relativistic hamiltonian dynamics i. classical mechanics

ANNALS OF PHYSICS 117, 292-322 (1979)

Relativistic Hamiltonian Dynamics,

I. Classical Mechanics

F. ROHRLICH

Department of Physics, Syracuse University, Syracuse, New York 13210

Received April 19, 1978

A new Hamiltonian theory is proposed for the dynamics of relativistic systems. It is presented here by means of a closed system of N spinless point particles. The potential energy may depend on the velocities and is assumed to be a sum of two-body interactions. The dynamical variables are chosen on the basis of realizations of the Poincark algebra 8. Only manifestly covariant realizations are considered here; the position four-vector of the whole system is not in the enveloping algebra of 8. The canonical formulation requires an 8N + &dimensional phase space and appropriate first-class and second-class constraints. Reduction leads to noncanonical algebras for the variables in 6N + 6- and in 6N-dimensional phase spaces. Only the latter do not admit a description of the world- lines of interacting particles (but only of free ones.) This incompatibility between the description of individual particles and the presence of interaction is proven to be the reason for the no-interaction theorem by Currie et al. which holds when individual particle variables are used. Interactions linear in the velocities lead to Lorentz-type forces between the particles. Applications of the theory to other physical systems are indicated. Canonical quantization is possible and will be presented in a future publication.

1. HISTORY OF THE PROBLEM

The special theory of relativity has come of age; it has passed 70 years, and it has become an integral part of physics. Yet it is still incomplete in at least one very important respect.

Of the various formulations of dynamics, Newtonian, Lagrangian, and Hamiltonian, the most fruitful seems to have been the Hamiltonian formulation. It permits that heuristic induction procedure called “quantization” which allows one (at least in many cases) to “derive” the quantum-mechanical description of a system from its classical description.

Special relativity still does not have a Hamiltonian dynamics. To be sure, there have been many attempts in this direction and we shall mention some of the most important ones below. But it is evident to everyone acquainted with the literature that the theory is in this respect in rather poor condition. Apart from the relatively trivial one-body problem we look in vain in such encyclopedic works as Pauli’s relativity article [l J or in the standard textbook literature, For example, a relativistic analog of the Newtonian reduction of the two-body problem to the effective one-body problem, or the separa-

292 0003-4916/79/020292-31$05.00/O Copyright 0 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.

RELATIVISTIC HAMILTONIAN DYNAMICS 293

tion of the center-of-mass motion of a closed N-body system with given internal interaction, does not seem to exist.

The present article is presented as a proposal to fill this void. We shall present a manifestly covariant, canonical, relativistic dynamics for closed N-particle systems in which the N particles are in interaction with one another. This dynamics can be quantized as we shall show, and it can also be extended to continuous systems of finite extent.

But the work to be presented was made possible only by the many previous efforts that either failed or proved unsatisfactory in one respect or another. I have learned greatly from them.

The lack of a relativistic Hamiltonian dynamics has encouraged studies based on the Tetrode-Fokker action principle [2]. Further encouragement in the same direction was provided by Wheeler and Feynman [3] who showed that electrodynamic interactions, usually described by fields, can also be cast into such an action principle. And when the “no-interaction theorem” (see below) was published by several authors in the mid-sixties, the Wheeler-Feynman dynamics was generalized further by Van Dam and Wigner [4]. Unfortunately, the well-known methods of canonical quantization cannot be applied to these classical theories.

When one tries to generalize the separation of the center-of-mass motion of Newtonian systems to special relativity one immediately encounters ambiguities. The definition of that center of mass is not unique and a variety of physically reasonable possibilities exist [5]. Thus, the question of the position of the system as a whole was raised and was combined with the much more difficult question of position opera- tors in relativistic quantum mechanics. The quantum-mechanical position operator proposed by Newton and Wigner [6] corresponds to one of the choices suggested by Pryce. It is one in which the three-vectors of position and associated momentum are rotation covariant and satisfy canonical bracket relations; but these three-vectors are not the space-parts of Lorentz four-vectors. Their adoption thus necessarily leads one to discard a manifestly covariant formulation. In fact, of the six alternative position variables defined by Pryce only one transforms as a Lorentz four-vector,

The first clue to a relativistic Hamiltonian dynamics was given by Dirac 171. He pointed out that one must choose a particular realization of the Poincart algebra in order to determine the appropriate variables. The three choices he elaborates are the instant form, the point form, and the front form. The instant form is the conventional time development using spacelike hyperplanes in Minkowski space; the point form is based on mass hyperboloids; the front form uses null hyperplanes, was introduced into particle theory by Weinberg [8], and plays an important role in high energy approximations as the “infinite momentum frame” formulation.

What has not been fully recognized, however, is the fact that for each of these “forms of dynamics” there exist still many alternative realizations of the Poincare algebra and, correspondingly, there exist many alternative choices of variables. This subject will be taken up in the following section.

An explicit realization of the Poincart generators for a many-particle system was used by Bakamjian and Thomas [9]. These were expressed as functions of the three-

294 F. ROHRLICH

vectors of position and momentum of the “center of mass” and of the three-vectors of relative (internal) position and momentum of the particles; all pairs of position and momentum satisfy canonical bracket relations with each other. The variables result in the free case from a contact transformation starting with the individual particle positions and momenta. The “center of mass” is here defined as the canonical conjugate of the total three momentum of the system. This scheme seems to carry through, at least for a certain class of interactions, but leads to considerable algebraic complexity.

However, the work of Bakamjian and Thomas was seminal to the development of the subject since it contained all the difficulties, explicitly or implicitly, which occupied the attention of most authors in this field since that time.

(1) The choice of variables. In BT the individual particle variables (symbolically denoted by q, p) are replaced by the variables for the system as a whole and by the relative variables (collectively denoted as the CM variables). The transformation between these is not unique and, in fact, not clear when interaction is present. Which variables are better ? The representation of the generators in terms of the CM variables seems to depend on the interaction. This would be an argument against the CM variables.

(2) Invariant world-lines. It was assumed by BT that their formulation would not permit an invariant world-line description; the internal variables do not keep track of each world-line in a covariant way. This assumption was elevated to a theorem by Currie and others [IO] to the effect that if the q, p variables are canonical and transform correctly under Lorentz transformations, then no interaction can exist between the particles. This “no-interaction theorem” thus does not permit the canonical q and p variables if the world-line is to remain invariant. But since the CM variables are related by a canonical transformation to the q, p variables, neither of these can be maintained in interaction with invariant world-lines. This appears to be a very serious difficulty. A selection of typical papers that addressed themselves to this question were edited by Kerner [l 11.

(3) Cluster decomposition. When the closed system of N particles separates into NI + NZ particles such that all NI particles are far enough removed from all NZ particles then each of these two clusters must be describable as a closed system. This property seems difficult to satisfy in BT. Foldy [12] and others have constructed suitable interactions which would permit cluster decomposition and for this purpose have used the realization of the Poincare generators that corresponds to the Newton- Wigner definition of position in quantum mechanics [6].

The literature on this subject is much too extensive to do justice to it here; nor is this the place for a full review. Suffice it to say that to order (v/c)” it has recently been possible to satisfy both the cluster decomposition and the invariant world-line condition for interacting particles [ 131.

In a recent paper Fronsdal [14] tries to get around the no-interaction theorem by using noncanonical bracket relations. But he also encounters the ambiguity problem of relating the q, p and the CM variables.


An important technical tool has not been exploited to the advantage of the present problem until very recently. It is the theory of Hamiltonian systems with constraints which has been known to general relativists for a long time [ 151. A spinning particle has been treated in this way by Hanson and Regge [16]. We shall see that this tech- nique is indispensible for a manifestly covariant Hamiltonian dynamics.

Constraints are also used in a recent work by Todorov [17] in order to ensure that the mass shell condition be satisfied asymptotically for each particle, as it escapes the interaction with the rest of the system. It seems to lead to a restricted class of allowed interactions. Upon quantization it leads to the quasi-potential equation for the two- body problem which he and others have studied previously [ 181. For the two-body problem this work is related to the recent work by Komar since both use two first- class constraints (see below) [19], but the problem of cluster decomposition seems to persist in the latter’s papers.

Section 2 is devoted to the choice of variables, the first of the above list of problems. Section 3 gives a manifestly covariant action which leads to a canonical Hamiltonian that then must be constrained by a suitable number of constraints. The constraint formalism will dominate that section. It will lead to an 8N + 8-dimensional phase space which is reduced to a 6N + 6-dimensional one in Section 4. The relation to the no-interaction theorem of results thus obtained is then explored in Section 5. It will be shown that one can have invariant world-lines (problem 2 above), canonical CM variables, as well as the no-interaction theorem, in complete harmony with each other. The key to the solution of this apparent contradiction lies in the fact that, as we shall demonstrate, no transformation exists,from the CM to the q, p wriables unless there is no interaction. This result clarifies a lot of past confusion. The theory is generalized to velocity-dependent interactions in Section 6. Lorentz-type forces can thus be obtained. The last section is a summary of the results obtained. The Appendix contains realizations of the Poincare algebra.

2. THE CHOKE OF VARIABLES

The history of the subject has shown the importance of the “right” choice of variables. We shall be guided in this choice by three types of considerations: physical, mathematical, and simplicity considerations.

Physically, a spinless particle is canonically described by a pair of three-vectors qu 2 Pa 9 i.e., by six variables, where a labels the particle (a = 1, 2,..., N). The three- vector notation ensures an easy way to keep track of rotation covariance. In a relativistic theory we must also have Lorentz covariance. Considerations of simplicity suggest four-vector variables. This simpficity is well known from formulations of special relativity; and the large number of papers on our subject matter that use three-vectors and that are then forced to contend with unreasonably complex algebra, as for example BT, supports this point. But if the pair of three-vectors q,, , pu for particle a is replaced by a pair of four-vectors, qau , pau , then we have eight variables instead of six. This implies that a manifestly covariant formulation must have two constraints for each

296 F. ROHRLICH

particle to make up for the extra variables qaO and pa0 which we added for simplicity. We would lose this simplicity again if we were to eliminate these extra variables. But if we keep them as a canonical pair, we are forced to use the constraint formalism of Dirac and others [15]. The theory is simpler if one uses more variables and suitable constraints than if one uses the minimum number of variables. This argument speaks against the Newton-Wigner choice.

The Poincare generators Pa and A4fiv can be expressed in terms of the q and p variables by

P” = -f pa’“, MuY = C (qn”pav - q,“p,‘9 = C (qa A paw (2.1) a=1 a a

The Poincare algebra 3,

{Pfi, P”> = 0, {p”, MD”) = ~“‘p” - ~“pp”, (2.2)

with trace 7 = $2 will be satisfied as a consequence of the covariant canonical bracket relations (CBR)

It is understood that q and p are time dependent. The algebra (2.3) must therefore refer to “equal time” whichever way this will be defined.

This realization of 9 is, however, suspect. The. no-interaction theorem was proven for a three-vector realization, but there is good reason to expect that this four-vector realization may also lead to a no-interaction theorem, the particle positions being treated as canonical variables.

At this point we recall a lesson from long ago. Copernicus taught us that the description of a system of N bodies is much simpler when their motion is referred to their center rather than to some arbitrary point. (For Ptolemy that arbitrary point was one of the bodies and he surely was forced to contend with a lot of complexity.) In Newtonian physics one uses the center of mass. In special relativity a suitable center must therefore be defined. This is an argument of simplicity.

From Pryce [5] we learned of the ambiguity of defining a center. But his result has only one definition of the center which is a Lorentz four-vector and we are thus tempted to adopt it. But a simple mathematical argument shows that none of his definitions can be correct for the purpose of implementation of our present philo- sophy. If the center hasposition four-vector Qu and momentum P@, the total momentum of our closed system, then the canonical bracket relations (CBR) must be

p, P”} = y, {Q”, Q”, = 0. (2.4)


The third relation, commutativity (in the sense of Poisson brackets) of the components of P, is already ensured by 9, (2.2). But the first relation (2.4) implies

{Q”, P”: # 0,

i.e., Q does not commute with one of the Casimir invariants of Y. Consequently, Q cannot be in the enveloping algebra 9 of 3’. But all centers proposed by Pryce are constructed in terms of elements of 3.

This mathematical argument thus leads us to the following deeper question: which realizations of .f lead to a position Q which (a) is a Poincart four-vector and (b) is not in 9 7

This purely algebraic problem has a unique answer. But first we must specify exactly what we mean by a realization of 9. The antisymmetric tensor Mu", the Lorentz generators, can be expressed without loss of generality in the form

This form has the obvious physical meaning of a decomposition into orbital and spin angular momentum, the latter being presumably due to the internal structure of the system. in the case of spinless particles S is then the total internal angular momentum resulting from a superposition of all internal orbital angular momenta.

By a realization of the PoincarC algebra 3’ is meant an algebra of the Q, P, and S (QRS algebra in short) which induces the Poincare algebra (2.2) via (2.5). For each realization there exist representations associated with specific values of the Casimir invariants, i.e., the system as a whole is given by an irreducible representation [M, S] of the Poincare group. Here we restrict the considerations to M ) 0.

One can classify the various QPS algebras on the basis of four criteria [20], Depend- ing on whether or not these are satisfied one obtains 24 = 16 different classes of realizations of 3’. These criteria are

(a) Q belongs to the enveloping algebra of .Y

(b) S is translation invariant

(c) Q is a Lorentz four-vector

cd) Q is local (its components commute with one another).

One observes that this classification is necessarily complete being based on dichotomic choices. Every realization belongs to one or the other of these 16 classes. This holds in particular for the recent canonical realizations discussed by Pauri and Prosperi [21] and by various other authors quoted in their extensive tist of references.

A covariant canonical formulation must insist on (c) and (d), while (a) cannot hold. This leaves (b). The mathematical result can here be made plausible by a physical argument.

The physical significance of the decomposition (2.5) as indicated above is a separation of the dynamical variables into those referring to the system as a whole, Q, P, and internal variables which make up S. Since the interaction between the particles should

298 F. ROHRLICH

not affect the uniform motion of the system as a whole, the internal variables and the system variables should be independent of one another. They must therefore commute; (b) must hold.

It is not difficult to prove that it is impossible to satisfy all four requirements (a)-(d). The three most important realizations satisfy three of the four and are listed in Table I. They are labeled CCR, the covariant canonical realization, CNR, the covariant noncanonical realization, and NWR, the Newton-Wigner realization.

TABLE I Three Realizations of B

CCR CNR NWR

QEB no {S, P} = 0 Yes Qfi Lorentz Yes tQ“, Q”) = 0 Yes

Yes

Yes Yes no

Yes Yes no yes

We shall adopt the CCR. The CNR corresponds to the only Lorentz covariant center suggested by Pryce. It is canonical only in the hyperplane orthogonal to P. While it could also be used for a Hamiltonian formulation, it seems to lead to more complicated expressions. The NWR is also one of Pryce’s centers and was used by Foldy [12] in a particular representation and by many others. It leads to considerable complexity. The QPS algebras for all three realizations were given explicitly in a recent paper by this author [22]. The CCR algebra involves the following nonvanishing brackets:

{Q”, P’) = y”, {SW, SW} = qws”o + ,,““SUP _ ,,uos’” _ ?7v~Su”; (2.6)

all other brackets between the components of Q, P, and S vanish. The internal variables must now be introduced explicitly. Again, these should be

canonical pairs of four-vectors if we wish to have a manifestly covariant formulation. Let us call them 4,~ and n,“. They must satisfy the CBR:

Gus %“) = 3”” &iJ 3 (iTGU. m = 0, h,@, %9 = 0. (2.7)

The algebra (2.6) of S can be implemented if it is defined in terms of the internal variables by

P” = c (& A 7ra)uv (2.8) a

as can easily be verified. The vanishing of the brackets of S with Q and P will be ensured by

{Q”, L”) = 0 = tQ”, ~a”>, {P, .$,y} = 0 = (P”, Tr.‘}. (2.9)

One also verifies that the .$, and rra are Lorentz four-vectors, i.e., they satisfy the same bracket relations with M as P does.


In making this choice we have restricted our attention to spinless particles. Addi- tional degrees of freedom are required to describe intrinsic angular momenta of particles [23]. We have thus restricted the following consideration to a particular set of representations.

This way one obtains N + 1 pairs of four-vectors, Q, P, and 5, , VI=~ (a = 1,2,..., N), i.e., a total of (2)(4)(N + 1) = 8N + 8 variables. The number of independent degrees of freedom is 6N. We shall need 2N + 8 constraints. These will be introduced in the next section.

How are these CM variables related to the 4, p variables ? The positions are clearly related according to Copernicus: at any one instant of time (to be specified later) the position of particle a is to be found by first finding the center of the system and then following the vector of internal position (relative to the center). We thus make the Ansatz

9a u = Q” + fau. (2.10)

A similar linear relation betweenp, , P, and 7ra may, however, not exist. The momenta are dynamically related to the positions so that the wanted relation would emerge by time differentiation of (2.9) and insertion of the relationship between the momenta and the velocity. This is surely an interaction-dependent relation.

But (2.10) suffices to specify the world-line of particle a; it is given by a Lorentz four-vector which is the sum of two other Lorentz four-vectors. The “invariant world-line” requirement can thus be satisfied if (2.10) can be proven to hold. This condition will be answered in Section 5.

Finally, one observes that (2.10) is part of a transformation that relates N + 1 pairs of four-vectors to N such pairs. It, therefore, cannot be expected to have an inverse: the expression of Q in terms of the qa and pa may not exist! An explicit expression of the center position Q in terms of the individual particle positions qa, therefore, cannot be expected. Section 5 will clarify this point.

3. CANONICAL DYNAMICS WITH INTERACTION

Any interaction V between the N particles which leaves the system closed and the motion of the system as a whole undisturbed necessarily depends only on the internal variables. In this section we assume, in addition, that it is independent of the nTT, . But since V cannot depend on the choice of origin, only a dependence on the difirences of the {, can be tolerated; .$= - tb = qa - qb . We write symbolically

and if we further assume an interaction of particle pairs only,

This expression ensures the scalar transformation property of V.

(3.1)

300 F. ROHRLICH

We already know that the canonical structure of N + 1 four-vector pairs will require constraints. These constraints cannot be allowed to hold in the 8N + 8- dimensional phase space because they would then contradict the CBR. The constraints will hold on a subspace only, viz., that subspace to which the physical trajectories are restricted. This subspace is called the constraint hypersurface because it is in fact determined by the constraints. Mathematically, the constraints are functions of the Q, P, & , and V, (a = 1, 2 ,..., N) and thus define that hypersurface [15],

GtQ, P, 6 4 - 0 (i = 1, 2 ,..., NC>. (3.2)

A special equality sign (“weak equality”) indicates that the Ci vanish only in that subspace. Of course, we will have to require that the dynamical development of the system preserves the constraint hypersurface, i.e., preserves each constraint: the bracket of each Ci with the Hamiltonian must vanish weakly,

{Ci, H} M 0. (3.3)

Since the C, vanish only weakly, they cannot be equated to zero until after the bracket has been computed.

If one wishes to start with an action principle, i.e., a Lagrangian L that leads to the canonical formalism just indicated, one must have L depend on N + 1 independent four-vectors Q, 4, together with their time derivatives. And one must have N, constraint relations. For heuristic reasons we shall start with such an action and derive from it the canonical Hamiltonian. But we shall not attempt to find the N, Lagrangian constraints. These will be provided only in the canonical form, i.e., in terms of positions and momenta rather than in terms of positions and velocities.

In writing down a Lagrangian action one must specify a time parameter. In fact, in relativity one often specifies separate time parameters for each particle world-line. Typically, in terms of the variables q&i,), the kinetic energy term would be

-pl( -P2”(~a))1’z 4 , (3.4)

where the dot indicates differentiation with respect to Aa . This expression exhibits a gauge invariance: N arbitrary functions

AZ -+ x = faOa> (a = 1, 2,..., N)

are involved in this invariance group. Complete generality favors such an Ansatz. However, this is undesirable for a Hamiltonian formulation, When interactions are

added, this invariance is broken; in fact, the only way to preserve it would be to introduce double integrals for the interaction terms as is the case in the Wheeler- Feynman action [3]. This would prevent a Hamiltonian formulation.

But beyond this the Hamiltonian formulation requires one single congruence of hypersurfaces in Minkowski space which label the successive stages of development of the dynamical system. In special relativity these are most commonly and most


naturally chosen to be hyperplanes. We shall choose them to be spacelike hyperplanes corresponding to Dirac’s [7] instant form of dynamics. Intuitively, these are certainly the easiest to work with.

The hyperplanes are characterized by an invariant parameter T, the proper time of the inertial observer who moves in a straight world-line orthogonal to the hyperplanes. This is the characteristic picture of Hamiltonian dynamics as described by one single inertial observer. It fohows that the action integral refers to only one single time.

Consequently, the action integral will have the form

A == s d-r UQ, 8; k, 6. (3.5)

But in order to make the correct statements concerning the kinetic energy in terms of 8 and the da one must note that according to (2.9) the fa are relative positions: .$, is a spacehke vector which (at least on the basis of physical intuition) would lie in the hyperplane 7 and whose dynamical changes in time will keep it that way. This means that gn is also a spacelike vector. In a suitable reference frame it will have no component t,“. Therefore, i, corresponds to the three-velocity v, and not to the four- vetocity v,u; its square will have a value between 0 and 1 and will not be a constant fixed by the choice of the time gauge. This situation is in marked contradistinction to Q, whose time derivative is the four-velocity of the center motion, Consequently, we make the following Ansatz for L:

L = gA4p - f m,( 1 - &2)1/z - V. (3.6) a-l

ln the standard way one finds

P” = MQ”, nau = m,.$,ul(1 - <,2)112

which can easily be solved for the velocities

Qu = PJqM, jaw = ~~~J”l(n-,~ + ma2)l12 (a = 1, 2 )...) N). (3.7)

The canonical Hamiltonian is, therefore,

H ran = PQ + c na . i, - L = & + c (n-,2 + m,y + v. (3.8) a a

This is, of course, just the result one expects. But we note that P and 7r, are four- vectors.

Now the constraints must be specified. One of these is very easy to write. It is just the statement that the system as a whole behaves like a single particle with momentum P and rest mass M,

c z po - (P” + My2 M 0. (0

The remaining constraints have to do with our choice of the spacelike hyperplanes

302 F. ROHRLICH

which parametrize the dynamics. There is, of course, only one inertial observer which is preferred over all others in a relativistic sense: the observer who is comoving with the center. He is characterized by the timelike unit vector

Pu E pq - Py. (3.9)

The spacelike planes labeled by 7 must have this vector as their normal if they are to be invariantly characterized.

An invariant constraint must therefore be defined relative to p, i.e., relative to the “center of momentum” frame (CM frame). In that frame the interaction takes place instantaneously between all particles. The internal variables 5, and ra are three- vectors in that frame and have no components Sao and n,O since these are variables relative to the CM frame observer. This requirement of 6, and rG to lie in the hyperplanes orthogonal to P can only be made weakly since 4,” and ~~~ are independent canonical variables. Thus, we must have the constraints

c,= rr,*PR50

c;5$/PPo (a = 1, 2 ,..*, N).

Since the total momentum of the system is P the sum of the relative momenta must vanish. This is trivial for their components parallel to P because of (C,). But for the remaining three components we must have (I is relative to P),

In the CM frame the three constraints (C,), (Ci), and (CL“) reduce to

t-a0 w 0, s?,O m 0, pZ,,-O (CM frame). (3.10) a

Constraints fall into two classes, first-class and second-class constraints. First- class constraints commute with all other constraints; second-class constraints do not commute with at least one of the other constraints. In our case the constraints (C) and (C,u) are first class, (C,) and (Ci) are second class.

The general Hamiltonian for a system with constraints consists of H,,, and a sum over all the constraints (since they vanish on the physical subspace, the constraint hypersurface) with coefficients wi . These wi can be arbitrary functions of the variables [15]. Thus,

H = Hcan + WC + c ~6, + ~4C; + w,,LCl” (3.11) a a

with Hcan given by (3.8). The conservation of the constraints, Eq. (3.3), must now be ensured. For C this is

easily checked; one has


The brackets may not vanish since the wi may depend on Q. Similarly, one has for C,‘”

(CLfJ, H} w {C,“, H,,,) = {CL&, V) e 0 (3.13)

smce v,~ and .$G ,, commute. The proof of the Iast (weak) equality proceeds as follows. The separation of vectors like n, into components parallel and perpendicular to P is easily established by means of the projection to the plane perpendicular to P,

so that

(3.14)

Therefore

The last (weak) equality is a consequence of (Cl) since the omitted term is proportional to [,, . P. But our system is closed; the net force on the system vanishes; therefore,

This equation is a direct consequence of the structure of V, Eq. (3.1). This concludes the proof of (3.13).

The conservation of the remaining constraints yields

since 5,,, and 7~~~ commute. The first term is proportional to 7~, P and therefore vanishes weakly. But the second term vanishes only if

w, m 0 (all a).

In the same way one finds that C, is conserved only provided

(3.16)

w; R? 0 (all a). (3.17)

The conservation of the second class constraints thus fixes the arbitrary functions w, and WA. But w and We@, the coefficients of the first-class constraints, are still arbitrary.

The equations of motion for the CM motion are

pg = (P”, H) m 0 and

e” = {Q”, H} M P”IM + (Q”, C>w + {Q”, C,: wL .

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The second term does not vanish, but the third one does;

(-P”)‘~“{Q~, C,“} = PT”c 7ra * P + 1 Tr:p w 0. a a

We can now choose the gauge in which T is the proper time in the CM frame,

@‘ M PuIM, s” w --I, (3.18) by choosing

w M 0. (3.19)

The center motion is thereby completely determined;

Q‘(T) = Q‘fO) + TP”/M.

The interesting dynamics is, of course, the internal dynamics;

(3.20)

(3.21)

In the CM frame this means

One concludes that in that frame 8, can be identified with the three-velocity only when the coefficients oL vanish, i.e., the physical interpretation requires

WI u R3 0. (3.22)

Thus, all the wi vanish and we can take

H = Hcan .

The internal dynamics is, therefore,

and

These two equations can be combined to

(3.23)

(3.24)


or, by expressing rra2 in terms of ga2,

d (

Ina& 1

%V

z (1 - gay2 = - at,, (a : I, 2,..., N) (3.25)

This is the four-vector generalization of the usual equations of motion in the CM frame. Equations (3.25) are, of course, equivalent to the canonical equations (3.23) and (3.24).

We must now check that the number of degrees of freedom for our N-particle system is indeed 6N. To this end we recall that the theory of Hamiltonian systems with constraints [IS, 241 teaches us that a first-class constraint is like two constraints since it affects a canonical pair of variables. Thus, counting the first-class constraints twice we have the following number of contraints: C and C,w are 2 and 6, respectively; C, and Ci are N each. We thus have 2N + 8 constraints. This is exactly the correct number that reduces the 8N + 8 variables to 6N degrees of freedom.

In the theory presented so far no mention was made on how the constant M is to be determined. Clearly, it is the mass equivalent of the total energy of the system when it is at rest. Thus, we have one more weak equality,

M m z (ra2 + n1,2)~12 + V. (3.26) a

or, equivalently,

& + Hcan m 0. (3.26’)

This is not a constraint, but a determination of A4 on the constraint hypersurface. It is the classical equivalent of the Schriidinger equation in the CM frame of the system.

In order to see that (3.26) is not an additional constraint but only the specification of M which occurs in C, one can proceed as follows. Instead of the Hamiltonian (3.8) which was derived from a Lagrangian one can base the theory on the Hamiltonian Ansatz

fj,,, = -(-py + c (r,2 + ))),2)1,‘:! 2 v (3.27) a

independent of M. The constraint (C) is now replaced by

R,,, 25 0; (3.28)

the other constraints are unchanged. One now observes that (-Pz)rjz is a constant of the motion and so, therefore, is the right-hand side of (3.26). This constant has the numerical value M. The discussion of constraints and the equations of motion based on (3.27) and (3.28) leads to the same dynamics as before.

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4, REDUCTION TO THE MINIMUM NUMBER OF VARIABLES

The canonical Hamiltonian dynamics presented in the preceding section involves 8N + 8 variables for a system of N spinless particles. If first-class constraints arc counted as two constraints each and second-class ones counted as one constraint each, the system has 6N degrees of freedom. In the present section we shall investigate the problem of the elimination of the constraints so that the system is eventually described by only as many dynamical variables as there are degrees of freedom.

Second-class constraints (like C, and CA) can be removed by a standard procedure due to Dirac [24]. This method modifies the BR but does not change the dynamical variables. The modified BR are such that they are compatible with the constraints when they are extended from weak to strong equations. These strong equations then permit elimination of some of the variables: one canonical pair of variables for each canonical pair of second-class constraints.

First-class constraints can sometimes be replaced by canonical pairs of second-class constraints so that the above procedure can also be used here. But not all first-class constraints permit such a replacement. Those that do not most be removed by use of a different (smaller) set of variables which commute with the constraints. We shall encounter all these situations.

First we replace C by a pair of second-class constraints. We begin by casting C into a different form. It consists of two parts. One is the mass shell condition

crp2+wmo (4.0

and the other is the positive energy condition

PO> 0. (4.2)

We shall find an appropriate C’ for (4.1) and incorporate (4.2) in C’. One must turn to physics again in order to “guess” the constraints to be added.

As mentioned earlier, the dynamics takes place on the hyperplane while all components of four-vector variables which are perpendicular to the hyperplane undergo only uniform motion or are constant. The following reduction to 3N + 3 variable pairs thus consists essentially in the eleimination of all these components, i.e., of all projections of four-vectors onto P. These projections are all linear functions of 7- or constants.

The projection of P itself is given by (4.1). This suggests that in agreement with (3.18) and (3.19) the constraint

(4.3)

be added. The term MT is simply the negative of the value of Q * P in the CM frame,

Q 'PjCM = -Q"PojCM m -TM. (4.4)


The origin of the vector Q is thus chosen to lie in the T = 0 hyperplane. The same conclusion can be reached from (3.20) if we assume that Q (0) is orthogonal to P. We note that condition (4.2) has here been incorporated into (4.3) by our choice M > 0.

Similar to (3.1 l), the question of conservation of the two constraints (4.1) and (4.3) must be based on the Hamiltonian

One finds

H = H,,, + WC + w’C’. (4.5)

and

t = {P”, H) s -2w’P? (4.6)

In order for C and c’ to vanish weakly one must have

w’ 25 0 and w e 0. (4.8)

The constraint terms in (4.5) will, therefore, not contribute to the weak equations of motion. The replacement of the first-class constraint (4. I) and (4.2) by the two second- class constraints (4.1) and (4.3) leaves the equations of motion invariant.

We shall now use this pair of second-class constraints together with the N pairs

c, = Tr, . P w 0, c:, r+ .& . P .a 0 (4.9)

of the last section to eliminate N t 1 pairs of the 4(N I- 1) pairs of CM variables Q, P, 5,) and T,.

In order to achieve this, the CBR must be modified into (necessarily no longer canonica1) BR so that the constraints commute with all dynamical variables. If that is the case the weak constraint equations become strong equations, valid everywhere in the new phase space.

One could now apply Dirac’s procedure to find these modified BR. But a simple argument will lead us to the same result; the correctness of the result can then be checked easily. The argument is as follows.

The new variables will be P, QLU = Qu - rP”/M, clL = faU, and n& = v,~. PO is given by (P2 + M2)lj2. The BR of these variables must form an algebra which is a covariant realization [20] of 8. The four criteria of realizations (see Section 2) can be specified; they must be (in the order of Table I) yes, yes, yes, no; it is the CNR of Table I. To see this we observe that when S * P = 0 and when Q * P and P2 belong to the center of the algebra then QU can be expressed in terms of P” and Mu” using (2.5) (See Eq. (A6)). Swill still be translation invariant. That the algebra has a covariant Q with noncommuting components is less obvious. But when the CNR algebra is written out (see (A5)) these relations become apparent.

308 F. ROHRLICH

The algebra of CNR is as follows:

{Q,‘, Q;: = -M”“lM2, tQ,“> P3 = PY, (4. IO)

{Q,“, cl% = &P*lM2, {Q,? S,} = &P”/M2, (4.11)

Q% 9 7r;,> = G,,Py. (4.12)

All other BR vanish. Suy is given by (2.8). One can easily verify that in this algebra all variables commute with the constraints

C, c’, C, , and Ci . These constraints can thus be taken as strong equations. With p = P/M,

P2 + M2 = 0, Q++T=O (4.13a) 7-ra * P = 0, & * P = 0. (4.13b)

The phase space is now 6N + 6 dimensional; the Hamiltonian is Scan and the equations of motion are formally identical with (3.23)-(3.25).

The only constraints on this system are now the set of three first-class constraints:

CLU = C” f 1 ?rau = 1 r& w 0. (4.14) a a

We shall now proceed to show that these constraints cannot, in general, be used to reduce the number of variables further by the above method.

It is easy to find a constraint C’* which is canonically conjugate to C”,

c’” 3 c pa& w 0. (4.15) Q

Terms involving the rra can be added to C’u but will not change the argument. “Cano- nically conjugate” means that

{C, cv} = Py (4.16)

since we are here working on the plane orthogonal to P. But (4.16) can hold only if

(4.17)

This would impose additional (secondary) constraints on 5, and rra if the pa depend on these variables. Such constraints cannot be tolerated since it would reduce the number of degrees of freedom to less than 6N. Thus, the pa cannot depend on the internal variables. For the sake of generality we shall assume, however, that they may depend on 7.

Parenthetically one notes here that (4.15) and (4.17) seem physically extremely reasonable. Combined with (2.9) they lead to an expression for the CM position Q in terms of a weighted average over individual particle positions qa ,

(4.18)

RELATIVISTK HAMlLTONIAN DYNAMICS 309

In fact, we know that this relation is correct in the nonrelativistic case where pu = rn,/xb mb and qaO = t for all a.

The Hamiltonian now has the form

H = Hcan -I- wJY i- Cq~C’“. (4.19)

Conservation of these constraints leads to

i.e.,

and

i.e.,

The term w,C” of H, therefore, does contribute to the equations of motion. But only f,, is affected,

(4.20)

(4.21)

This is of course the same result as was obtained in (3.21). Except that now the function wIU = wU is specified explicitly by (4.21). But as we have seen there, the physical interpretation requires that this additive function vanish;

6J @ = UP R3 0. (4.23)

This must hold without secondary constraints in order to keep the 6N degrees of freedom. Therefore, (4.23) must hold as a consequence of (4.14) and (4.15).

Two cases are possible:

(A) All p,, are constant and

(4.24)

Here xi is the numerical value of the dynamical variable n,. Since the p., must be constant, the rra must be constant. The particles must, therefore, be free, Y m= 0. The first term in (4.21) then vanishes and the second term reduces to (4.14).

310 F. ROHRLICH

(B) All pa are

pa = ma Cm,. I (4.25) b

In the nonrelativistic limit (7~:~ + maz)1/2 -+ m, so that the second term in (4.21) again reduces to (4.14).

If neither (A) nor (B) hold, Cl” cannot be accepted as a constraint and the first- class constraint C,fi cannot be replaced by a pair of second-class constraints. The method that removes constraints by changing the algebra of the variables has been exhausted.

Let us see exactly what prevents C, @ from vanishing everywhere in phase space. It commutes with P and with the rr,,; it also commutes with Q, ,

{QLu, CLv> = CL”PYIM2 m 0. (4.26)

The only dynamical variables which do not commute with the remaining constraints are the 5,, ,

{& , c,y) = c Py8,, = P;ly. (4.27) b

These variables are therefore not “first-class variables” (in Dirac’s nomenclature) or “observables” (in Bergmann’s) [15]. Such quantities by definition commute with all constraints. An observable is, in general, not a constant of the motion. This occurs, however, in general relativity when H is a first-class constraint; in our case H,,, F$ 0 so that we can have nonconstant observables.

Now the minimum number of dynamical variables, 6N in our case, must all be observables. It is easy to construct these. P, Q, , and 7ral already are observables. Since the right side of (4.27) is independent of a, the difference variables of position

(4.28a)

commute with the C,u and are, therefore, also observables. There are exactly N - 1 independent difference variables. Similarly, there are N - 1 difference variables of momentum

n-ll,bL = qL - rr;, .

The constraint CLU can now be taken as a strong equation,

(4.28b)

if one considers only observables and their algebra. There are now 6N observables which span this minimum phase space: P, Q,u, [Lb1 , and nib1 . Their algebra is (4.10) and,

{Q,*, f:bl) = &uPv/M2, lQ,Y dibl) = db,P’/M2, (4.30)

&%bL 9 ddI> = p?@ac - &3 + a,, - sbc>- (4.3 1)


Because of (4.29) we now have

S@” = c (f, A ?Tp = $ cc (& A 7T,ip (4.32) a a b

and it is easily verified that the algebra of observables induces the CNR of 9 just as the 6N + 6 variable algebra does.

We have thus obtained the 6N-dimensional phase space; but we have done so at very great cost: we no longer have the variables f, . This has two consequences. First, the trivial one that the equations of motion (3.23) and (3.24) should be rewritten in terms Of gabi and 7j,bL;

u “al.

C maz)l12 - (n2,, + mb2)ljz ’ i-f&l = -qp+s-)>

aelu

(4.33)

where the 7~,,~ are defined in terms of the r&l by

b

Since the i,, are observables, one can define ial by (3.23) which is consistent with (4.33) and (4.34). Thus one is again led to (3.25).

The second consequence of having lost the IJ, variables is the loss of the individual particle trajectories. Since there is no way to express the 5, in terms of the tab , the world-line expressed by qa ,

which is just Eq. (2.9) written in 6N + 6-dimensional phase space, is no longer in our 6N-dimensional phase space. The individual particle trajectories cannot be expressed entirely in terms of observables. This result clarifies the problem (2) raised in the first section. However, we shall return to this question once more in the following section.

It is important to emphasize that in 6N-dimensional phase space the internal variables tab1 and n&,l give complete infOITnatiOn Only about the relative InOtiOn of two particles but no information about the motion of any one particle relative to the center of momentum position Q: the translation invariant 7ral can be obtained by (4.34), but the translation covariant .$,, cannot be obtained unless there exists a linear dependence between the t,, such as (4.15). As was proven above, in the presence of interaction, V # 0, such a relation is not available: the individual particles cannot be localized by vectors from the origin. No world-line exists in 6N-dimensional phase space. The only exceptions occur when V = 0 or when one is describing the nonrelativistic limit. in these two special cases

(4.36)

takes its place in parallel with (4.34).

312 F. ROHRLICH

5. THE NO-INTERACTION THEOREM

Having reduced our Hamiltonian dynamics to a covariant formulation of the minimum number of variables, an associated BR algebra, and an appropriate realization of 9, we are now ready to explore the questions raised by the no-interaction theorem [lo]. That theorem is concerned with the Hamiltonian dynamics in terms of 6N individual particle variables qa , pa which satisfy canonical BR. It asserts that a relativistic system so described cannot involve interaction between the N particles. As we shall see, the essential part of this theorem lies in the individual particle variables rather than in their BR. It requires these variables to be both canonical and observable. The nature of the BR depends on the choice of the realization of 9: the no-interaction theorem in the literature uses the NWR of 9 and consequently involves CBR; we shall use the CNR of 9’ and consequently shall have noncanonical BR. The no-interaction theorem is independent of the choice of realization.

The result of the preceding section can be regarded as a “no-interaction theorem” for CM variables: a closed relativistic system of N spinless particles can be described in terms of 6N CM variables which permit localization of individual particles (world- lines) only when their mutual interaction vanishes. This fact can also be expressed in terms of covariant individual particle variables q. , pa as we shall now proceed to show.

For V = 0 the first-class constraints (4.14) can be replaced by the second-class constraints C,@ and Cp, (4.14) and (4.15) with the pa satisfying (4.17). Proceeding as before, one finds that the representation of the CNR given by (4,10)(4.12) is changed by a replacement of (4.12) by

(5.1)

the other BR remain unaffected. With this algebra the last constraints are removed, i.e., they become strong equations,

(5.2)

reducing the phase space to 6N dimensions. A transformation to the individual particle variables is now possible;

4a “=Q~+~~,=~P~+Qe,“+~~~, (5.3a)

PaU = POP -+ =L * (5.3b)

The qa , pa are N pairs of Lorentz four-vectors, i.e., 8N variables. But they are restricted by the strong equations which follow from (4.13),

qs4+7=o, p. - P + Qf = 0. (5.4)

Since these are 2N equations, the q,, , p,., phase space is also 6N dimensional.


The inverse to (5.3) is

Q” = c paqa“, P” = cp,u’, (5.5a)

<au = ;. qn”@ab

a

- pb), xa” = 1 PbU&b - &I). (5.37) b b

The CM position Q is here seen to emerge as the weighted average of the qa , the weights being the mass equivalents of the kinetic plus rest energies of each particle, as follows from Eq. (4.24). We note, however, that the numerical values of the pa are not known until the initial conditions are specified;

T” e3 7Tfu a a for each a at r = 0. (5.6)

Interestingly, a formula for Q in terms of the qa exists only for free particles. The Poincare generators emerge in the form (2.1) by substitution of the transforma-

tion (5.5) into (2.5) and (2.8). The strong equality (4.17) plays an important role. The physical meaning of relation (5.4) that restricts the four-vectors qa and pa to

only three independent variables each is easiest seen in the CM frame P = 0:

400/CM = 73 P~OICM m (P” + m la Y2/ CM * (5.7)

They express the common time components of the position variables as the CM time T, and they give the mass shell condition for each particle.

The algebra of the individual particle variables follows from that of the CM variables;

(qau, qb”) = [qF/P” - qo’PJJ - M~“]/W, (5.8a)

&“, P;) = PI;y&, + P:~PYIM~, (5.8b)

{PaU, Pb”) = 0. (5.8~)

They are obviously not canonical unless the system consists of only one particle (N = 1). The Hamiltonian for our V = 0 case is

H = Hcan = PWf + 1 (pzl + ma2)1’2 (5.9) u

and leads to the equations of motion

p,u = 0. (5. lob)

This concludes the discussion of the individual particle variables in the V = 0 case. There is a simple argument to show that the individual particle variables are un- suitable for the V # 0 case. One considers 6Sdimensional phase space in which the

314 F. ROHRLICH

CM variables P, Q, , cab1 , and rrabl do describe the system in the presence of interaction. If the qa , pa are admissible there must exist a transformation SO that the former are expressible in terms of the latter.

From the general relation (2.9) one has

J%J. = q&. 9 q&l = cl:1 - qb”l (5.11)

Similarly, from the decomposition of pa into components parallel and perpendicular to P (similar to (53b)),

=L = P$al f (5.12)

The total three-momentum follows from (2.1)

p =CPa. a

(5.13)

This leaves Q,@, and as we have already seen there is no way in which the center position can be rxpressed in terms of the individual particle variables unless V = 0. The reason follows from (2.10): such an equation requires a linear relation among the faal which is available only in the V = 0 case (in the form (5.2)).

One concludes generally that in our formulation the individual particle variables do not permit a relativistic Hamiltonian dynamics unless V = 0. As already demonstrated at the end of the preceding section these variables can be used for V # 0 if one restricts the theory to the nonrelativistic limit. But that is, of course, well known.

Finally, we note the interesting relationship of the CM variables and the individual particle variables in the CM frame characterized by P = 0. The 6N + 6-dimensional phase space is then effectiveIy restricted to the 6Wdimensional subspace of the internal variables 5, , x, in which the canonical BR

Ezk, &“I = 0 = {vak, %V, bL3k, GZ> = Sk’ L , (5.14)

hold together with the first-class constraints

p+O (5.15)

and the Hamiltonian

Hint = c (x.” + may2 + V[Sl. (5.16) a

In the same frame of reference the 6Ndimensional phase space of the individual particle variables qa , pa which satisfy cununical BR

kdk, 4bZl = 0 = {Pa*, Pa’19 {qak, Pb? = skz hb (5.17)

are restricted by P = c p. M 0. (5.18)

a


And the motion is governed by the Hamiltonian

Hint = 1 (p,” -t %v + VII. (5.19)

It is evident from these equations that in the CM frame the internal variables 5,, x, of the CNR (6N + 6) are equivalent to a CBR algebra of three-vector individual particle variables, qa , pa . This algebra of the qa , pa is not the CNR algebra in individual particle variables representation, (5.8), with P = 0. Rather, it is a representation of a subalgebra of the NWR of g in the CM frame (see the Appendix).

6. VELOCITY-DEPENDENT INTERACTIONS

The canonical dynamics of Section 3 can be generalized to include interactions I; which depend on the 7~, in addition to the 5, . Let tab = 4, - eh and let

The latter case can be regarded as the difference of velocities in the sense of (3.23), although this equation will no longer hold with the present interaction. In any case, C’ of (3.1) is now generalized to

The important symmetry property

is still present. vb, = vu, (6.3)

The CCR used in Section 3 of the 8(N + 1) variables is supplemented by the constraints C, CmLu, C, , and Ci. The Hamiltonian (3.1 I) conserves these constraints also with the generalized interaction (6.2) provided W, ‘v 0, wh w 0 as before. (3.i6) and (3.17). The coefficients w and wlU of the first-class constraints are again left undetermined by this requirement of conservation. Physical requirements determine these: w * 0, (3.19), identifies 7 as the time of the CM frame. Equation (3.21) obtains in the generalized form

It is a matter of consistency to require here that when V is independent of ma Eq. (3.23) should result. Thus, the physical interpretation of $, leads as before to wlU = 0, (3.22). The other equation of motion, (3.24), is unaffected by the generalization of V. Elimination of n7, is no longer possible without the knowledge of an explicit form of I..

316 F. ROHRLICH

Reduction to 6(N + 1) variables proceeds exactly as in Section 4. Further reduction by use of (4.14) and (4.15) leads to the requirement

(6.5)

instead of (4.21). But the condition (4.23) of wIU = wu = 0 leads again to cases (A) and (B) given there as the only alternatives which satisfy this condition: the reduction from 6(iV + 1) to 6N variables is possible only by giving up the world-lines and by restricting the variables to fnbl and natii . World-lines exist only in cases (A) and (B). In the latter one also has

z-0 &7,~

(NR limit).

Of special interest are the velocity-dependent interactions which are linear and homogeneous in the velocity. We shall demonstrate that such an interaction is formally identical to an electromagnetic interaction in which radiation is neglected, i.e., it is a Lorentz force.

For this demonstration it is most instructive to start with the Lagrangian (3.6) and the interaction energy

a b a#b

The four-vector A& must be antisymmetric in ab. If we define

A,” = 1 A$, b#a

V can be written in the form

V= -x&Aa. (6.9) a

The resulting Lagrangian,

L = #f@ - C m,(l - &2)1i* + C 8, . A,, (6.10) a a

leads to the canonical momenta P and 7ra in the standard way. Solving for the velocities one finds

@’ = P”/M, s’,“ = (n-,“ - Aal&)/(r, - A,)2 + m,2)1i2.

The canonical Hamiltonian is, therefore, the invariant

(6.11)

H can = P2/2M + c ((na - A# + m,2)1/2. (6.12) a


The complete dynamics now requires the addition of the constraints C, C-I’, C, , and Ci , and one verifies that H = Scan conserves them. The equations of motion reproduce (6.1 I) and also yield

(6.13)

Elimination of n,, between (6.1 I) and (6.13) leads, after an easy calculation. to

with grrb = [,, -- &, and

A coupling constant r can be considered included in each Aab . This result is notable since no assumption of a “field” between the particles was

made. Nor can this FuY satisfy Maxwell’s equations. A consistent application of these equations would lead to the Lorentz-Dirac equation and not to (6.14) [25].

Finally, one observes that by symmetry

so that the constraint C,” implies the constraint

(6.16)

The four-vector r,(mech) follows from (6.11) to be

T,,u(mech) = ~~g,~~/( 1 - gn2)1/2 (6.17)

and occurs on the left-hand side of (6.14). The conservation of the constraint (6.16) is ensured by the vanishing of the total “Lorentz” force,

which is a consequence of the symmetry of the interaction.

7. CONCLUSIONS

(6.18)

It is, of course, well known that not every relativistic theory can be cast into Hamiltonian form. The existence of generators of the infinitesimal Poincart transformations is ensured for every dosed relativistic system. This includes in particular

318 F. ROHRLICH

the time translation generator PO. But this generator by itself is not sufficient for a Hamiltonian description of the internal dynamics of this closed system. Specific examples of non-Hamiltonian relativistic theories are those by Wheeler and Peynman [3] and by Van Dam and Wigner [4].

The no-interaction theorem [lo] makes three basic assumptions about a physical system’s description:

(4 relativistic invariance,

(b) Hamiltonian dynamics,

(c) independent particle variables.

These three assumptions are proven to be incompatible in the presence of interactions. Obviously, if (a) is dropped, there is no longer an incompatibility: (b) and (c)

combine to give nonrelativistic Hamiltonian dynamics in canonical form. If (b) is dropped, the compatibility was proven by Van Dam and Wigner [4]; this is the situation described above in which one has a relativistic theory without Hamiltonian dynamics.

The present paper investigates the third alternative: assumptions (a) and (b) without assumption (c). This, too, is shown to lead to a consistent theory. And beyond this it is demonstrated in some detail (see Section 5) why assumption (c) cannot be added to the other two.

If the independent particle variables which are so well known from nonrelativistic mechanics are not available, it seems that no obvious alternative variables present themselves. This is, however, not the case because the desire of representing the ten Poincart generators in terms of the new variables imposes rather strong restrictions on these variables. The classification of the realizations of the Poincare algebra g leads one to select specific realizations with only very limited choices of representations for each realization (see the Appendix.)

After deciding on manifest covariance only two realizations become important, the CCR and the CNR, the CCR with the CM position Q nof in the enveloping algebra of ,ip (Sections 2-4). The CCR is represented in the 8(N + 1)-dimensional phase space and permits a canonical algebra. The CNR is represented in the 6(N $ I)- dimensional phase space and does not have a canonical algebra. Another representation of the same CNR is in a 6N-dimensional phase space. Of course, for all phase spaces of more than 6N-dimensions the appropriate number of constraints are present.

All these representations are in terms of what we call the CM variables. Trans- formations to independent particle variables are proven to be impossible in the presence of interaction. This answers the first of the three basic problems stated in Section I, the problem of the choice of variables.

The second problem is that of the existence of world-lines. It has been conjectured [9] that one cannot describe an interacting relativistic Hamiltonian system and at the same time have a description of individual particle world-lines. This conjecture is not quite correct. It is correct only for the minimum phase space of 6Nvariables. But one can use a larger phase space (for example 6N + 6-variables) with suitable

RELATiVlSTlC HAMILTONIAN DYNAMICS 319

constraints in which one can have world-lines. In fact, the 6(N + I)-dimensional phase space is much more convenient to use, especially because of the obvious physical meaning associated with the variables e, which permit the world-lines. This meaning is present, however, only in a suitably chosen gauge (Eq. (3.22)). One can say that the world-lines are not describable in the minimum phase space (which is spanned by observables) because they are not gauge invariant. The similarity with electrodynamics is apparent: it is much more convenient to use potentials than field strengths, but the observables will not include potentials because these are not gauge invariant.

The third problem mentioned in Section 1 is the cluster decomposition for separable potentials. The existence of this property of the theory requires a proof that the dynamics of the system is independent of the reference hyperplanes, i.e.. that the CM hyperplanes we have used are not necessary. This difficult problem will be studied in the second part of this work.

The closed system of N spinless particles which was the basis of discussion in the present paper is only an example of a physical system that can be described in this way. Various generalizations suggest themselves: open systems emerge as the limit when the mass of one or more of the participating particles becomes very large; spinning particles can be treated by use of more sophisticated representations such as were used by Van Dam and Biedenharn [23]; continuous systems can also be treated: the descrete index on the particle variables then becomes a continuous one and the Kronecker deltas in BR like (2.7) become Dirac delta functions. An explicit example of such a continuous system in the simplest (one-dimensional) case is the relativistic string. It has been treated essentially by the method described above in several publications [22, 261.

One of the advantages of a Hamiltonian dynamics lies in the possibility of canonical quantization. While this process is only a heuristic means of obtaining a relativistic quantum dynamics it has proven extremely useful in many instances past and present. Consequently, the quantum dynamics of the refativistic Hamiltonian theory given here will be treated in the second part of this work.

APPENDIX

As mentioned in Section 2, the classification of realizations of 3 separates into classes of types Q f g and Q 6 g. The former type is commonly used and two of them have been given in an earlier publication [22]. If we use the notation of Table I with + and - standing for “yes” and “no,” then the realizations of type Q E :3 given in Ref. [22] belong to the classes (+ + - +) and (t- + + -). These were called the Newton-Wigner realization (NWR) and the covariant noncanonical realization (CNR, Q E @). Representations of the former were given by Shirokov [27] and by Foldy [28] with generalizations to reference hyperplanes with arbitrary normal n by Fleming [29] and very recently by Lorente and Roman [20]. The CNR, Q E d, can be found in papers by Bacry [30] and also by Shirokov [27].

HEAVY-ION TRANSFER REACTIONS 327

The Hamiltonian for the system is

H = H, + V, = Ha + V, = H, + K,, , etc., (4)

where H,, . HB ,..., are the channel Hamiltonians whose eigenstates are the channel states @, , / Qslt,... . These satisfy

H, @,, ’ = (w, + h”ky3/2m,) 1 oy,i = E ! @.,., , y E \‘X, p,...;, (5)

where -w,, --wg ,..., are the binding energies of the A emu b -- c. B =I a $- c,.... i systems; and k, . k, , are the relative motion wave numbers for r’,

Coordinate representations of the channel eigenstates are

2.2. Distorted- Watre Approximatiotls

DWRA t-matrix elements [2-51 for the reaction N --f /3 are

(7a)

(7b)

where C,; is the potential acting between u and c in the exit channel; and Vi acts between b and c in the entrance channel. The distorted-waves 1 EA+” and @L--) are solutions of the Schrodinger equations,

(H,-- r/,)!& = EjEi) , (8a)

(H,3 f I/,,+) ! @A-’ y = E : @a-’ , (8b)

with outgoing-wave and ingoing-wave boundary conditions, respectively. These solutions satisfy the Lippman-Schwinger equations,

G:,-’ = l/(E - H./ -- i0)

is a channel propagator. These equations take into account the possibility of U, and UO leading to transitions to other channels (y # iy or y # /3). Most choices of U, and rl, restrict the number of such open channels, often to just the single channel, (1 or p. In such cases, equations in which y denotes a closed channel reduce to trivial identities and may be ignored.


All other BR vanish. Since the internal variables are constrained to 5, ::- E,, ,

TCI == xai , this realization is denoted by CNR, , Q $9. This algebra would result from the algebra (2.6) if the second-class constraints

(4.9) were eliminated by Dirac bracket construction, leaving all other constraints. Here we have an example of two different realizations that belong to the same class.

As the minus signs in the class characterization increase so does the number of different realizations that belong to a given class.

If S . F --_ 0 and P2 and Q . P are to belong to the center of the algebra, one finds that one returns to a type Q E g algebra, viz., the CNR mentioned in Section 2 which we can now characterize by (+ + + -).

The cluss (t t J- -) has the following algebra:

(p, Qy = -M”“IP’ {Q”, P”: = Py.

{Qfi, So”) = (S1~PO - Su”P~)/P2, (A5)

is”“, ,yy = py,y + pysuo - pyy .- py,y*

Its realization by QPt,r, was given in (4.10)-(4.12). The position Qu is a Lorente four-vector and is one of the choices proposed long ago by Pryce (his choice (d)). It can be expressed in terms of the Lorentz generators by

Qu = IW’P,,/P~ + ~lju CA@

with -T the value of the invariant Q . p as given in (4.13a). We denote this algebra by CNR (Q E <g).

ACKNOWLEDGMENTS

I want to express my thanks to Joshua Goldberg, Arthur Komar, and especially to Peter Bergmann for illuminating discussions on the theory of Hamiltonian systems with constraints. The last of these also provided valuable criticism to an earlier version of Sections 2 and 3.

REFERENCES

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given.

relativistic hamiltonian dynamics i. classical mechanics

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