, u + ax, and a. Under right transformation by a group-element (a, M), we have AUR-AU

AuA-~--%

Aafl-l

U + ax --% M-‘(u

(1.11) + ax) + M-laMa

o -% M-laM. Under left transformation,

we find Au -h AaA-l

M(Au

- (la/l-k),

-h

MflablM-‘,

u + ax -5

u + ax,

(1.12)

u -5 a. Thus, as we could deduce directly from Eq. (1.9) Au flak1 I are right-invariant

(1.13a)

u + ax are left-invariant. uI

(1.13b)

Physically, right transformations rotate and translate the reference system describing the top, while left transformations alter the body-fixed frame of the top. The usual meaning of Poincart-invariance for physical systems is that the physics should not change under rotations and translations of the reference system. We now choose to allow our Lagrangian to depend only on left Lorentz scalars formed from the right-invariant quantities given in Eq. (1.13a) and not containing other

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explicit (1~~‘s. This is our definition of the relativistic spherical top. There are therefore four invariants which may be used in the Lagrangian:

a4 =: Det (T = (1/16)(~Y’a,*,)~ E (1/16)(~. u*)~,

where we define 0: = U/2)

l ,,d0

(1.15)

with co123= + 1. Our free Lagrangian is therefore some function Lo = Loh , a2 , a3 , ad) of the invisriants. B. Canonical Momenta The canonical momenta Lagrangian,

are now defined as the following

Pu = -8L,/&&

- 2dw,@L3

= -2UUL,

w = -aLo~ao,,= -44auvL, - 2(u%%4~ - z4”U%dA)L, - &7*yu

variations

of the

. u*> L,)

(1.16) (1.17)

where (1.18)

-&(a, , 4, , a3 , ad = @la4 Lo(al , 4 , a3 ,d.

(,P is actually a combination of coordinates and momenta: see Section 3.A.) The minus signs in the definitions are chosen to give the correct nonrelativistic limit. For example, in the spinless case, we might have a Lagrangian like L = -m(l

- v2)1/2 M $mv2 - m

so the momenta would be pi = mfjl(l

- 9)1/2 = aLlavi = -aLlzvi

Using SZP = Sku and the variation

.

Suuy given by Eq. (1.6), we find the following

Euler equa.tions as the coefficients of the arbitrary variations 6x, and MU,, , respectively: Pu = 0, (1.19) & + &SAU - S”AUAV= 0. (1.20a)

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We now use Eqs. (1.16) and (1.17) with the identities in Appendix Eq. (1.20a) may also be written

A to show that

9" + U'LP" - ll"PU = 0.

(1.20b)

Thus the quantities (1.21) are constants of the motion which will be identified as the canonical generators of translations and of Lorentz transformations, respectively. Taken as a whole, the set of quantities (1.21) are the canonical generators of the Poincare-group transformations. We note that Eq. (1.20a) can be integrated directly using the definition (1.3) of uuy to cast it in the form (d/dT)(Ll,,S~“fl,“) Taking the integration

= 0.

constant to be S,, , we have P(T)

= LF~ss,BA~”

(1.22)

so that P”(T) S,,(T) = PS,, This also follows directly from multiplying and identity (AS) imply that

= const.

(1.23)

Eq. (1.2Oa) by S,, . Since Eq. (1.2Oa)

$kU” + uuAs*A” - s*uAuA” = 0,

(1.24)

we also have s;“(T)

= fl”,S,*,lP”

(1.25)

Szv(7) = S@“S,*, = const,

(1.26)

where $$ is a constant. Thus F”(T) as we sequel of the For which tensor

could confirm by multiplying (1.24) by S,, and using identity (A.7). In the SuV will be related to the body-fixed components of the angular momentum top. later use, we exhibit the form of the useful four-vectors and invariants can be made from the momenta (1.16) and (1.17). One is the Pauli-Lubanski [ 161

W" = FU"P" = (l/2) E~"%Y"hPc = 40*u"u"(211L2 + Lda, + a&,1) + u""u,(u . a*)(-LIL, - 2L,L, + a,(L#

+ (l/2) a,L,L,).

(1.27)

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Another

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is the vector

:= 4n”‘si,(2L,L,

- L,[a,L,

+a *~vzI,(u . u*)(L&, We define the symbols

+ a&, + a:&, -t a,L,]) - 2L,L,).

(1.28)

M* and J? as

Ma = PUP, = 4u,( L,)” + 8u,L,L,

- 2a,a,( L.$ + 4a,u,( L,)”

(l/2) J2 = (l/4) SUVSU,= 4u,(L,)”

+ 8a&&,

- u,L,(u,L,

16L).

+ 4u,L, -

(1.29)

- 2u,u,(L.J2 (1.30)

Also of use are (l/4) ST;

= (u

cr*)(2Lz[2L,

- UJ,]

- L,[u,L,

+ u,L,

and W” := wu w, = -(l/2) = -M2J2

P~P,S~%,,

- PTsU,syAP,

+ V” VU

M? and W2 are the analogs of the two Casimir invariants group [ 171.

(1.32) of the abstract PoincarC

C. Construints As our theory stands, freedom. In order for our degrees of freedom in its of SUv. The classic work this, namely

the canonical momenta contain too many degrees of spinning particle to have only the three physical rotational rest frame, we must eliminate three of the six components of Pryce [lS] singles out three sensible ways of achieving (a) (b) (c)

V@ = Su”P, = 0, &youzzz 0 MSO@-

(1.33)

vu = 0:

In each case, only three of the four components of the equation are independent. Depending on the condition chosen, the particle position variables have different physical meanings. A summary of Pryce’s results concerning the classical and quantum-mechanical properties of these alternative position variables is given in Appendix B. Since, in the end, the set of variables corresponding to the choice of any one of the conditions (1.33) can be reexpressed in terms of any other set, it will suffice to treat only one.

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The significant convenience of having a Lorentz-covariant subsidiary condition motivates us to choose to restrict the Lagrangian so that condition (a) is satisfied [S, 10, 121: l/u = S”“P Y E 0 * (1.34) An immediate consequence of (1.34) and identity (A.7) is that S@“S,*,= 0.

(1.35)

How shall we now force the Lagrangian to obey the constraint (1.34)? Equation (1.28) tells us that a sufficient condition is to require the Lagrangian to satisfy simultaneously the two differential equations 2L,L,

= L,(a,L,

2L,L,

= L,L, .

(1.36)

+ %!L, + a,& + a,L,),

(1.37)

However, contraction of Eq. (1.28) with CY”U, and a*%, (1.37) are equivalent if

reveals that (1.36) and

a1a2a3 + 2as2 - 2a12a, = 0. We shall see shortly that this indeed is true for the free top when the Euler equations are satisfied. In the interacting case, this expression no longer vanishes and Eqs. (1.36) and (1.37) must hold separately to maintain VU = 0. We actually want to be able to give properties of the Lagrangian which are independent of the Euler equations, so that Eqs. (1.36) and (1.37) will generally be treated as independent conditions even in the free case. We therefore will describe the relativistic spherical top by means of a Poincart-invariant Lagrangian satisfying Eqs. (1.36) and (1.37) in order to guarantee that our relativistic system has the right nonrelativistic limit. There is another constraint to be imposed on the Lagrangian to ensure the arbitrary nature of the parameter T. In order to have the analog of the r-reparametrization-invariant spinless particle action S = -m we require that the Lagrangian (1 /a

j dr (uw,)1/2,

be homogeneous

of degree one in the velocities:

-Ma1 9a2 5 a3, 4) = alLI + a2L2 + 2a3L3 + 2a,L, .

Let us now analyze the constraints Vu = 0. Differentiating (1.34), we find PPy

on the velocities

+ S~VP” = 0.

which

follow

(1.38) from (1.39)

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It is easy to use the Euler equations (1.19) and (1.20) to show that Pw = MW/(PVU”)

= c--w,

(1.40)

L!+’ = 0.

Thus we have the following constraint among the velocities: u”(2a,L, - a,a,L, + 2a,a,L,) = 2@%,~U~(UJ1 + u,L,). Contracting

(1.41)

(1.41) with 21, , we find that if L, is not zero, then U$2U3 t 2u,2 - 2u,“u, == 0.

(1.42)

Substituting Eq. (1.42) back into (1.41) we find that if u,(u,L, + u,L,) is nonzero, the velocity constraint may be compactly expressed as 24’1= (Ul/U3) UL”“U,,$I~.

(1.43)

It is important to remember that Eqs. (1.40)-(1.43) are valid after the application of the Euler equations of motion, while Vu = 0 followed from the form of the Lagrangian alone. Bearing this in mind, we observe that the differential equations (1.36) (1.37) and (1.38) alone imply that w2/h&)2w3L2 - w&3 = -(WL,/L,(L,)3)[2u,L,

- w4-u - u,“L, - L&l,

+ &z,u,)]

= (u1u2u3 + 2u,2 - 2&Q,).

(1.44)

Applying the Euler equations, we find that Eq. (1.42) makes the square brackets in Eq. (1.44) vanish. Equation (1.42) may be applied again to show the equivalence of the two vanishing square brackets when the Euler equations hold. It is then a simple exercise to show the equivalence of (1.36) and (1.37) directly. Equation (1.44) and a number of other useful relations following from the differential equations alone are listed in Appendix A. The solution (1.40) for ZP in terms of P” is completely fixed once a scale, say u2 = 1, is chosen for ZP. There is a similar ambiguity in the expression of a~” in terms ,.P which will turn out to be important when we introduce interactions. We first observe that if u”y = AS”” + BS*uu, (1.45) then a, = C2P2 ,

u2 = (A2 - B2)(S . S),

a3 = +B2C2P2(S * S),

u4 = $A2B2(S. S)2

and Eqs. 1(1.42), (1.43), and (1.17) can be shown to hold identically.

(1.46)

There is a

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HANSON

AND

relation among A, B, C, P”, and P” Eqs. (1.29) and (1.46) we have

REGGE

which

may be found as follows.

l/C2 = (2L, + L,B2S,,S”E)2.

From (1.47)

But L, is of the form

so that L, and L, can be calculated in terms of derivatives of 2. When we substitute these expressions in Eq. (1.47) and replace the a,‘~ by the expressions (1.46), we find a relation of the form

A2 - B2 S -S p2

1 B2 S . S -__ 1 A2B2 (Se S)2 -_-

’ 2

c2

p2

)

4

c4

p4

)

p2 =

*

(1.48)

From this equation, we see that if PJ‘ and SUy are given, only two of the functions A, B, C are independent; we can thus fix all three by choosing, say, A and B/C. The scaling of A (and hence of B and C) corresponds to the reparametrization invariance of the system in T. A/C is then related to the angular velocity of the top. The arbitrariness in B/C will be clarified by later discussions using the Dirac formalism for constrained Hamiltonian systems. Note that the conventional description of the top corresponds to setting B = 0 and C = l/M.

nonrelativistic

D. Trajectories We now argue that the constraints of the previous subsection show that M2 is a function of J2 which can be regarded as the relativistic analog of the relation between the total energy and the angular momentum of a classical nonrelativistic top. We first observe that Eqs. (1.36)-(1.38) combined with expressions (1.29), (1.30) for M2, J2 imply that we may eliminate the derivatives Li to find consistency conditions among L, , M2, J2 and the ai . The resulting condition is a rather lengthy algebraic expression of fourth order in L,2. (See Appendix A.) There seems to be no evident way of simplifying this relation. As a4 ---f 0, however, one root of the equation for Lo2 is given by

L,2 = u,M2 + (l/2) a,P + 2MJ((1/2)

u,u, + u&2.

(1.49)

If we set M = A, J = B where A, B are constants, the resulting L, satisfies the differential equations (1.36) and (1.37). The resulting motion corresponds to a system with separately Jixed M and J, whereas in general one can only infer the existence of a trajectory function

M2 - f(J”) = 0.

(1.50)

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relating W to P. Details are given in Appendix C. The basic idea is that if we define

4 = a214,

rl = a31a12, e =

a,/a12,

then

and an explicit computation

yields the vanishing of the Jacobians:

a(w, ~2) a(f, 7)

a(M2, J”) act, 8)

aw2, ah

J”> = 0 0) *

From this result it follows that (at least locally) there exists a function f(J2) such that (1.50) is true. On the submanifold a3 = a4 = 0, Eqs. (1.29) and (1.30) tell us that we may choose M = 2‘59’ - 3 J = 2(2tp2

(I .51)

2’

where 2’ = aP([, 0, O)/a.$. We now compute the derivatives respect to .$, finding

of M and J with

(1.52) $ = ;f’(J’) = g (g)j’ /a3za4=o = ($)l” la3=ar=0. This equation tells us that there is a simple intuitive relation between the trajectory slope and the angular velocity which may be extended beyond a3 = a4 = 0 by the techniques of Appendix C. Finally, Eq. (1.51) implies Z(f)

= -M

+ ([/2)112 J.

(1.53)

Equation (1.53) can be viewed as a family of straight lines in the M-J plane parametrized by 5. The envelope of these lines (see Appendix C) is just the trajectory function (1.50). The next obvious task is to find explicit solutions of the differential equations (1.36) and. (1.37) corresponding to a given trajectory. The general method is to reconstruct L, from the trajectory using Eqs. (1.51)-(1.53) on the boundary submanifold a3 = a4 = 0 and to use (1.36) and (1.37) in a generalized eikonal procedure to continue L, for any a,, ah. The resulting calculations although straightforward are exceedingly complicated even for the simplest trajectories. A nontrivial example of an explicit solution is L.,2 = (1/2)(Aa,

- Ba, + [(Au1 - Ba2)2 - 8B(Aa, - 2Ba,)]‘12),

(1.54)

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for which the trajectory has the form (1.55)

BM2 - (l/2) AJ2 = AB.

E. Poincare’ Group Generators For completeness, we give a simple argument to deduce the generators of the Poincare group from the action principle. In terms of the canonical momenta (1.16) and (1.17), the variation of the action is 6 s” L(a, , a2 , a3 , a3 dT D z-

sab (P” Su, + (l/2) SLY &,,) d7

+ ues~v - Sub /\” dT 1)

P“(b) 6x,(b)

- (l/2) S”(b) M,,(b) + P@(a) 6x,(a) + (l/2) s”u(a) M,,(a). Now if we make an infinitesimal

(1.56)

translation of the end points, we have Sx”(a) = Ed, Sk+(a) = 0.

The coefficient of E, is P” = Translation

If we make an infinitesimal

(1.57)

Generator.

right Lorentz transformation

of the endpoints,

sxqu> = oJ~yX,g~‘l - xogul,), SAAv(a) = c@(L+, gus - flAB gy,),

so that WV(a)

= A,@(u) SAAv(a) = c@( gumgvs - gus g”J,

the coefficient of oU, is 4411”= xuPy - xvPu + Suv = Right Lorentz Generator.

(1.58)

If the action is invariant under translations and Lorentz transformations, P” and Mu” are conserved and generate the canonical transformations of the PoincarC group.

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2. F~AMILTONIAN

DESCRIPTION

OF A CONSTRAINED

CLASSICAL

SYSTEM

In order to find a consistent quantum-mechanical operator systemcorresponding to the classical Lagrangian description of the free spherical top given in Section 1, we must first understand how to set up the Poisson brackets and the Hamiltonian description for a constrained system.If we are fortunate, the commutation relations of the operators may be deduced directly from the Poisson brackets of the independent canonical variables. A. Dirac Brackets We now give a brief outline of Dirac’s procedure [2-51 for constructing a consistent set of Poisson brackets corresponding to a Lagrangian system with constraints i = I,..., 72. @ii% P) w 0, (2.1) The symbol “w” is read “weakly equals” and is used to indicate a constraint which vanishesformally but may have nonvanishing canonical Poissonbrackets. Dirac [2, 31 originally considered only constraints which followed directly from the form of the Lagrangian (“primary constraints”) or which followed from the time derivatives of the primary constraints (“secondary constraints”). The distinction. between primary and secondary constraints does not seem to be important. Dirac then divides the constraints into two subsets, the “first-class constraints” whose Poisson brackets with all other constraints vanish weakly, and the “second-classconstraints” which have nonvanishing Poisson brackets with at least one other constraint. (Remember that one must compute the canonical Poisson bracket first, and set the constraints weakly zero only after the computation.) The first-class constraints imply the existence of arbitrary functions in the equations of motion. Jn his work on gravitation [4], Dirac introduces another kind of constraint (“gauge constraints”), not implied by the Lagrangian or the Euler equations, and which have nonvanishing Poisson brackets with the first-class constraints. These gauge constraints have the effect of eliminating the arbitrary functions in the Hamiltonian corresponding to some or all of the first-class constraints (which have now actually become second-class).As we shall see, the gauge constraints may depend explicitly on the time. This will introduce changesin the Hamiltonian, which we can treat using Hamilton’s action principle. If all arbitrary functions are not eliminated from the Hamiltonian equations of motion by the obvious gauge constraints, there is yet another type of constraint which may be used to accomplish this elimination, the “invariant relation” [5]. A set of invariant relations. i = l,..., k

(2.2)

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HANSON

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is a set of relations among the canonical variables obeying (2.3) so that if z+$= 0 at t = 0, Eq. (2.2) holds for all times. (The symbol FZ is used in Eqs. (2.2) and (2.3) to indicate that all other constraints besides the +!Qhave already been set to zero.) lnvariant relations are part of the definition of a canonical system in that they restrict the phase space of admissible motions. Invariant relations can actually be treated as secondary constraints if we introduce a Lagrange multiplier hi corresponding to each invariant relation Z/Q z 0, i = I,..., k. We now express the momenta in terms of the velocities 4, so vu% PI - $uA 41, and define a new Lagrangian

The equations of motion are now

But then the new equations of motion are exactly the old ones, except now the constraints &. = 0 appear as secondary constraints following from the primary constraints PAi = aL/8Xi * 0. The net effect is to implement the I/Q as ordinary constraints while the hi disappear completely from the dynamics. Now we consider a second-class subset {&(q,p); 01= l,..., m> of the available constraints. By definition, each +ol has nonzero Poisson brackets with at least one other &, and the matrix of Poisson brackets G, = {da 3 +a1 is nonsingular.

(2.4)

We denote by C$ the matrix with the property

We have found it useful to define the prime variable

t?= ic- {E,A>GA3

P.6)

SPHERICAL

corresponding to each ordinary bracket can then be defined as

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canonical variable;

note that 5 w t. The Dirac

(2.7) and is seen from its definition in terms of 5’ and 7’ to satisfy the Jacobi identity. The point, of course, is that while 5’ = 5 and 17’ iv 7, the Dirac bracket is roof weakly eq.ual to the Poisson bracket. The Dirac bracket is constructed in just such a way that the brackets of each constraint with any canonical variable are strongly zero. To make the transition to quantum mechanics, aside from ordering problems, we take

if3 7>* - i[4, 71 = &%j - g>.

(2.8)

The operators C& can then be taken effectively zero on the Hilbert space of physical states because all their commutators should vanish. The question of whether or not there actually exist quantum mechanical operators realizing the algebra of the Dirac brackets apparently must be dealt with on a case-by-case basis. One useful property of the Dirac brackets is that they are iterative; if you have computed the brackets for a subset of the constraints and wish to add others, you may use Eq. (2.7) with the first set of Dirac brackets replacing the ordinary Poisson brackets. The final Dirac bracket is the same regardless of the order in which the constrained brackets were computed. (See the next subsection for a proof.) This technique can be used, for example, to display the covariant nature of a set of relativistic brackets before setting PO = (P” + M”)lp, x0 = t and destroying the manifest covariance of the system of brackets. In the (case when only constraints following from the Lagrangian and their derivatives are considered, Dirac singles out the first-class constraints xi for special treatment. Since the Poisson brackets of the xi with all other contraints are weakly zero, they may be multiplied by arbitrary functions 11~and added to the liirst-class Hamiltonian H,’ without changing the motion of any of the constraints. Thus Dirac takes as the total Hamiltonian H = H,’ + vixi

(2.9)

where H ;CYHo, but the Poisson brackets of H are not equal to those of Ho . The function of the vi’s is to specify a complete solution of the Euler equations for the motion of the system. We will give examples of the use of Eq. (2.9), but in the end we adopt a procedure in which all gauges and invariant relations are specified so that there are no first-class constraints, and hence no arbitrary functions in the equations of motion.

514

HANSON

B. Hamilton

Variational

AND

REGGE

Principle with Constraints

We now develop the ideas of the previous subsection from a slightly different viewpoint which clarifies the meanings of some of the concepts. We begin by considering the Hamilton variational principle in phase space, so the p’s and q’s are considered as independent variables [19]:

o=ss=s

f pi dqi - H dr i-1

(2.10) 4 $ (pi dqi - qi dpJ - H dT).

As usual, we find Hamilton’s

equations,

aH -=+!h,

!+!@L.

i!Pi

Suppose relations We then pendent

2

now that we know the entire set of constraints, gauges and invariant which restrict the phase space truly available for the particle motion. reexpress the 2n canonical variables pi , qi in terms of the 2m truly indevariables i = l,..., 2m zi 3

and the 2n - 2m constraints .q = +f--2m M 0,

i = 2m + l,..., 2n.

Thus we may expresspi and qi as functions of the z’s and of r, which we write Pi = PiG, T), 4i

=

4iCz,

(2.12)

T>-

Note that while pi and qi are by definition not explicit functions of 7, the zi may be explicitly T-dependent; the explicit T'S in Eq. (2.12) are necessary to compensate for any T-dependence of the zi . Now we consider the zi’s and T = zOas a set of independent variables, so that (2.13) where

c, = g i (P,z - qi+)ea i=l

(2.14)

SPHERICAL

We immediately

515

TOP

see that

is just the Lagrange bracket of the new set of variables z, , including constraints, with respect to the old set of canonical variables qi and pi . If we now define the Poisson bracket as

we find the following

properties,

zk

> z&k,

zj)

=

(2.17)

hj

az. Equation

(2.18) follows

(2.18)

from the fact that

Q,P

-gq).2

917

917

(2.19)

It is now easy to show that the equations of motion for zi are dzi/dT = (Zi ) H} + where (qi , pi , T) are treated as the independent right-hand side of the equation. The action principle now becomes

0=6S=6

j(?

azjpT variables

(2.20) when computing

Ck dz, + (C,, - H) d7 .

the

(2.21)

I;=1

However,

we require that the constraints sg-,,

= 6z, = 0,

& w 0 hold throughout k = 2m + l,..., 2~7.

the variation,

so

(2.22)

Thus the restricted action principle is 0 = 85’ = 8 j ( Fl C; dzi + (Co - If) dr).

(2.23)

516

HANSON

AND

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The variables zz, i = l,..., 2rn are independent variables whose Lagrange brackets are given by Eq. (2.15). We now demonstrate that the Poisson brackets of the independent zi are just the Dirac brackets. If we define the matrix Carl = t&7 7 ztl, we find that the Dirac brackets

a, b = 2m + l,..., 2n,

are (2.24)

Now we multiply by the Lagrange bracket, so (2.25) where k, j = l,..., 2m and the sum can be extended from 2m to 2n because, by Eq. (2.24), {zi , zj>* = 0 when i = 2m + l,..., 2n. Thus the Dirac brackets are the inverse of the restricted Lagrange brackets following from Eq. (2.23), with only 2m variables; by definition, the Dirac brackets must therefore be thePoisson brackets of the restricted system. In other words, a simple restriction in the number of variables appearing in the Lagrange brackets causes drastic changes in the inverse of the Lagrange bracket matrix; the canonical Poisson brackets are changed to Dirac brackets, which can be expressed in terms of the canonical Poisson brackets only by using Eq. (2.24). It is now trivial to prove the iterative property of the Dirac brackets mentioned earlier. Indeed, successive restrictions on the range of variables of the Lagrange brackets give the same final restricted Lagrange brackets, and hence the same inverse. Note also that if all gauge constraints and invariant relations are not imposed, the Lagrange bracket matrix will be singular unless enough variables are removed to give a nonsingular matrix and well-defined Poisson brackets. Dirac’s treatment without gauge constriants amounts to grouping the leftover variables with the Hamiltonian. Finally, we observe that by known theorems [20], we can find local canonical coordinates, & and (llc , with Dirac brackets & 2 ai)* = --6,j * In general, the 2m independent local canonical coordinates,

zi’s will be certain T-dependent functions

(2.26) of these

SPHERICAL

517

TOP

In terms of & and & , the action principle (2.23) can be written

0=6S=6 (P, d!i, -6;4%) -17 0.

Repeating the entire argument of Eqs. (2.13))(2.21),

where now everything

(2.27)

we also have

is expressed in terms of the new variables & and & . Thus

(2.29) E?-= c,, -

C, + H,

where CL = O,..., 2m and z,, = T. The integrands of Eqs. (2.27) and (2.28) differ at most by an exact differential which can be removed by a suitable canonical transformation on & and j, . A specific choice of the variables ijB and j& determines c0 and hence, from (2.29), g. This c.hoice fixes the explicit T-dependence of the zi appearing in the equation of motion (;!.20) with (q,p, H) + (q, 6, 8). Conversely, we may always perform a suitable T-dependent canonical transformation on the variables & and & which changes the Hamiltonian into any desired function. If the Dirac brackets of the zi’s do not depend explicitly on 7, then the best choice for the Hamiltonian is clearly the one which assigns no explicit T-dependence to the zi’s. C. Exanlple: Spinless Point Particle As a simple example of how the Dirac brackets can be used to find a quantum system, we treat the spinless point particle in the manifestly Poincare-invariant Lagrangian formalism, where the Lagrangian is (2.30) Here xu is a function of 7, an arbitrary parameter describing the displacement of the particle along its world line. The action is the integral over the particle world line, b s

=

-tn

sa

ds = -in

lb (dx” dsyJ1/2 = j”’ a’7 L(x, u) a a

(2.31)

518

HANSON

AND

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and is invariant under a reparametrization T -+ T’(T) of the integration If we define the canonical momentum corresponding to X“(T) as P”(T)

=

-

s

=

m

(g2

=

“1

dXW 7

parameter.

(2.32)

7

II

the variation

of the action is 6s = j-” dr 6x, $ n

With fixed end-points,

- Sx+z) P,(a) + kP(b)

P,(b).

the vanishing of 6s implies the Euler equation dPujdr = 0

(2.34a)

m(d”xu/ds2) = 0

(2.34b)

which can also be written

For infinitesimal trandations of the endpoints, &‘(a> = E“‘, the coefficient E” is the generator of the contact transformation for translations: Translation

Generator

= P”.

of

(2.35)

For infinitesimal Lorentz transformations, 6x“(a) = 2x,wVU with wuy = -wyU, we find Lorentz Transformation Generator = x@P” - x”PiL. (2.36) From Eqs. (2.32) and (2.34) we find that both generators motion dPu/dr = dWV/dr = 0

are constants

of the (2.37)

and are therefore expected to generate symmetry transformations of the theory. Next, we define the canonical Poisson brackets of the system to be

(Our signs are chosen to give the correspondence of Eq. (2.Q conventions with no Lorentz metric in Section 2.B.) Thus

in contrast

{P”, P”> = {xu, xv} = 0,

(2.39a) (2.39b)

{P”, X”> = - g”“, {pu, Mm”) z guBp” - gua$‘s, (x’1, M”fi} = g@@Xa - g”“xs, {MU”, M”B> = gwarMv8 - gvnMdd +

to the

(2.39~) (2.39d) guBMav

_ guf3J,fa~,

(2.39e)

SPHERICAL

are the canonical Poisson brackets canonical coordinate a?. The canonical Hamiltonian is

519

TOP

for the Poincare

Ho = -Pw,

group augmented

- L

by the

(2.40a)

and is conserved, dH,/dT = 0.

(2.41)

The canonical Poisson brackets of H,, would normally However,, a direct evaluation of (2.40a) gives

generate translations

Ho = 0.

in T.

(2.40b)

To resolve the problem of the vanishing Hamiltonian, we first observe that there is one constraint which follows from (2.32) and hence is a direct consequence of the form of the Lagrangian: x(x, P) = P2 - nz” e 0. We use the symbol w0 because this constraint

has nonvanishing

(P’ - m2, x”} = -2P”

+ 0.

(2.42) Poisson brackets: (2.43)

If we consider no other constraints, we may follow Dirac’s prescription and add to the canonical Hamiltonian an arbitrary multiple of (2.42). The new Hamiltonian H = H,, + D(T)(P” - m2) w 0

(2.44)

has a chance of giving the right equations of motion by virtue of its nonzero Poisson brackets like (2.43). In fact, V(T) may be related to uU since (H, x“} w -247)

Pu

(2.45)

while we define {H, x”> = w.

(2.46)

Hence D(T) = - (@/“/(2nz)

(2.47)

and the Hamiltonian H = -[(~~)~~~/(2nz)](P~

-- mz)

(2.48)

generates the equations of motion in 7. We need not know the Poisson brackets of (u~)~/~, as they are always multiplied by (P” - n?) m 0. Since uU = dxu/dT, the arbitrariness present in the definition of the integration parameter 7 remains in the Hamiltonian (2.48).

520

HANSON

AND

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Now we will eliminate all arbitrariness from the Hamiltonian system by imposing a gauge constraint which fixes the definition of T and makes (P” - nz2) m 0 into a second-class constraint. Our choice is (2.49) Note that we are free to make other convenient choices for 4% if we desire [21]. The matrices needed to compute the Dirac bracket are

(2.50)

According

to Eqs. (2.6) and (2.7), we find P’” = P@ - guO(PZ - m2)/(2PO), x’u = .P - (P”/P”)(,Y”

(2.51)

- T),

and the Dirac bracket ~~~4” Equation

= it, 4 - WPO)E,

P”HxO,4 - cc,x”w2, 711.

(2.52)

(2.39) is then replaced by (Pu’, p”}*

(2.53a)

= {X@, x”>” = 0,

{P”, x”}* = -g””

(2.53b)

+ gti’oP”/PO,

{P“, M-B]* = (Pu, M&B), ix@, ME”)* = g~6Xa - guaxa - (P”/Po)(x”g”fi (Mu’“, M”B}* = (Muu, M”Bj.

(2.53~) - xflgoq,

(2.53d) (2.53e)

The Poincare algebra of Pu and Ma6 is preserved, while P2 and x0 have vanishing Dirac brackets with all variables. x0 now plays the role of the time parameter describing the evolution of the system, while the Hamiltonian is H = (P” + mZ)1/2. Hamilton’s

(2.54)

equations of motion become A = dA/dx” = (aA/axO) + {PO, A}* = @A/8x0) + {H, A}

(2.55)

SPHERICAL

521

TOP

where the second expression may be used after eliminating PO in terms of P and interpreting x0 as a parameter instead of a canonical variable. As a check, we verify the equations of motion: PO

xz

A

=

0,

$’ = - aHlax 3i.ozzz 1>

= 0,

(2.56)

j, = aH/aP = P/(P” + ??z”)liz, &p=O. Note that since x0 is now a parameter, Moi does not have vanishing Poisson brackets with H; Moi is nevertheless a constant of the motion due to the compensating partial derivative with respect to x0 in Eq. (2.55). D. Quantum Mechanics of Spinless Point Particle The traditional abstract system of quantum operators for the spinless point particle can be deduced almost directly from the Dirac brackets (2.53). With the definitions H

=

PO

so = we postulate the commutation

t

=

(p”

+

1$)‘/“,

(2.57)

= parameter,

relations

i[Pi, Pj] = i[H, Pi] = i[xi, xj] = i[t, xi] = 0, i[Pi, t] = i[H, t] = 0, i[pi, xj] = @, i[H, xi] = Pi/(P”

(2.58)

+ m2)l12 = dx”/dt,

corresponding to Eq. (2.53). The only ordering problem occurs in the realization of Me8 in terms of xU and P”. This is resolved by simply requiring hermiticity, giving Mii

=

xip.i

-

xjpi,

Moi = rPi - (1/2)(Hx” + xiH).

The commutators

(2.59)

of the Ma0 with PO .G H, Pi, and xi are then

i[pu,

M”0]

i[xC,

Mik]

= =

gulp” -@kxj

i[xxi, MOj] = -@it

-

gutif’s, +

(2.60)

aijx%,

+ (l/2)[(Pi/H)

xj + xj(Pi/H)].

522

HANSON

AND

REGGE

There is a potential ordering problem in the computation of i[Moi, MOj], but the troublesome pieces cancel to give the usual completely Poincare-covariant theory:

The Hamiltonian

equations of motion are dA/dt = (2A/2t) + i[H, A]

(2.62)

so that dMoi/dt = Pi - (i/2) H[H,

xi] - (i/2)[H,

dMQ/dt = dPildt = dH/dt

= 0.

xi1 H = 0, (2.63)

Thus P” and MaB are constants of the motion; this would not be true if we had omitted the explicit t-dependence in Eq. (2.59). The wave equation in coordinate space is found by using the realization P = --2/2x of the algebra (2.58) so that H = (-V2

+ M2)‘/2.

(2.64)

The Schrodinger wave equation is then H+(x) = i(2$(x)/2t)

(2.65)

and has a nonlocal, positive-energy character. The Klein-Gordon equation follows by iteration of Eq. (2.65):

((ayw)

- v2 + WZ~+(x) = 0.

(2.66)

We should also note that the norm in our Hilbert spaceis chosen as (2.67) so that the operators (2.59), for example, are formally Hermitian. For different choices of the inner product, Eq. (2.59) would take different forms. The presence of external fields causes the Klein-Gordon wavefunction to develop negative frequency components. The norm (2.67) is then not conserved and one finds instead the conservation of a charge which is not positive definite. Therefore, one should really think of (6(x ) as a field and develop a second-quantized theory. Then one would discover that our $(x) is nonlocally related to the conventional scalar Klein-Gordon field.

523

SPHERICAL TOP

E. Remark Finally, we observe that the same classical description of the spinless point particle that we have derived laboriously from the Dirac bracket formalism also follows from the action S=

-m J” dr(1 - vZ)ljZ = Ib L dt a a

(2.68)

where v = dx/dt. The canonical momenta are

P = aqav = mv/(i - ~)1/2

(2.69)

and the Hamiltonian is H = p . v - L = m/(1 - y2)lP = (p2 4 m2)l12.

(2.70)

The relevant Poisson brackets are {pi, ~3) = &ii (2.71) {H, 2) = Pi/(P2 + tn2)1/2 so that the canonical system is just that of the Dirac brackets (2.53). The corresponding quantum system is of course identical. Why, then, bother with Dirac brackets ? The point is that in some casesit will be much harder to write down a PoincarC-invariant Lagrangian involving only the independent variables. The nature of the physical system can often be seenmore clearly by writing the equations in a manifestly covariant form involving redundant variables. In order to impose constraints which eliminate the redundant variables to give a consistent Hamiltonian system, we must use Dirac brackets. The Dirac brackets then provide the starting point for the development of a consistent quantum system.

3. HAMILTONIAN

DESCRIPTION

OF FREE SPHERICAL

TOP

We now apply the techniques reviewed in Section 2 to develop a Hamiltonian formulati.on of the free top, which was described in Section 1 from a Lagrangian viewpoint. A. PoissonBrackets The first step in constructing a Hamiltonian description of a system is to define the cano:nical Poisson brackets. This must be done with care since our angular coordinates AU, have been defined throughout to obey AAT = g, so only six of

524

HANSON

AND

REGGE

the variables are independent (before further constraints are applied). Suppose that the $i, i = l,..., 6, are the six independent combinations of the Au, which we take as canonical coordinates. Returning to Eq. (1.5) we define 8131Lv in terms of the $i by the equation se”” = AAL”SA”” = a;“($) SC&

(3.1)

where a?” 2 = --up. From Eq. (3.1) we also have &I”” = A”,al”($q S& .

(3.2)

The angular velocities oUvcan then be expressed in terms of the canonical velocities q$ as ouv = ayy+> q$. (3.3) The form of a? is not completely arbitrary. In order for Eqs. (3.1), (3.3), and (1.6) to be compatible, we require (au;“/ac$j) - (au;“/a$&) + uj”Au\ai,AV - upujy The canonical momentum

= 0.

(3.4)

conjugate to c$~is defined as (3.5)

To express SUyin terms of Ti , we introduce a function b? with the properties @‘b;o = g@orgvD- g”flg”m,

aybj,uv = 26i, 3

(3.6a) (3.6b)

so that S”” = br(rj) Ti .

The spin part of the canonical Poisson bracket is found by considering expression

(3.7)

the

The derivatives with respect to AU” and Suv on the right-hand side are to be taken with SUvand AUv held constant, respectively. We next note the identity * a6 ab: - (-g”“g”“ga’ + gwRg”~g~T - gvRgrL”ggar + g”“g”“gR’>biso7 (3.9) K” a+j - bi a9h

SPHERICAL

TOP

525

which incidentally means that by is a realization of the Lore&z group Lie algebra. From (3.8) and (3.9) we find the following expression for the canonical Poisson bracket:

(3.10) Applying

Eq. (3.10), we find the canonical Poisson brackets {P”, x”} = --g”“, {P”, P”} = {X”, X”} = 0, {hv,

Lw}

(3.11)

= 0,

{/p, &pi?} = /pugr,B _ /pBgve,

We therefore see that 5’~~obeys the Lorentz group algebra all by itself, while AUV transforms as a four-vector under S’@ only on its second index. This indicates that the S”B ar1: the infinitesimal generators of right Lorentz transformations on flu”. B. Constraints Now we must examine the constraints on the system so that we can make the transition from the canonical Poisson brackets to the self-consistent Dirac brackets of the classical system. First, let us study the consequences of requiring the Lagrangian to satisfy Eqs. (1.36) and (1.37) so that vu s pp,

w 0

(3.12)

but with no gauge constraints imposed. The Poisson brackets of the three independent components of VU are nonzero, but there are actually only two independent second-cI.ass constraints among them. The constraint ST,*,

m 0

(3.13)

noted in Eq. (1.35) is a direct consequence of (3.12) and is first-class. The other

526

HANSON

AND

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constraint which is first-class when no gauge constraints are imposed trajectory relation (1.50): M2 = PUP, 68 f(.P = (l/2) S,,P”).

The canonical Hamiltonian

is the (3.14)

is

Ho = - PW, - (l/2) P”U,” - L, = 0

due to the homogeneity condition imposing gauge constraints, is

(1.38). The full Dirac Hamiltonian,

H = Ol(T)[P” - f((1/2) SU”S,,)] + C3(T) SYs,*,

(3.15) before (3.16)

with u1 and v2 arbitrary functions. Now suppose we compute the velocities = 1P, if6 xuj = -2u,P {H, A““} = 2t1~f’il‘YS~” - 4~~fl‘YS$” FE li’:

(3.17) (3.18)

From (3.17) we find v1 = -zP/2(u

. P) = -(u”)‘l”/2M.

(3.19)

The arbitrariness in the function u1 can be eliminated by choosing a gauge for the parameter T, as was done in the example of Section 2. Equation (3.18) is more interesting. Multiplying by .A we find

uuv= 211, f tsuu- 4u2s*~*.

(3.20)

The arbitrariness in ZIPmeans we may add to CY any multiple of S*@“. This will be of more significance when we introduce interactions. Rather than pick one kind of gauge condition conjugate to S @v S,,* w 0 and two others conjugate to the two second-class parts of Vu GZ0, we shall take advantage of the fact that SUVSzVM 0 follows from Vu m 0 and choose a gauge which closely resembles VU M 0. Since vu c3 0 means that our top has only the three usual components of angular momentum in its rest frame, we seek a gauge condition which constrains the six independent components of A py so that only the three Euler angles appear as independent coordinates in the rest frame. We begin by defining p” = &P,/M

(3.21)

where p2 = 1 and pu satisfies the differential equation p + BS*““p,

= 0.

Recall that suV = LIu~S’&~‘B = constant of motion,

(3.22)

SPHERICAL

The condition

527

TOP

(3.12) implies (3.23)

and also (3.24)

Since SU* is a constant of the motion which is a function only of the canonical coordinates and momenta, we may choose &qoiw 0

(3.25)

as an invariant relation. As a consequenceof Eq. (3.24), only two of the conditions (3.25) are independent. The condition S’Oi

=

COPS

p w 0 CL”

(3.26)

is of course physically equivalent to (3.25), where the role of n is now played by II’ = CL! (seeSection 4, Eqs. (4.49)-(4.58)). If we d.efine Si = (l/2) &Sjk, Eqs. (3.23) and (3.25) imply that ,ijkSipk w 0 so pi = &i = ,poi

(3.27)

for some CLNote that Si is the body-fixed angular momentum. Introducing Eq. (3.27) into Eq. (3.22), we arrive at p” - olBJ2 = 02

(3.28)

ci - Bp” = 0. By the sametoken that we can make the gauge choice x0 m r, we can choose P OR3 1.

(3.29)

From Eq. (3.28), we find B = 0 and LX= const. Then since 1 F p~pu = 1 -

$Ja

we must have 01= 0 so that p” = APP,/M

5%g=o

(3.30)

(Note that the gUoappearing here is the left metric tensor and does not transform

528

HANSON

AND

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under right transformations.) Multiplying Eq. (3.30) by (1, we get the most convenient form for our combined gauge and invariant relation constraints: x*” = /IOU - Pu/M a 0.

(3.31)

Equation (3.31) says that AUy reduces to a pure rotation matrix in the rest frame of Pu and thus has the correct physical properties. Just like VU = 0, xU w 0 is a four-vector constraint with only three independent components. To see this, we note that

~~(4,

+ P,/M)

= P./M) Pux* + xux” = 0.

We emphasize that by choosing (3.31), we have fixed v2 . From Eq. (3.30) and the Euler equations, we find A,~(d/d~)(-A~~x,)

= fl,“~nYPv/M

w 0

or CFP “- ‘- 0 . But this means that ouy is proportional diately have

to P,

(3.32a) so v2 = 0. Furthermore,

we imme-

5.1Ly5 * * 0 II”

(3.32b)

a”‘yuY w 0 *

(3.32~)

and from Eq. (1.40)

Our constraint choice (3.31) is therefore consistent with our interpretation a gauge condition which fixes the value of v2 . C. Preliminary

of it as

Dirac Brackets

Since we may impose constraints successively, we shall now display the properties of the covariant brackets which result from imposing only Eq. (3.12) and its conjugate gauge constraint (3.31). We first take the six independent constraints

where M = (PUP,)‘/“,

Vi

=

siup

xi

=

AOi

Y 63 _

0,

pi/M

w

0

3

and combine them into the six-vector

p = (Vl, v2, v3, x1,x2, x3>,

i = l,..., 6.

SPHERICAL

529

TOP

The matrices needed to compute the Dirac brackets are -(MS’

SUM2 C’i

=

{+i,

jj}

=

(MW

f

PiPjIM)

+ PiPjIM) 0

(3.33)

and its inverse 0 (C-l)0

= )

csij

+

pi97cpk

_

pjsikpk

’

(3’34)

(PO)”

We maly cast the Dirac brackets into a very convenient manifestly covariant form by observing that C.. z T. C”ET. 2, 201 373 3

i, ,j == 1 6 a, p == 1::::: 8

where

Tim= 2”

T” , 7cLL

j/c= 1,2,3 lp = 0, 1, 2, 3

(3.36a)

and Pi/PO P2/Po P3/P0

Tk, =

--I

0

0

O-l 0

0. 0

(3.3613)

--I

The matrix Pa is fy

M2Pv

E

-Mg""

Mg"" 0

(3.37)

while its inverse has the form I gLAY (c-y

=

gr” _ M

M . P”

(3.38)

The effect of T,, is to convert VI, and xr into VUand xU by virtue of the relations P@V,, = 0 and P”xu = -(l/2) Mpx,, asfollows, {A, v”j Tk, W {A, vkTkrr}

= {A, vu}.

(3.39)

530

HANSON

AND

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Defining the eight-vector p = w”,

vl, v2, v3, x0, x1, x2, x3),

we may write the preliminary

a = I,..., 8.

(3.40)

Dirac brackets as (3.41)

where we use a prime to distinguish the present preliminary final brackets. From Eq. (3.38), we have the explicit form

A straightforward 1.

computation

brackets from the

gives the following primed brackets:

{P”, xv>’ = -gL”“,

2.

{xu, x”}’ = -Pv/M2,

3.

{P”, Pv}’ = 0,

4.

{P”, suy

5.

(x’1, s”y = (WPA - SuApV)/M2, {Su”, SUB}’ = Sua(g*B - pvpB/M”) - S@B(g*a - PvP*/J42) + Sav(g”fl - PuPfl/M2) - SBv(g** - P”Pa/M2),

6. 7. 8.

= 0,

{A”“, xq = A”,(PvgmB- PBg”“)/M” 9 {Lb”) P@>’= 0,

9.

(A~“, Lv) = 0, 10. {&, SaB}’ = /@( g*B- pvp0/M”) - &B( g"" - p*f'"/Mz) + guo(g”l,PB- gBvP”)/M.

(3.43)

As opposed to the canonical Poisson brackets, these brackets are all compatible with setting the constraints Vu and x” identically zero. We furthermore see that the position variables xu have nonzero brackets. This is a familiar phenomenon, noted for example by Pryce [18]. Recalling that the generator of Lorentz transformations is Mu* = xwp” - x”pu + su”, we compute the brackets (Mu, MaS}’ = gvR,,,f‘m _ gvaMwB+ g“8Mav _ guuamM6v, (Mu”, pa>’ = gu”pv - g”“pu’.

(3.44)

SPHERICAL

531

TOP

The system of primed brackets is therefore Poincare-covariant.

Note that

{&f”“, p}’ = g”LYx”- gaxLL so that xu is still a four-vector. At this point, the trajectory function could consider the Hamiltonian to be H = V(T)[P”P,

(3.45)

remains as a first-class quantity,

so we

- f(g!$“sq].

(3.46)

We compute the velocities to be (3.47) so that ZJ= -(U”)ll”/(2M),

cv* = 2vf’S~~,

(3.48)

in agreement with Eq. (3.32). D. Dirac Brackets The final Dirac bracket system is found by selecting a gauge constraint conjugate to the trajectory constraint and thereby eliminating all arbitrary functions. Making use of the iterative property of the Dirac brackets, we choose the final constraints Cl = PUP, -f((1/2)

S,,S~~) 5%0,

42 = x0 - 7 w 0,

(3.49)

and impose them on the primed brackets. The required matrices are simply

cij = {CJi)jlj)’ = [2”po .-‘,‘“1 (3.50)

c;’ = & [J

;I.

Hence Eq. (2.7) with all Poisson brackets replaced by the primed brackets (3.43) gives the form of the final Dirac brackets, which we now denote with stars. The results are:

532

HANSON

REGGE

1. -.3

(Pu’, y}*

3.

{PU) P”}* = 0,

4.

(P~,S~q*

5.

(xu, s”“}* = (1/M2)(S”“PA - SudPv) - (P”/PoM2)(So”PA - SOAPY), {SW, piD}* = Sm( gd _ pvf’B/M”) - suo( g”” - P”P”/Mz) + Sav(g@B- PuPB/M”) - SBv(gu= - P”Pa/M2),

6. 7. 8. 9.

10.

=

AND

+ g~opv/po,

-g””

(p, xv}” = - P/M2 + (PWY - PY?y/(WPO) = -SuuJMZ + (p”Suipi - pupipi)/(MpO)“, = 0,

{A”“, xa}* = ( l/M2) &(P”gaA - PAgay) - (I /POW) P”/IqPvg”” - P^g”“) - (f’/P”M2) {/p, pm>* = (f’/PO) gOa&SAv,

sOah%,“,

{LPU) LP}”

= -(f’/POW) &(PAgO, - Pyg”“) h”,S~B + (f’/P”M2) (la,(P”gos - PBgoy) klS~y, {~uv, p3>* = (IucsgvB _ &Bgut2 + (l/W) /I@,(PuPag~a - PvPSghe + PAPBgY&- PAPag”B) - (f’/pOM2) k,S~y(P”So~ - PBSoa), (3.51)

where M = (P”P,)“” J2 = (l/2) S*“S,, . The Hamiltonian

= [ f((1/2) SU,S~“)]1/2 and

f’

= af(J2)/a(.F),

with

is H = PO = (P” + f((1/2)

(3.52)

swwy))l’2

and generates time translations via dA - s + ((P” +f((1/2) -isat

SYS~J)~I~, A}’ = g

+ (PO, A}*.

(3.53)

Here we have used the traditional notation x0 = t to emphasize that x0 is considered as a parameter and not as a canonical variable, in accordance with the gauge constraint (3.49). The Poincare group generators are now written: PO = H, pi = pi, MOi

=

tpi

~ij

=

-jpj

-

xiH _

+

SOi

=

xipi + Sij.

tPi - xiH - SijPj/H,

(3.54)

SPHERICAL

The star brackets of these quantities theory is again Poincare-covariant. generators (3.54) are constants of the The transformation of XG under the x0 = t is now just a parameter, while

533

TOP

obey the Poincare algebra (3.44) and the Applying Eq. (3.53), we confirm that the motion with respect to x0 = t. PoincarC group is altered in the star brackets. x transforms as follows:

(P”, xi;* = (H, xi>* = Pi/H, {Pi, xjj * = gij 7 {J,fij,

.k}*

=

-@kp

{MOi,

x7c] *

=

t@L

(3.55) + _

&lxi,

xip7c/H.

Finally, we give several properties of the Pauli-Lubanski tensor, W” = S*u”Pv. Since VU = S”“Pv is strongly zero with respect to the star algebra, we have {Wu, w”}” {W”,P

= E~V~~W~P* = -,M?Y”“,

q* = 0

(3.56)

( W’“, xv>* = (;“!M2)(

WY - (P”/H)

WO).

The square of W” is w2 = W” w, =: -(I

/2) P,PY!&p”

= --M2J”

(3.57)

while with Si = (l/2) &%j7i, we find wo=

-P*S,

(3.58)

W = -(l/H)(M2S

+ P(P es)),

S = -(H/Adz)

W + P(P . W)/HM2,

and

SUi

=

(1/M2)

(3.59)

,i?‘kpiWk.

By virtue of Eq. (3.59), we may express the Poincart generators (3.54) in terms of W if we wish. One can now consider all kinds of rearrangements of the independent variables 18, Pi, Sij and Aij and compute their brackets using Eq. (3.51). We note here a set of variables which has particularly simple brackets and corresponds to the Pryce-Newton-Wigner variables [22, 18, 231. We define Si = (l/2) ,ijkSjk .595/87/z-18

534

HANSON

AND

REGGE

and then examine q = x-(S

x P)/H(Hf

J = (M/H)S Rij = Aij _ =

/lij

-

M) = xi -sy(H+

+ P(P . S)/H(H

M),

+ M),

(3.60a) (3.60b)

+ M)

fliOpj/(~ AikpkpjlH(H

+

M).

(3.60~)

We retain the same variable P and the Hamiltonian H = PO = (P” + f(J”))““, where J2 = (l/2) SuvSu, , and repeat the warning brackets with A*, . Observe that RikRjk

=

,Sjij

RkiRki

=

gii

that MZ = f(J”)

2

(3.61)

,

so that only three components of Rij are independent. the following brackets for the new variables. 1.

{p”,

qj)*

=

has non-zero

From Eq. (3.51), we find

+@i,

{H, qj}* = Pj/H, 2.

{qi, qj}* = 0,

3.

(P”, z-j* = 0,

4.

{Pu, Ji>* = 0,

5.

{qi, J’}*

6.

(Ji,

7. 8. 9. 10.

Jj)

= 0, *

=

-&kJk,

{Rij, qk}* = 0, {pj, pk}* = 0,

{Rij, H>* = (f ‘/H) p&X Jl 9 @ii, Rim>* E 0, (Rij,

Jk}*

=

ckjlRil.

(3.62)

The PoincarC group generators become H, P, and &fii

=

qipj

MOi

=

pi

_ -

qipi qiH

+ +

,$jkJk, .$kJjpk/(H

+

(3.63)

M).

Since pi&k1 HW

Jkpl +

MY

(3.64)

SPHERICAL

535

TOP

and (MOi, Rjk~” = 8ikRjzPz - PkRji + f (qiRjzE7~zmJm_ H H+M qj and R@ do not transform four-vectors to transform.

&knJnRizJz ) M(H + M) ’

(3 65) ’

under boosts as we would expect the space parts of

E. Quan turn Mechanics The system of single-particle positive-energy quantum-mechanical operators describing the relativistic spherical top can be deduced almost directly from the Dirac brackets for the final constrained system. For the sake of sidestepping the ordering problems inherent in Eq. (3.51), we will define the quantum system in terms of the variables (3.60) and their Dirac brackets (3.62). Most other ordering problems can be resolved by choosing the operators to be Hermitian with respect to the norm of the Hilbert space, for which we choose the convention

We then take t = q” = c-number, M2 = PUP, =f(J”)

(3.67)

= q-number,

and the Hamiltonian H = (P” +f(J”))““, where the twelve independent

operators

(3.68)

are

qi, Pi, Ji and three of the Rij. From the Dirac operators to be

brackets i[pi,

(3.62), we take the commutation qj]

=

relations

of these

@i,

i[pi, Jj] = j[pi, pj] = j[pi, Rjk] = 0, i[qi,

qj]

=

i[qi,

i[Ji,

Jj]

=

-&kJk,

Jk]

=

.kjlRil,

i[Rij,

i[Rij, R1”] = 0.

Jj]

=

i[qi,

Rj”]

=

0,

(3.69)

536

HANSON

The Hamiltonian

AND

REGGE

(3.68) then has the commutation

relations

i[H, qi] = Pi/H, (3.70)

i[H, P”] = 0, i[H, Ji] = 0.

The explicit computation of i[H, Rij] poses nontrivial ordering problems and will not be reported here. The Poincare group algebra is generated by the operators H, P, and Mi; = qipj _ @pi tMoi = tPi - (1/2)(qiH The Hamiltonian

&lcJk,

+ Hqi) + &“PP”/(H

(3.71)

+ M).

equations of motion for an operator A are dA/dt = i[H, A] -/- aA/at.

(3.72)

Thus, we see, for example, that dMoi/dt = -(i/2)(H[H,

qi] + [H, qi] H) + Pi = 0,

so that the Poincare group generators are constants of the motion. We remark that the algebra (3.69) can be realized with the differential

operators (3.73)

The eigenfunctions

of J’ for the spherical top are just the matrix elements

of the unitary representations of SU(2) [24]. It is well-known that these functions provide a complete basis for the Hilbert space of square-integrable functions on SU(2). Ordinary rotations act on the right index m only. The left index m’ is necessaryto represent the Euler anglesasoperators on the Hilbert spaceof physical states. The additional label m’ implies the same (2j + 1)-fold degeneracy of the spectrum as occurs for the nonrelativistic spherical top. It is obvious that any given top will be either a fermion (j = half-integer) or a boson (j = integer). The Schrodinger wave equation corresponding to the Heisenberg picture equation (3.72) would be H#(x, t; IF) = (-V2 +f(J”))‘~“#

= i(t),

(3.74)

SPHERICAL TOP

537

where Ji is understood to be the operator in Eq. (3.73). One might thus conjecture a Klein-Gordon field equation of the form (0

+ f(J2))

#(x, t; P’)

= 0.

(3.75)

It would be amusing to seeif this equation could be realized in a second-quantized field theory, to study the nature of the Green’s functions, and to try to introduce interactions. 4. LAGRANGIANDESCRIPTION OFTHERELATIVISTICSPHERICALTOP INTERACTING WITH ELECTROMAGNETISM We will now develop an approach to the inclusion of electromagnetic interactions in our Lagrangian for the relativistic spherical top. At this point, we must simply postulate that the basic electromagnetic interaction appears in the form

LI = e(dxu/dT) k(X).

(4.1)

Then, under a gauge transformation, the Lagrangian changesonly by a divergence

LI’ = e(dxx,/dT)(Ayx) f aA(x)/ax,)

= Lr + e dA/dT

(4.2)

and the action remains the same. In addition, we assumethat if any derivatives of A@(X) appear, they occur only in the gauge-invariant combination F”“(X) = A@+-- ‘4Y.U= (aAqJX,) - (aA”/ax,).

(4.3)

We will defer the question of whether or not the field acts exactly at the point xu appearing as a Lagrangian coordinate in our treatment of the classicalfree top, and will simply examine the consequencesof our assumptions. A. Example: Spinless Particles For comparison with the spinning particle, we give the Lagrangian formulation of a spinless reIativistic particle interacting with electromagnetism. We take the Lagrangian of a spinlessparticle in an external vector potential AU to be

L = --n~(u~)~~~ - eu,A”(x)

(4.4)

so the canonical momentum is

Pu = -aaL/atr, = w~(w/(u~)~~) + ek(x).

(4.5)

We define the mechanical momentum as

TU = rn(~~/(u~)~~~) = P” -- eA”(x).

(4.6)

538

HANSON

AND

REGGE

The Euler equations (4.7a) may thus be written in terms of n-u as 7j’l

=

-eFUl’u

(4.7b)

Y

where Afi = u YAu*“. Note that (d/dT)(7T2) = 0

(4.8)

so that 9 is a constant of the motion. In fact, the constraint n2 - m2 = P3 - 2e(P . A) + e2A2 - m2 = 0

(4.9)

is a constraint among the canonical variables P” and xu following from the form of the Lagrangian. B. The Bargmann-Michel-Telegdi

Equations

We now turn to the relativistic spherical top. Suppose that, in addition to the interaction (4.1) our spinning particle has a magnetic moment. The Euler equations should then contain the equations of Bargmann, Michel, and Telegdi (BMT) [l]. If we make the obvious choice L = L, - eu,,A“ - (l/2) guuVFUY

for the interaction

(4.10)

Lagrangian, we find the canonical momenta Pu = -(aL,/&,)

+ eA” = ?T~+ eA”

Py = -(aL,/lb,J

+ gFuy = .Zuv + gF”*.

(4.11)

The Euler equations are fr@ = -eFUYu, + (l/2) guar4Fa4*u, SW + &&~

- SUAQAY= 0 (4.12)

= L+ + u”rr* -

U”T? + g(&FAy

Now if we take L, to satisfy Eqs. (1.36)-(1.38) z+rv = 0

- F“bAV) = 0.

then (4.13)

so 2xY~

= 0.

(4.14)

SPHERICAL

However,

539

TOP

from Eq. (4.12) SU”S,+, = (F”

Since the constant field, the constant

-t gFU”)(,q

+ gF,*,) = const.

vanishes when the field vanishes, but is not a function is zero. Applying (4.14) to (4. IS), we find

(4.15) of the

In the presence of a pure electric field, we find (E . X) = 0; this is clearly an unphysical restriction, so Eqs. (4.13) and (4.14) are physically unreasonable. We must thus find some alternative to Eq. (4.10) in order to couple the magnetic moment in such a way that we can derive the BMT equation in a consistent manner. It is possible that there are several methods of accomplishing this. The method which we found most successful was the following: make the substitution arr”

--j

&UY

=

ue” + gL,(zr, 6) Fuv,

(4.16)

where g is a constant and the presence of L,, maintains T-reparametrization invariance. Equation (4.16) is a highly nonlinear, implicit definition of 5 that amounts to a kind of minimal coupling principle for magnetic dipole moments. Now we set a, = 2.2,

a2 = f? * 6,

a3 = u&31,

a4 = Det 0,

and take L(a,) = L,(aJ - eu,A”.

(4.17)

Let us define

*” = P&&4, P” = Sg&d,

6) = - _aLO %I a=const’

(4.18)

6) = - - ah,

aeu,'

so that ilr” and P” have the same functional form as the free momenta, except that cv is replaced by ?‘. The canonical momenta are Pfi = -8Lli?u,

27~~= -aLpa,,

= vu + eA”

= -aL,p~,,.

(4.19)

540

HANSON

AND

REGGE

The mechanical momenta can be expressed in terms of Eq. (4.18) as (4.20) (4.21) We now observe that

aL,la6ea = -P, aemB/au,= --gF,Bn~,

(4.22)

2(Gu, - u6,#3cr,, = -gF,,P”. Thus the choice of the coefficient L, in Eq. (4.16) results in the following remarkable simplifications in Eqs. (4.20) and (4.21): (4.23) so that 7-1”= (1 - (l/2) gF * S) 4”

(4.24)

S”” = (1 - (l/2) gF . S) P. If L,(ai[u, 61) is chosen to satisfy the same differential Eqs. (1.36)-(1.38), then l-l), = 0 and (4.24) gives V” = s?r, = 0.

equations in ai as before, (4.25) (4.26)

The Euler equations for PU are easily found to be (4.27) where

aLo---- 1 aLL, a(@ + ax,

2 azu,

gL,,F”O)

%X,

so that aLo/axu = -(1/2)gL,s,,FeB+u.

(4.28)

The mechanical Euler equations are thus + = -eF”Lyu, + (l/2) gLoSmBF”B~“.

(4.29)

SPHERICAL

We now usual, is

show how the BMT

equation

541

TOP

arises. The Euler equation for 9~“. as

&J + &SAV - p~gAu xx 0. This can be rewritten, and P, as

using (4.24) and the known

sup = rr’w Contracting

- u’w

(4.30) forms

(1.16)-(1.17)

+ gLo(FuASAY - SUaFA”).

with r, and using the derivative of the constraint

for 4” (4.31)

(4.26),

S%” + 557” = 0

(4.32)

n2uu = TP(U . TT)+ S”“frv - gLoS~nFhp~y.

(4.33)

we find

The antisymmetric

Inserting

product of (4.33) with 7~” gives

this expression

in (4.31) gives one form of the BMT equations,

n-22% = gL,,sr2(F%A”

-. PF,“)

- gL,(x+‘*FADrr~

- rrvPAFA,~~)

+ 7T~syG-A- rr”S”%,) . From Eq. (4.34), we can derive the following

(4.34) equation of motion for W” = S*+ry

~~66’~ = -TP W,+ - gL,&r2F~“Wv The identification

+ rr” W,Fm%B).

:

(4.35)

with the standard BMT equation is complete if we set gL0 Iux = +sPn rrTT”Ius = muu

IBMT /BMT

(4.36)

.

The BM’T method, which does not use a Lagrangian, could not distinguish between 7~” and ww, which are not necessarily proportional in our treatment. Since (d~7)~ = dx, dxu in the standard BMT equation, direct comparison of our result to BMT requires a definite choice of the gauge relating dr to dx and dA. Choosing L, to be a numerical constant on the equations of motion, as implied by Eq. (4.36), is precisely the condition which makes T the ordinary proper time in the spinless case. Clearly it would be desirable to derive the BMT equations by introducing magnetic couplings into our formalism in a simpler way.

542

HANSON

AND

REGGE

C. The Evolution Equation Finally, we show that all the equations of motion can be written general form. First, we multiply (4.30) by nV and use (4.32) to get

in a simple

S”“(Yr, + UJT”) = 0. Since W@= LP%, and rrU are the only null vectors of Suv we must have

+ + (Ag”” + BS*@” + a”“) 7~” = 0. Contracting

(4.39)

with 7~~and W” we find

(4.40)

Thus we may write (4.41) where QU” = I+‘” + BPuv. Similarly, if we multiply

(4.42)

S *Uv by n, and use Eq. (4.41), we may show (4.43)

It is also obvious that Eq. (4.30) can be written

d & [ (S scL” * sy2 ] + gun [&)T]

-

[ (s y;)l,2-]

Q,” = 0.

(4.44)

Thus L& looks like a generalized angular velocity, giving the change in the momenta as a function of the momenta. We observe that we may use Eqs. (1.22)-(1.26) to rewrite the coefficient B. If we define p” = A ““Tr”/(“2)l~2, (4.45) where p2 = 1, then Eq. (4.39) can be written

,Y + BS*u”p, = 0.

(4.46)

SPHERICAL

543

TOP

Thus (4.47) At this point, we notice that the arguments developed in Eqs. (3.21)-(3.31) for the free case hold for the interacting case as well when P@ is replaced by TP. In particular, we are free to imitate Eqs. (3.25) and (3.29) and impose the invariant and gauge relations Soi = 0 and p” FZ 1. This correspondsto UO 3 P '""g B = 0,

(4.48)

QLY = #-pv Now, however, u@”is not proportional to P” and in general a3 # 0, a4 # 0. When we fix B, we naturally fix the time evolution of fl, since A-l/l = u and (Tdepends on B. There is a convenient factorization which may be used to exhibit the B-dependence of /l and to show that the same physical motion follows from different choices of B. (In the language of Dirac’s formalism describedin Section 2, B depends on the arbitrary function multiplying the first-class variable SY$ in the Hamiltonian, so different B’s should be physically equivalent. Compare Eq. (3.20) and (4.42).) Now let A, and fl, be two Lorentz matrices resulting from different choices of B, with fl,=MA,,

(4.49)

where M is a Lorentz matrix to be determined. Then, since ffl = &l/i, u2 = cl;l~&, we have

and

(4.50) But from Eq. (3.20), we can deduce that the difference between g1 and ug is given even in the interacting caseby (4.51)

u2 = crl + S* db/dr

for some function b(7). But then M-‘&I

= (db/dT)

A,S*A;’

= (db/dT)

S*

(4.52)

where s* is the constant matrix given in Eq. (1.25). We may integrate (4.52) directly to get Mu” = CuA[exp(S*b(T))]AY (4.53)

544

HANSON

AND

REGGE

where C is a constant matrix. Thus A 2 = MA

1

= CeS*bA 1 .

(4.54)

Now we examine the expressions (4.55) where p1 + B,S*p, pz + B,A,S*A;lp,

= 0, (4.56)

= 0.

But from Eqs. (4.54) and (4.55), ,bz = i@pl + Mb, = (db/d~) MS*p, = ((db/d~) - B,) -‘l,S*&pp

- MB,s*p,

.

Thus B, = B, - db/dr.

(4.57)

We see that by choosing db/dT = B, , so that Bg vanishes, we may express any Lorentz matrix fl, as A,(T) = M(T)-~~~,(T) where A~“7rJ(7r2)“”

(4.58)

= const.

is then physically irrelevant, while A, is physically identifiable as a pure rotation in the rest frame of @, so A, defines the Euler angles of the top.

Mu”(T)

D. Constraints The velocity constraints treated in Eqs. (1.39)-(1.44) complicated analogs in the interacting case. We only one would proceed to study the constraints in the g = of motion, (4.29) and (4.30), together with Eq. (4.33)

for the free case have more show a brief sketch of how 0 case. From the equations we find

@(T-I * u) = u~M2 - eS”*F,AuA. Substituting

(4.59)

the explicit forms of the momenta, we find 0 = ufiL,[--2e(uuFu)

- 4a,L, + 2a,a,L, - 4ala,L,]

+ 4u%AuAL3(alL1 + a,=%) + 4eL.@‘F,,$ + (l/2) eL,(o * u*) u*~“F~~u~.

(4.60)

SPHERICAL TOP

545

The contractions of (4.60) with CYU, and ZPgive two scalar constraint equations (UOOFU)SEu’Lu,,cY~F~,,z~~ = 0

(4.61)

and 2L,2(ala,a, + 2ax2 - 2a12a,)+ 2e(uoFu)(ZL, - al&) + UP) e&Co - a*)(uu*Fu) = 0.

(4.62)

All other contractions give equivalent expressions.The corresponding expressions for g # 0 are easy to derive. We mention at this point a similar topic, the trajectory constraints for g f 0. Starting with Eq. (4.25) and proceeding by analogy to the free case,we deduce the existence of a trajectory constraint qIi”qG,-f((1/2)

Pr,,)

= 0

(4.63)

or (1 - (l/2) g(F . S))-%r2 -f((1/2)

S . S[l -- (l/2) g(F . S)]-2) = 0.

5. HAMILTONIAN DESCRIPTION WITH No MAGNETIC

OF INTERACTING MOMENT

(4.64)

TOP

We now examine the Hamiltonian formalism for the electromagnetically interacting relativistic spherical top of Section 4. For simplicity, we will set the magnetic moment coefficient g equal to zero. A. Spini’essParticle For comparison to the spinning case,we again outline the treatment of a spinless particle. With the Lagrangian (4.4) and momenta (4.5) and (4.6) we get the canonical Hamiltonian Ho = -P%,

- L = 0.

(5.1)

The constraint of Eq. (4.9) is now interpreted as the solitary first-class weak constraint 7r2- m2 w 0 (5.2) of the system, so that the Hamiltonian may be chosen as H = v(T)(T” - m”).

(5.3)

546

HANSON

The canonical Poisson brackets

AND

REGGE

are

{P@, P”) = (X’, xv> = 0, (5.4)

(Pu, x”} = -g”“.

The Poisson brackets of the mechanical momenta, 7~ = Pu - eAu(x), are then {nil, VT”} = -e((&P/i3x,)

- (i3A”/8x,))

= -eFuv, (5.5)

( +, X’> = -g”“. The Hamiltonian

(5.3) generates the following

motion for x:

(H, xU> = -2~5~~ = ~‘1

(5.6)

so that TP = -llq2v (5.7)

u = -(u~)‘~2/2nz. The equation of motion for 7rU is ?ip = {H, .rrU} = -2em,,JrylFyu = -eFuvu,

(5.8)

in agreement with Eq. (4.7). Now we choose a gauge constraint to fix the parameter and make the constraint (5.2) second-class. Then, as usual, the Dirac brackets following from $1 = 2T2- m2 M 0, $2 = x0 -

7 NN 0,

(5.9)

are (5.10)

{xu, x”}*

= 0,

{7P, x”}* = -gL”” + g97+70, {w, %q* = - eFUy + (e/r”)(F%rA The Hamiltonian

go” - I;Y%T~go*).

(5.11)

is now H = CTO+ eA”(x) = (x2 + nz2)l12 + eAO(x) = ([P - eA12 + m2)lj2 + eAO(x).

(5.12)

SPHERICAL

547

TOP

B. Prelinzinary Spinning Dirac Brackets We take the Lagrangian and the canonical momenta for the interacting g = 0 spherical top from Section 4, Eqs. (4.17) and (4.19). The canonical Hamiltonian vanishes as usual, H,

=

-PW,

-

(l/2) SYJU,. --

L =

(5.13)

0.

The canonical Poisson brackets are taken from Section 3, Eqs. (3.10). Given our assumption that the argument of the external potential Au is the can’onical coordinate xu, we find that the mechanical momentum, rrU = P” - eAU(x), has the canonical Poisson brackets (5.14) (5.15) The Lagrangian is chosen as before to obey the constraint If@ =

S%”

e

(5.16)

0

among the canonical variables. As noted before, there is a combination of the three independent constraints which is first-class and which can be taken to be ST,*, By imitating our previous arguments, class trajectory constraint

Before fixing any gauge constraints,

w 0.

(5.17)

we find also in the interacting

we may thus take the Hamiltonian

H = h(T) (r2 -f(@,"~"")

+ 7122~~($:&)

case the first-

to be

+ U2(T)S~~S~". (5.19)

Thus u” = UK ~‘9 = -24

(+

-

eSuvFv,d 7T2_ e(s . F),2).

(5.20)

We also find that Eq. (3.20) for ouy is now replaced by - n*FuAz-A) u”” = VI (2f ‘Suv + 2e(rrLLFYAr,, - 4v2s*u~. rr2 - e(S . F)/2 1

(5.21)

548

HANSON

AND

REGGE

The equations of motion are thus 7j” = {H, +} = -eFuvu, , ,‘+” = {H, s”%)

= pfl,B

(5.22)

- g@$p

where we use Eq. (5.20) for TV and Eq. (5.21) for gMlly.Equations (5.22) thus agree with Eqs. (4.29) and (4.30). (Remember that g = 0.) Now we could compute Dirac brackets using the two independent second-class constraints in VU = 0. We shall instead proceed directly to impose constraints conjugate to Vu = 0, as we did in Section 3. By the arguments of Eqs. (4.45)-(4.58), we have in our system a gauge freedom which corresponds to seY+ILyM 0 and two integral relations corresponding to the second-class parts of Vu M 0. We showed in Section 4 that a consistent constraint was p = fl@Y7T”/(4u2 m

guo

which we choose to write as x” = /p

- 7p/(+y’l”

m

(5.23)

0.

Since p2 = 1 (or +xu = -(l/2) Mx2), only three of these constraints are independent, just as only three of the V@are independent. We now compute the Dirac brackets consistent with setting VU w 0 and with eliminating the corresponding arbitrary functions by setting x” M 0. The set of constraints is summarized in the eight-vector @ = (VO, vl, v2, v3, x0, x1, x2, x3>,

a = l,..., 8.

(5.24)

For convenience, we define M = (77%-p, N = M - (e/2M)S,,Fa~, Puy = F”” f W2(xuFv%rA

(5.25) - rrvFP”rrA).

The arguments of Eqs. (3.33)-(3.39) are then repeated to give the Dirac-bracket matrices MNP” Mg”” + & PAPAY C=%zz _Mgul’ _ 4jYLq M (5.26) --q”” -5 (c-y = Njj P” M

SPHERICAL

The preliminary

Dirac brackets

549

TOP

may then be written

explicitly as

(5.27) The Dirac brackets can now be computed. We first define the following occurring quantity,

frequently

(5.28)

Guv = g“” - (e/NM)S~T,l*, where GJJvis not symmetric

and G, o = 2(M + N)/N.

1.

{TP, a?}’ = -gu”

2.

{X“, X”>’ = -S”“/MN = -(I (+, +]’ = -eFuv + (e’/NM)

3.

+ (e/NM)

4.

(+, JP}’ = (e/NM)(F%‘,%rY

5.

(x“, LPI\’ = (I /NM)(S’+

PFeu

The resulting brackets

are:

= -- GYu,

/M2) GunSA” = (l/M-) F~“SwBF~Y = -ePG,V

GAS,,“, = eF”*G,u,

- F“4Sm”~A), - SJ+Y),

Since the constraints (5.16) and (5.23) are strongly zero in the prime system of brackets, they may be used wherever convenient to re-express the right-hand sides of Eqs. (5.29). The symbol A4 = (++,J 1/2 is not a constant in Eqs. (5.29), but has nontrivial prime brackets. Among the prime brackets of the PauliLubanski tensor are {W”,

595/87/2-‘9

I+“‘)’ = -MTP

- &

S*“” [M(S*

. F) - &

(S . S)(F* . F)],

(5 30)

5.50

HANSON

The Hamiltonian just the trajectory constraints:

AND

REGGE

for our new, partially constrained, prime bracket theory is constraint, which is first-class because there are no other H = a(~)+? - f((1/2)

S,,P)).

Now we must compute the equations of motion We find l.P = (H, xu}’ = -2v(TP

(5.31)

using the prime brackets

- (e/NM) S~~F$rA)

(5.29). (5.32)

so that now D = - (~/~)(u~)~~~NM(N~M*

+ e2~“F,BSavSSdFsf,~)-1i2.

(5.33)

Also o’- z< &{H,

A”“].

= 2vf ‘Suy - (2ev/M2)(+nuF,aG6L’ After using the identities in Appendix the form (5.21) with

- z+PF~~G~~).

(5.34)

A, the above equation can be rewritten

V,(T) = +(e”v(-r)/SNM)

F”“F$

in

(5.35)

,

Our gauge choice thus implies that Eqs. (5.22) continue to hold in the new system and that (dK@/dr) + vK,, where Ku = +/M or Ku = WU/((4.44) hold with QU” = CF.

= 0,

IV2)1/2. Thus the evolution

equations

(4.41)-

C. Fixed Mass Dirac Brackets Now we impose our last available gauge constraint functions from the Hamiltonian system. We take $61= n-2 -f((1/2) cj2 = x0 -

to remove all arbitrary

S,,SI*‘) M 0,

7 m 0,

(5.36)

where {c& , +2}’ = -2E = -2(n”

= -2Gouzrp - (e/NM)

SoyFVu~u).

(5.37)

SPHERICAL

551

TOP

The final Dirac brackets are thus

7):‘. + & {6,x0:’{7r2- ,f(&“S~‘“),

(5.38)

With a little labor, we arrive at the final system of brackets for a charged spherical top consistent with the constraints (5.16), (5.23) and (5.36):

2.

P" {Y, x")" = - ~ kN MN +

3.

(xU, 7r"}" = -eFLlaG," + 5 (F""Ga,nBG"" - FYaGa~n'Gou), E

4. {7rW,YA]* = -&FUSS,

(G‘%rASoY- G"A~ASou),

A$ _ Fuw.&l'nA)

+ +M(FY"G,pB(So'nA

- So%")

- G""(x"SAmF,B~" - T&Y~F,~~'~~~)), 5.

(xU, s""}* = &

(Suvn-A- YAd') - &

+ + 6.

GTr,(SoV~A - SoAr")

Sou(~"SA"F,,rr5 -- rATF,grB),

{S”“, Sa4}* = S”” ($8 _ ?&?)

_ S”fl (f’” - g)

+ S”” (g”” + &&T%~" - h 7. {A"", x"}* = &

- T"S~~)(~S~~F~~~-~ - &YAFA,,d') (TT"S~" -

TT~S~~)(X~(S"~F~~~T~ - T"S~"F,~~~"),

(IuA(~"GaA - nAGa") - &

/luAGn8QrYGoA - mAGo")

T~F~,,(~T~G~~- &GvA) - f's"],

552

HANSON

AND

REGGE

e Lt ~,FaRGR,,d’(~“GoA -+ EM2

+ &

AuA(7r”Goh -

-

dG”“),

n”G”“)[z-“SaYF~,~”

-

dSaYFys7is]. (5.39)

For the convenience of the reader, we repeat the definitions of the symbols in (5.39): J2 E @,“S@“,

f’ = q(P)p(J”), N = M - +MS,,FmB, Gu” = g”” - & E = G%,

Su’“F,“,

.

The constraints (5.36) are now strongly zero and have vanishing star brackets. Thus wherever r” appears in (5.39), we may replace it by Tr” = (57”+ f(( l/2) SU”S,“))‘/2.

(5.40)

SPHERICAL TOP

553

The problem is now to find a revealing form for the Hamiltonian; we are faced with the problem alluded to in Section 2.B. If we use PO = 7~0+ eAO as the Hamiltonian, the explicit time dependence of our variables is complicated. The ideal solution would be to express all variables in terms of Newton-Wigner type coordinates (3.60) as done in the free case. The time-dependence of these coordinates would be entirely implicit and this would select a Hamiltonian unique except for time-independent canonical transformations. This Hamiltonian would correspond to the Erikson and Kolsrud [25] treatment of the Dirac electron (see also Dixon [26]). This appears to be a difficult task and we have not carried it out. In the static case, however, the choice PO = no -+ eA” assignsno explicit time dependence to all variables and the corresponding algebra has time-independent Poisson brackets. It is clear, however, that the theory contains no intrinsic magnetic moment. This conclusion is reasonable because our classical theory applies to systemsof particles only, without the introduction of antiparticles. We will discuss this point further in the final section.

6. CONCLUSION There are many unanswered questions and new subjects for investigation which are suggestedby the treatment we have given for the charged relativistic spherical top. This final section is devoted to a discussionof a number of these topics and to sugge:stionsfor directions in which further work might proceed. The first question to which we addressourselves is the consistency of taking the electromagnetic potentials to act at the “position” xU of the particle. In the oneparticle quantum theory with an external field, we would presumably be led to awkwarcl functions AU(x, t) where the x’s would not commute. Furthermore, we should include the energy and momentum of the electromagnetic field in the Lagrangian in order to make the entire system Poincare-invariant, with Poincare group generators which are constants of the motion. Then, however,we would have to develop simultaneously the constrained canonical formalism for the electromagnetic field. (This is not too difficult, and is essentially equivalent to the mlethod of Bjorken and Drell [27]. See also Dirac [3].) But if Au(x) is treated as a canonical coordinate, the variables x” have the meaning of labelson the infinite degreesof freedom of A“(X), rather than canonical coordinates. We conclude that we cannot have local interactions with a canonical electromagnetic field unlessthe x” can be interpreted as commuting four-vectors labeling the field. One approach to resolving this problem would be that of Jordan and Mukunda [28]. They double the space of quantum-mechanical states by introducing antiparticles. The new degreesof freedom enable them to derive Abelian four-vector coordinates which are essentially a Foldy-Wouthuysen transformation [29] of

554

HANSON

AND

REGGE

the Newton-Wigner coordinates for the particle and antiparticle. It appears, however, that the currents which result from their method are local only for spin 4. This limits the usefulness of the Jordan-Mukunda method for the introduction of local interactions. It therefore appears that the concept of the quantum-mechanical interacting top is not well-defined unless we enlarge the quantization scheme to a secondquantized system including tops and antitops. It is possible that the methods of Weinberg [30] or of Bhabha [3 l] could be used. The present theory would then be a classical limit of the quantum electrodynamics of tops for slowly varying fields and with pair-production neglected. Even so, there may be difficulties for particles of spin greater than 4, as illustrated by the work of Velo and Zwanziger [32]. [We remark that the finding of Velo and Zwanziger, that high-spin wave equations may violate causality in regions of intense fields, has in fact a classical counterpart in our theory. If the quantity E = no - e(NM)-l

SouFuy~*

vanishes at any point on the path of the top, then the corresponding four-velocity can become spacelike while the mechanical momentum remains timelike due to rr%-u = M2 = const. One could then find particle trajectories bending backwards in time. The present Hamiltonian formalism would then probably be inadequate to deal with the problem, since ordinary time would no longer be an appropriate label on the particle trajectory.] There are many things which might be clarified by a better understanding of the quantum theory of spinning particles. In addition to clarifying the problems of electromagnetically interacting high-spin fields [33], some light might be shed on the question of the intrinsic magnetic moments; is g = llj as indicated by Hagen and Hurley [34], or does it depend, for example, on the nature of the trajectory function? Equation (3.75) raises other intriguing questions. Can it be interpreted as a field equation for a quantum field operator representing a family of particles lying on a trajectory? What are the properties of the Green’s functions? Are they related to the Gribov calculus [35] or to the dual resonance models? How do fermions and bosons differ? And how could interactions and currents be introduced? We mention a number of topics which we have left open even in the classical theory. One is to understand better how the intrinsic magnetic moment g enters the classical theory and to treat the g # 0 case in the Dirac-Hamiltonian formalism. There is also no reason why a formalism could not be developed to include higher multipoles beyond the pure dipole case we have treated. An extension of our methods to the asymmetric top would be interesting because the known

555

SPHERICAL TOP

elementary particles do not possessthe same degeneracy as does the quantized spherical top; the asymmetric top could provide a description of elementary particles (e.g., the electron and muon could from a basis for a spin 4 representation of an asymmetric top). Both the asymmetric top and the top with higher multipolles will probably involve Lagrangians which depend explicitly on /lU, . Since the spinless particle action has a clear interpretation as a classical path length, it would be interesting to find an analogous geometrical interpretation for our spinning action function and to develop the quantum theory using Feynman’s path-integral approach [36]. Finall:y, our treatment of electromagnetic interactions should be extended to get the “true” Hamiltonian for time-dependent fields, and to reach a point in the quantum-mechanics (or quantum field theory), where the connection between our approach and treatments based on the Dirac equation is clearly apparent. In addition to electromagnetism, one might aIso attempt to include other interactions such as gravitation in our formalism.

APPENDIX

A: IDENTITIES

We list here a number of identities which we have used repeatedly in carrying out calculations mentioned in the text. All Greek indices run from 0 to 3 and are raised and lowered by the metric g’“” with diagonal elements (1, - 1, - 1, --I). The totally antisymmetric tensor @Yasis chosen to have c”lz3 = + I.

1. Antisymmetric Let SUv= -9’

Tensor Identities and TUV= - T”” be four-tensors and define the dual tensor S*‘r” = (l/2) f”%sUI@*

(A.1)

Then S **LIv zz-

U” s 3

04.2)

Sa4TmB = -S*aST* a8 2

FAT,” - T*“‘S;’ Sd“T,,sY4 = -(l/2)

= -(l/2)

g”B(SuyTu,),

Saa(S““Tu,) - (l/4) T*“$S‘=‘S,*,),

s*“2TA8= suAsy = -(l/4)

ga4(SY;“).

(A.3) (A.4) 65)

(A.7)

556

HANSON

2. Vector and Antisymmetric Let PLLbe a four-vector

AND

REGGE

Tensor Identities and SuV = -sY”

an antisymmetric

tensor and define

vu = J+vp Y,

(A-8)

W” = SuvPy = (l/2) E”%Y”~P, .

(A.9)

Then (PUVV - P”vu)* (PUWb, - puwU)*

= @OP~VO = WUpy - wupu - pzs%W,

(A.10)

= @up* w, zz.zPw v” - PVVU f PW”,

wu w, = - (I/2) P~P,S,,s”B 3. Three-Vector

(A.1 1)

- PY&S@yP, .

Identities

In three dimensions,

we have the following

if Si = (J/2) GikSik, S@ = @Sk:

Siipj = (p x S>i, pisikpk

_

pjsikpk

=

pi,jklpkSl

=

&kpk(p

From Eqs. (1.27)-(1.32) relations

_

&kSk(p

(A.14)

. p).

Equations

and the constraints

(1.36)-(1.38),

we find the following

- a,a,L, - a,a,L,]

= -(~2LJ-W33)Pa&2

- a,“L, - (a3 + (l/2) ala2> La], (AX) + a&J

= 2L,L2 + L,(a,L, = 4LlL2 - L3@J, -

pi,iklpkSC

. s)

a,a,a, + 2aS2 - 2a12a, = (M2/L,L,2)[2a,L2

-GIL,

(A.13) _

4. Identities Following From the D@erential

(l/2)

(A.12)

+ a2L2)

=

L,(2L2

=

&/L3)[L3(2L2

-

+

=

(~/J5oM~/2)

J2L, +

a,L,)

-

L,(& @‘3)

+

a2L3)

(1/2)

L,(2L,

-

a2L3)l

(A.16)

J+f2L21,

M2 = P&P, = 4ulL12 + 8a,L,L, - 2u2a3L32 + 4a,a,L32 = L,(2L, - u,L, - a4L3L4/L2) =

(LO/L2)(2LIL2

=

(LoL3/L2)hLI

-

L3(a2L2

+ a4L4))

+ a3L3)

= -(1/2)(L,L,/L,)(u,p~“),

(A.17)

557

SPHERICAL TOP

J2 = (l/2) S,,,SUv= 8a,Ls2 + 16a,L,L, - 4a,a,L,Z - 2a4L4(a2L4+ 4a,L, - 16&J = 2-L&L, - a,& + a,L,L,IL,) = GLolLlWlL2 - aA% + a4L4LJ = (2L,L,/L3(a2L2 + a,& -+ 2a,U = -(1/2>(LoL,/L1>(u,“~).

(A.18)

Finally, we give the algebraic equation which the Lagrangian must satisfy when the dependenceon the Li is expressedin terms of M2 and J2: 0 = L,S(a22+ 16a,) + L,6M2(-2a,a22 + 4a,a, - 48a,a,) + L,6J2(-a23 - 16a,a,) + Lo4M4(a,Za,2+ 4as2- 8a,a,a, $- 48a12ad) + Lo4M2J2(-a,a23 - 8az2a, - 4a,a,a, - 8Oa& + + + + + + + + + +

L,*J*((1/4) ao4+ 2a22a,- 32a,2) Lo2M6(4a12a2a,- 8aIas2- 16a13a,) L,2M4J2(-4a,a22a, - 20a,a,” + 20a12a2a4 + 88aIa,a,) L,2M2J4(a23a, + 4a,a,a, - 8a,a,2a4- 8Oa,a,2) L,2J6(a23a4+ 16a,a42) Ms(4a12as2) M6J2(-4a,a,a,2 - 16as3- 8a12a,a4) M4J4(a22a32 + 48a,2a4+ 8a,a2a,a4+ 4a12ad2) M2J6(-2az’a,u, - 4a,a2a42- 48ap,‘) J8(a22a,2+ 16aA3).

APPENDIX

B: SINGLE-PARTICLE

COORDINATE

(A.19)

OPERATORS

The classic work on the subject of single-particle coordinate operators for spinning bodies is that of Pryce [18]. As an aid to the physical interpretation of our theory Iof the spherical top, we give a summary of Pryce’s results. The three interesting types of coordinates for spinning particles are (a) Rest-frame center of momentum is the center of momentum calculated when the body as a whole is at rest, then Lorentz-transformed to an arbitrary frame. Classically, one would write

(B.1)

558

HANSON

AND

REGGE

where I?,-,$is the energy of the ith component particle computed when the total momentum is zero. x, transforms as the space part of a four-vector. The corresponding spin matrix obeys the covariant constraint

V” = s av u”p = ().

03.2)

(b) Center of momentum is the center of momentum frame of an arbitrary observer. Classically,

of a body computed in the

xi Eixi xb = Ci

(B.3)

where Ei is the total energy of the ith particle in the observer’s frame. This coordinate coincides with (a) only in the rest frame; in other frames, the relativistic mass-increases of diametrically opposite particles will differ, causing xb to shift away from x, . xb is defined in a frame-dependent manner, and so is not part of a four-vector. The corresponding spin matrix satisfies sR”=o.

03.4)

(c) Newton-Wigner Coordinate [23] is the name now usually given to the peculiar weighted average of (a) and (b) noticed earlier by Pryce [22]: X, = (MX, + EX,)/(M

+ E).

03.5)

A4 and E are the total invariant mass and energy of the system, respectively. x, is the only single-particle coordinate variable which commutes with all of its own components. x, is frame-dependent and is not a four-vector but its commutative properties have given it an important place in quantum theory. The corresponding spin constraint is

&fSO” c - ypC” = 0 . Our notation has the following

03.6)

relation to that of Pryce: Here x = x,

Pryce

I; (c)

q 2x,

(4 X Cc) Q (4 4

(a)

S = S,

(4

@I

(c)

x

Sb

cc>

s

J = S,

(4

5

SPHERICAL

Since only three components use the three-vector portion:

559

TOP

of S’“” are independent

in each case, we will generally

si = (l/2) ,ijkSjk. The simplest commutation

03.7)

relations are those of the variables (c), which obey i[xci,

x,j] = i[pi,

pj] == 0,

i[x?,

S,j] = i[pi,

S,j] == 0,

j[S,i,

S,j] = .-&kSck,

i[Pi, x,j]

= LW.

With thl: Hamiltonian PO = H = (P” + my,

03.9)

the Poincare algebra is generated by P” and the operators Mij

=

x ipj G

_

&pi

= tpi -

x jpi c

+

(I/2)(Hxci

&jLS

k c >

+ xciH)

(B.lO) + &SS,jPk/(H

+ m).

The commutation relations of the operators (a) and (b) can be deduced from their expressionsasfunctions of the operators (c). The following equations express the relations among the operators: x, = x, s, = ;sc

Tn;Hy;,7)

= Xb - pxs,, n?

-

m(H

- m) m2(H + m) ’

+ m)

PXS

P x s, xb

=

x~

+

H(H

+

(B.ll) P(P . S,)(H

P(P . SC> +-

f12)

=

x,

+

d,

~2

(B.12) sb = ;

P(P . S,)(H - m) s, + H;g :,“Fil, = Et H2 s, $. ’ H2(H + m)

x, = x, -

s, x P H(H

+ m) = xb + UP

sc=$sa+

H(H+m)

. s,>

s, ‘x P m(H

+ m) ’ p(p

m(H

* sb>

+ m) .

(B.13)

560

HANSON

AND

REGGE

By making use of the constraints (B.2) (B.4), and (B.6), one may show that the Lorentz group generators have the form Mif = x,ipj

_ x,jpi

+ ~2,

(B.14)

Moi = tPi - (1/2)(Hxmi + x,%) where 01= a, b, c. The interesting commutation

+ szi,

relations are those of the x’s, ij

i[xai, x,j] = - S,

m2

+

s:

pisoj _ pisoi a a m2H

EijkPk(P

,,,“H”

H2

s

(B.15a) ’

i[xbi, x,j] = Slj/H’,

(B.15b)

i[x ci, .x,q = 0,

(B. 15~)

and those of the x’s with Moi: (B. 16a) i[MOi,

Xbj]

=

t@j

-

k

(xbi

;

+

;

xbi)

-

&ikS

i[Moi,

+ ;

(B.16b)

&kSb7~/H,

xCi) - G

k

pj&klS

-

H(H

kpl

; ,n)Z -

(B. 16c)

It is evident that x, is a noncommuting four-vector, x, is a commuting non-fourvector, and x,, is neither commuting nor a four-vector. We note that the commuting four-vector coordinates of Jordan and Mukunda [28] can be written (B. 17) In Section 3, we give a prescription for augmenting the Pryce system of operators x, S and P by including the angle operators Rij in the algebra. APPENDIX

C: TRAJECTORIES AND THE DIFFERENTIAL EQUATIONS

In this Appendix, we develop a method for finding solutions of the differential equations (1.36)-(1.38) which we require the Lagrangian of the spherical top to

SPHERICAL

561

TOP

satisfy. To solve the equations, we can begin by specifying the boundary conditions the surface a3 = a, = 0 corresponding to a given trajectory function. In particular, we give the function

on

Lo@, 9 a2 70, 0) = (a,)l’2 2[[,

0,O).

This function is known once the trajectory function M2 = f(J2) is given. In fact, from (1.51), we have 215 = 2a,/a, = (dJ/dA4)2 == #(Je)

cc. 1)

where #(J”) is a known function of J”. Therefore by inverting this equation, we find a function 4(t) = J2 so that M2 = f($([)). From Eq. (1.53), we have J 2(2tp ---=z

a9

-= x

1

445) 1’2 ( 2f 1

(C-2)

Therefore we assume that &,(a, , a2, 0,O) is known once the desired trajectory is specified. We now introduce new homogeneous coordinates xi, i = l,..., 5 such that ai = xi/x, ,

i = 1, 2, 3,4.

In terms of these variables, we find

(C-3)

Thus Eqs. (1.36) and (1.37) may be written 2x,x, + h,X, = 0, (C.4)

h,X, - 2h,h, = 0.

In x-space we now consider a family 2 of surfaces xi(a, differential equations axipo

=

v~(x),

axflaT

Tj determined by the (C-5)

== wicx),

where d4

= A, 7

u2(x) = -2x, ) %(X) = Al ,

us(x) = -2h,

Q(X) == 0

)

562

HANSON

AND

REGGE

and WI(X) = 2x, ,

w*(x)

= 2x, ,

w3w ws(x) = A, .

wg(x) = 0,

= A, 3

A surface of the family lies on an L, = constant hypersurface. in terms of ~~(0, T), we find

If we express L,

2 = $ hiVi = 0,

by virtue of (C.4). We also find aux(u~ au

41 = $ vi a~~~=oxl

a*L, I h

=

h43g’

= &

a2L,

2h

1

a3L,

2Tg-2gax,

lax,-

2x a% 3 ax,ax,

@,A, - 2h,h,) = 0.

Similarly,

ahi/aa = axi/ar = 0

K-6)

for all i. Obviously we then have

avijaa =

aw,pa

=

avilar = awilaT = 0.

Notice that av,/& = awi/aa is trivially satisfied and therefore (C.5) is completely integrable, or else(C.5) would not define a surface. Moreover,

a2xi/ao2= a5qar2 = a2x,lao a7 = 0,

K.7)

and the xi must be linear in (J, 7 so that xi = pi + vi’s + wir where pi , vi and wi do not depend on D, T. Notice that ~~(0, 0) Vi(X) = Vi(P),

since vi, wi are independent of 0, T.

w&4

= w@>

(C-8) pi, so that

SPHERICAL

563

TOP

Through each point in x-space, there is only one surface belonging to our family of surfaces. Consider the 3-dimensional manifold JY defined by xQ = x4 = 0. Through each point P of J&‘, there is exactly one surface E(P) of the family. We now choose the point pi to lie on &’ so that p3 = p4 = 0. In this case, if we know L,h , P:?, 0, 0, PA we know 4 , h, , X, and, from (C.6), we know also &, h, . Hence we know z&), wj(p) and the corresponding surface through P is determined. Now suppose we are given a generic point Q, not necessarily on 4, with generic coordinates xi . If xi = pi + w(p)

+ TWi@)

(C.9)

for some u, T, then Q is on Z(P). But then, since L,, is constant on Z(P), we have Lo(Q) = 4#‘). The solution procedure now runs as follows: We give xi and solve the nonlinear system (C.9) of 5 equations in the 5 unknowns, p1 , pz , p5, u and 7, in terms of the xi. Therefore (C.9) defines pr(x), pg(x), p5(x), u(x) and T(X). Finally, we recall that

is explicitly &&

given by specifying the trajectory 3x2 9 x3 , x4 3 x5> = -w)

function.

= (Plc41P5wY2

Therefore =%3(x)/&>,

0, 0)

is the desired solution. Since Eq. ((2.9) may have more than one solution, there may be different Lagrangians corresponding to the same trajectory. We now turn to a discussion of the implicit relation between M2 and J2. From (A.17) and (A.l8), we have M2 == cw3/~2)W,

+

J2 ==2&&,/~,)(~,~, +

a3L3)

=

J%~,l~,)(xJ,

a,L,

+

2Q4L4)

+ =

x,h,),

2L,(h,/&)(x,&

+

x,h,

Consider now a surface of the family 22 through the point P. Setting it’f2(U, T) = J2(o,

T)

M2(Xi[U,

= J2(xi[a,

T]), T]),

we find that

and similarly aM2/aT = aJ2/aa = aJ2/aT = 0.

+

2X4x4).

564

HANSON

AND

REGGE

(Recall that L, and Xi are constant with respect to u, 7.) Therefore on Z(P), M2 and P are constants. But M2 and J2 are functions of the reparametrizationinvariant variables 4 = a21al ,

6 = a,la12.

7 = a3/a12,

On Z(P), t, 7 and 8 are also functions of 0, T. One can then let 1~~= u, o2 = T and compute

But now Eqs. (C.10) and (C.11) can be considered as two identical linear systems in three homogeneous unknowns, aM/&f, aM/i3y, aM/i30 and aJ/C$, aJ/aq i3J/ae respectively. The coefficients are the same in each case: @/‘laoi , a@o, , a8/aai . Therefore the corresponding solutions are proportional and we have aMla[

= y

aqag,

etc. We conclude that the Jacobians vanish, aM aJ ------=, at av

aM aJ 3 at

o

cyclic in c, 7, 8.

But this means that there exists a functional relationship M2 = f(J”),

with the same function f(J2) as was given on the manifold A, but which is valid at any point Q, insofar as there is always a surface S(Q) through Q and meeting & at some point P. A particular choice for L? is 2 = (A - Bly’2 SO

Lo = (Au, - Bap2. Using the techniques given here, one can reconstruct the general L, with the result Lo2 = (1/2)(Aa,

- Baz + [(Aa, - Ba2)2 - 8B(Aa, - 2Ba4)]1/2)

where BM2 - (l/2) AJ2 = AB

565

SPHERICAL TOP is the trajectory.

This choice of L, can be seen after a short calculation to satisfy the differential equations (1.36) and (1.37). The actual construction of L,, for a given trajectory is difficult to carry out in all but the simplest cases. The trajectory function can be viewed as the envelope of a family of curves in the M-J plane specified by 9 and parametrized by 5. A simpler example of an envelope than those given so far is provided by setting 9(t) = t/2 in Eq. (C.2). We then find a family of straight lines in the M-J plane given by the equation (Q/2) + M - (5/2)‘l” J := 0. The [-derivative

is then (l/2)( 1 - J(2&‘/“)

= 0.

When one eliminates 8 from these two equations, one finds that the envelope of the family of lines is just the parabola J2 = 4M.

ACKNOWLEDGMENTS A number of our colleagues have provided useful suggestions and information, and we wish especially to thank V. Bargmann, B. Durand, P. Ramond and A. Wightman. We are also grateful to G. Ponzano for his kind assistance with the differential equations and his thorough comments on the manuscript.

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