I L I~UOVO CIMENTO

VOL. 21 B, IN. 2

11 Giugno 1974

Motions of Relativistic Hamiltonian Interactions (*)C). I~. A ~ E ~ s Unive~~

el Cali/ornia - Los Angeles, Cal.

(ricevuto il 20 Giugno 1973)

Summary. -There is presented a class of completely Hamiltonian N-particle interactions, invariant under the Poincar6 group. The motions are not given b y N-tuples of world-lines in space-time, but b y certain N-dimensional submanifolds of the phase-space ~ for R4~. The set of these motions is a 6N-dimensional symplectic manifold and is determined b y the choice of JY functions on ~ which have zero l)oisson brackets on ~. This 6N-dimensional manifold is in 1:1 correspondence with the 6Ndimensional phase space for the system whenever an observer is specified. The phase-space transitions thus obtained for any p a i r of observers constitute the dynamics.

1.

-

Introduction.

T h e i n v a r i a n c e is e n s u r e d if t h e N f u n c t i o n s m e n t i o n e d i n t h e S u m m a r y a r e c h o s e n t o b e i n v a r i a n t u n d e r t h e a c t i o n in ~b of t h e s p a c e - t i m e g r o u p . I n t h e P o i n c a r d - g r o u p case t h e r e a r e sufficiently m a n y s u c h i n v a r i a n t f u n c t i o n s p r e s e n t so t h a t e v e n w i t h t h e r e s t r i c t i o n of c o m m u t i n g , t h e r e s u l t i n g d y n a m i c s does n o t h a v e t o b e t h a t of n o n i n t e r a e t i n g p a r t i c l e s . (The n o n i n t e r a e t i n g case is a s p e c i a l ease, of course.) T h e p o s i t i o n o b s e r v a b l e s c o m m u t e . T h e m o t i o n s a r e n o t g e n e r a l l y r e p r e s e n t a b l e as ~V-tuples of w o r l d - l i n e s b e c a u s e t h e n , b y t h e n o n i n t e r a c t i o n t h e o r e m of C u ~ g I E , JOI%DAN~ SUDAlCSHA~ a n d LEUTWYLEIr t h e p a r t i c l e s c o u l d n o t i n t e r a c t . T h e N c o m m u t i n g f u n c t i o n s a r c set e q u a l t o 0 t h u s d e f i n i n g a 7 N - d i m e n s i o n a l s u b m a n i f o l d ~ of ~b. T h e s y m p l e c t i c s t r u c t u r e of r u p o n r e s t r i c t i o n t o (*) To speed up publication, the author of this p a p e r has agreed to not receive the proofs for colrection. (**) The preparation of this paper was supported in p a r t b y NSF Grant No. GP-33696X. 395

396

~.

AR~NS

admits an N-dimensional set of singular directions in ~ . E a c h N-dimensional integral manifold of this N - v e c t o r field is a motion. There are further conditions t h a t h a v e to be m e t b y the ~V functions defining ~ . F o r one thing, t h e y m u s t be functionally independent. The equations defining t h e motions are generalizations of H a m i l t o n ' s canonieal equations and m a y h a v e only local solutions. Therefore, j u s t as in the classical ease, one m u s t postulate global solutions.

2. - Symplectic structures related to space-time.

L e t M be the space-time manifold. F o r the m o m e n t , it does not m a t t e r whether this is R" for some n, or not. One can construct T~(M), the cotangent bundle over M((~), p. 79; (2), p. 167). I t s dimensionality is twice t h a t of M. There is a n a t u r a l sympleetic structure (also called H a m i l t o n i a n structure) on TI(M) a n d a corresponding ]?oisson b r a c k e t ((1), pp. 143-144; (2), p. 168). There being a choice of sign, we specify t h a t in this paper, in t e r m s of canonical co-ordinates ((2), loe. eft.),

{/' g } - @, ~x,

~x, @~ '

where s u m m a t i o n on r e p e a t e d indices is intended. Consider t h e p r o d u c t of N copies of TI(M): ~ b = T I ( M ) • ... X TI(M). This is t h e same as t h e cotangent bundle of N copies of M (or at least isomorphic to it): ~b ~_ T I ( M x ... x M ) . As a result, there is a symplectie structure in ~b and a Poisson bracket. W e now suppose M has a space-time structure and t h a t there are co-ordinators ((~), pp. 155-156). This m a k e s M isomorphic to R a. Moreover, it enables us to t a l k a b o u t t i m e co-ordinates, since one of the four co-ordinates (the last, to be specific) in a co-ordinator can be called a time co-ordinate. N o w when a co-ordinator x ~--(x ~, x 2, x 3, x 4) is chosen, we obtain in an obvious w a y a co-ordinate s y s t e m Xl~ Xl, Xx~ X~, X2, X2, X~, ..., X~., X~, Xm, X~v)

in M •

x M ~--M ~.

F r o m this we obtain the associated m o m e n t a

(1) S. STEtCN~E~G: Lectures on Dif]erential Geometry (Englewood Cliffs, N. J., 1964). (3) R. A R ~ S : Trans. Amer. Math. Soc., 147, 154 (1970).

M O T I O N S OF R E L A T I V I S T I C I t A M I L T O N I A N I N T E R A C T I O N S

3~

and thus 8 N canonical co-ordinates in r

(2.~)

1 2 (p~, p ~ , p ~ , p ~ , ..., x~, z~, %8, x~).

L e t 5P~ be a submanifold of q) w i t h these properties: It-1. Given a n y point of 5P~, t h e n there are functions /F~, . . . , F ~ defined in q~ with 8(F~, ... ~ F~) ~ ( p ~ , ... ~p ~ ) ~ o such t h a t { F ~ , F ~ } = 0 for 1 < i < ] < ~ fined b y t h e equations

FI~O~

such t h a t 5z~ near t h a t point is de-

F2=O~...~-'W = 0 .

It-2. Given a n y co-ordinator x and a n y 4n co-ordinates a~, ..., a~, ..., a~, ~ ..., a~ there are points on 5 ~ which h a v e these as its x co-ordinates. I n greater detail, there are functions H~ ~ ..., H x defined for all x values near those a~, ~ ...~ ag4 and for all values of t h e p's~ b u t depending only on the p~. with j =/= 4, ~ = 1, ..., /V, such t h a t equations for ~fs can be written in the f o r m (but see (2.4) below)

(2.2) Then 5~ will be called a Hamiltonian surface (8). I t follows f r o m t h e second p r o p e r t y t h a t 5P~ has dimension 8 N - / V . W e will now show how the choice of a H a m i l t o n i a n surface defines an N-particle system. The space of motiotls of the s y s t e m shall be t h e m a x i m a l connected integral varieties of the singular distribution of the restriction of t h e sympleetic structure of q~ to t h e surface $zN. L e t us explain this with co-ordinates. The symplectic structure of ~b is (2.3)

o~ = dp~,z Adx~,

where we sum on ~ (as well as on k). As co-ordinates on Y~ we can use t h e list (2.1) with the p ~ deleted. Because of (2.2), the restriction ~5 of (o to 5Ps is expressible in these co-ordinates b y replacing p ~ in (2.3) b y - - H ~ . The singular distribution of ~ consists of those vectors X t a n g e n t to 5 ~ such t h a t ~ 0 for all other vectors :Y t a n g e n t to 5P~ at t h e same point as X.

(8) R. AR]~NS: Hamiltonian ]ormalism ]or noninvariant dynamics.

398

R. ~ R n ~ S

A connected integral variety of this distribution (*) ((4), p. 87) is a connected submanifold S of 5 f such t h a t every vector t a n g e n t to S is singular for r i.e. belongs to this distribution. W h e n 25 is 1, then ~ r ~ 5 f is 7-dimensional and the maximal integral of the singular distribution are the extremal curves in 5f~ ((5), p. 181) for p~ dx ~-]~- P: dx 2-k P3 dx 3 -- H~ dx ~. I n classic~l mechanics one c~n always infer t h a t solutions will exist for some positive time, beginning at a n y point of ~1. For a discussion relevant to physics one must assume b e y o n d this existence theorem t h a t such solutions exist for all time. W e m u s t examine briefly what these existence questions a m o u n t to for N particles when we h~ve N times. Consider the m a p p i n g z from ~ down onto M ~ wherein an element ~0 of ~5 (which is of course a eovector in M ~ located at some point of M ~ called the base point, or point of application, of ~) is ~(projected down ~>on its base point. We want to compose this with the map t~ which requires the choice of a coordinator x, is defined on M ~, has values in R~ and has the components x 14' ~2~ ' " " ~ x~). Thus we h~ve

Mx...xM

R•215 Denote the composite t~ o~ b y ~ .

Proposition. Let W be a maximal connected integral variety o] the singular distribution o / t h e restriction of o~ to 5fN. Then a) the image of W under x~ is a nonempty open set in ! ~ b) locally, the mapping 7~ is 1 : 1 on W. These are the local existence and uniqueness properties of the integTal manifolds. W e now prove them. At a n y particular point, we m a y take the functions / ~ which define ~ to have the form given in (2.2). Define the vector fields X~ b y X ~ ( ] ) = {F~,, ]}. Evidently 8

(*) Distributions are called di]]erential systems in ((1), p. 130). (~) C. CHEVALL~u Lie Groups (Princeton, N.J., 1946). (5) R. ARIses: Journ. Math. Anal. Appl., 9, 165 (1964).

M O T I O N S OF R E L A T I V I S T I C H~kMILTO:NIA~ I]~T]~I~ACTIONS

399

where Y ~ ( ] ) = {H~, ]} and so X~ has no x~-component for a n y fi, l ~ < f i ~ N . Thus t h e X~ are linearly independent. The n e x t observation is t h e crucial one t h a t each X~ is t a n g e n t to 5P~. This is because X~(F~) = 0 for all fi, b y It-1. An easy c o m p u t a t i o n shows t h a t each X~ is singular for ~ and t h a t e v e r y singular X is a linear c o m b i n a t i o n of these X~. Thus the singular distribution is N-dimensional. Now singular distributions for closed 2-forms are always completely integrable (also called involutive) b u t in this case the i n t e g r a b i l i t y is particularly evident because t h e c o m m u t a t o r [X~, X~], being generated b y ( / ~ , J~}, is 0. Therefore the manifolds W (which we h a v e already decided to call motions) are ~V-dimensional. N o w we come to a). L e t ~ lie on a m o t i o n W. E v a l u a t e t h e vector fields X~ at ~. T h e y are t a n g e n t to W at ~. Thus t h e y show t h a t arbitrary, although small, changes in the various x~ are possible w i t h o u t leaving W. This establishes a). Continuing with this W, let ~ be as a b o v e and let ~ also lie on W, near ~. T h e n t h e co-ordinates of ~ differ f r o m those of 7 (to the first order) b y the components of ~ X~-~ ... ~ ~ X~ ~ where ~ , ..., ~ are the p a r a m e t e r s effecting t h e infinitesimal displacement f r o m ~ to ~ . Now the difference between x~(~) and x~(~) is (to an a d e q u a t e approximation) j u s t ~ because the x~ comp o n e n t of X~ is 1. Thus ~ ( ~ ) = / = z~(~) unless ~ . . . . . , ~ = 0 and t h e n we have %~ as b) asserts. This Proposition being established, it is n a t u r a l to require global existence and uniqueness~ as follows. It-3. The mapping ~ maps each motion W onto R ~ in a 1:1 way. I f 5z~ has this additional p r o p e r t y , we call it a complete H a m i l t o n i a n surface. L e t us denote t h e space of motions b y [Sz~]. I n w h a t follows we will in c o n f o r m i t y with t h e n a t u r e of things h a v e to deal simultaneously with two co-ordinators at a time, since co-ordinators correspond to observers. The f o r m of t h e functions H~, ..., H~ in (2.2) depends on the co-ordinator x in question, and m a y be different when we change to another co-ordinator y. This will surely be t h e ease when the surface 5P~ is not i n v a r i a n t u n d e r t h e action in ~b induced b y t h e action of the space-time group in M. F o r this reason we m u s t a u g m e n t the n o t a t i o n to emphasize this depend e n c e i ' a n d accordingly denote each H~ in (2.2) b y (2.4)

x~ .

This x is a quadruple of co-ordinates in M and these should not be confused with the x's in (2.1) which are co-ordinates in r in t e r m s of which t h e function H~ (now n a m e d as in (2.4)) can p r e s u m a b l y be expressed. To repeat, if (2.4) is evaluated as some point where t h e co-ordinates (2.1) t a k e on some 8 N numerical values, one should not seek to replace the xX, x~, z ~, xa constituting x b y some four of these numbers. There is little danger of this error unless N = 1, of course.

400

R. A R E N S

3. - The system defined by a complete Hamiltonian surface. We base our discussion on the concept of dynamical system as defined in (~). See also (3). Accordingly, we need a space of states, K. For K we take the cotangent bundle T~(R3g). This is the usual phase space for N particles. The dynamics A is a family of transformations Ay os K onto K reflecting the change from one co-ordinator x to another y (2.~). To define these maps one m a y first define maps A * from the space of motions, onto K, as is done in ((2), p. 157), and in ((g), Sect. 7). We follow (~) which deals with the ease N =- 1. Let W be a motion. Let x be a co-ordinator. Then, according to It-3, there is an element ~ on W such t h a t ~,(~) is the point (0, 0, ..., 0) of R g. (This can be interpreted t h a t the times x~, ..., m~ for ~ are all zero.) :Now ~ has the following appearance: ~ = q~-~ . . . § q~, where ~ = b~ dm~ ~- b,~ dm~~- b~ dx~3 -- H~ dx~.

0, 1 ~ ~ 0,

~

~

The base point x(~) must be

~ 0).

F o r co-ordinates in /t a~ we must avoid letters like x, y, ..., because t h e y are used as co-ordinates in M. Therefore denote the Cartesian co-ordinates in R~ b y (~, ~ , ~ , ..., ~ , ~ , ~ ) . We define zJx(F) to be the covector

b~ d ~ ~- bx2 d ~ +

b~8d~] + b2~d$~ + ... + b~ d ~ -[- b~2 d ~ + b~v~ d ~

in /t ~ with base point (rl, 1 rl, ~ rl, 3 r~, 1 r2, 2 r~, 3 ..., r~). Thus d x maps the class of all motions onto K. We can now set dye= = Llyo (~y)-i and, as in ((3), Sect. 7), a dynamical system results. W e want to show t h a t this system is ~(completely Itamiltonian ~>(*) which means simply t h a t the ~(dynamorphisms )> z]~ preserve the sympleetic structure t~ of T~(R3~). The best w a y to prove this is to show t h a t the space [ ~ ] of motions has a natural symplectie structure [~] and t h a t ~]~ transforms it into no m a t t e r what x is used. The elements of [5~] are the W. A vector in [5~] m a y be regarded as a pair of elements W0, W~ just as a vector in R" m a y be regarded as a pair of points r, s (the tail and head of the vector, respectively). (This old-fashioned w a y of t~lking simply involves ignoring the difference between increments and differentials and can be made rigorous b y either taking line integrals or using

(*) See (2.~). These systems are not Hamiltonian system of the type considered in (6). (6) P. DRoz-VINC]~T: Nuovo Cimento, ]2 B, 1 (1972).

iV[OTIO:NS OF ]%t~LATIVISTIC ItA]YIILTONIAN INTEt~,ACTIONS

401

t h e m e a n - v a l u e theorem.) W e will write Wo W~ for t h e vector in question. Suppose Wo W~ and W0 W~ are two vectors with the same base point. W e define

<[~]; w0 w,, wo w #

(3.1) as follows. Select

q%

(3.2)

on W~ ~s~ on W , , q~ on W2. Consider t h e expression
~o} .

I t remains constant whenever a n y qq is displaced on its W~ because such a displacement is singular for go. I t s value m a y therefore be t a k e n as t h e definition of (3.1). (Incidentally, we h a v e just now used t h e connectedness of each W.) W e now select t h e ~0~ such t h a t = , is 0 for all of t h e m . As a result, each of the vectors ~ -- ~0o (tangent to 5z~r has Axe---- 0 for every ~. Consider the explicit f o r m of &

where s u m m a t i o n is intended for a f r o m I to 3 and for a f r o m 1 to iV. Repeating, Ax~ is 0 for both vectors in the expression (3.2). H e n c e (3.1) has the value

which is (the sum)

p~o(~0 - p~o(~o) x~(~1) - x~(~o)

x~(~) - x:(~0)

The lower right-hand e n t r y here for example is A ~ for the displacement vector f r o m A*(~o) to A~(~s=) and hence t h e entire expression is

<~; A'(Wo) ~*(w,), a*(w~) ~'(w~)>. This completes our proof of the s t a t m e n t m a d e above, which we repeat.

Theorem. For each co-ordinator x, the map A ~ of the space [Szz~] of motions onto TI(R ~) transforms the symplectic structure of the former into that of the latter. Corollary. The system is completely Hamiltonian. A consequence of this is t h a t t h e infinitesimal d y n a m o r p h i s m s ((3), p. 158) h a v e generating functions. The infinitesimal d y n a m o r p h i s m s are t h e vector fidds in TI(R ~) which give t h e linear a p p r o x i m a t i o n to d y when y is close to x.

402

n. iRONS

I n t h e n e x t Section we obtain f o r m u l a e for the generating functions in this sense: h is a generating function for Z if Z(])= {g, h}. W e close this Section with t h e r e m a r k t h a t to obtain a s y s t e m of N free particles, in t h e Einstein-lV[inkowski space-time scheme, one should t a k e

H~= [m~+ (p~,)~+ (p~,)~+ (p~,)~]~. The functions F~ m a y be t a k e n as (p~0)~ - ( p ~ ) ~ - - . . . -

4.

-

(p~a) ~ - m~.

Generating functions for its infinitesimal dynamorphisms.

Let Z ~ - - $ ~ ( 8 / ~ ) ~ - . . . - - ~ ( ~ / ~ ) be a vector field in R ~, where ~ , ..., ~ are the Cartesian co-ordinates. Suppose Z preserves the space-time s t r u c t u r e of R t (For a discussion of this see (~)). I f x is one co-ordinator a n d e is a real n u m b e r , we can define a m a p (to be called x,) f r o m M to R ~ b y m a k i n g x~(m) b e t h e result of letting t h e point x(m) of R ~/low for e units of t i m e in the vector field Z. A p p r o x i m a t e l y =

...,

This x 6 is a co-ordinator. Therefore A~ ~ defines a ]low in TI(R 3~) and this has a velocity field A'z.x in TI(R3N). The components of this field are the t e r m s linear in e when one computes the changes in the co-ordinates ~ , V~ (a = 1, ..., 2/; a---- 1, 2, 3) in T~(R ~) effected b y A: ~. H e r e the ~ are t h e mom e n t a corresponding to the $'s. The calculations t a k e only a b o u t two pages, b u t the c o m m e n t a r y would treble t h e space needed. However, t h e p r o b l e m is completely formulated, and so we give t h e answer. W e give first t h e coefficient of s in t h e T a y l o r expansion for t h e A ~ due to the m a p A: ~. One will see t h e r e symbols $~. These are functions in T~(R 3~) obtained f r o m ~ in the following way. P r e s u m a b l y ~ is some expression involving ~1, ~ ~s and ~ . To obtain $~ replace $~ b y ~ for b = 1, 2, 3 and replace ~4 b y 0. (This 0 originates in our having m a d e ~ ( ~ ) ---- (0, ..., 0) in Sect. 3.) I n t h e same sense~ we understand $~.~ to m e a n ~ / ~ b followed b y the substitutions which produce ~ from ~. One will also see t h e r e a s y m b o l H~ which is an a b b r e v i a t i o n for ~H~ (see (2A)). This is a function defined on T~(Rs~) as follows. Replace each x~ b y ~ for b = 1, 2, 3 a n d replace x~ b y 0. Replace each P~b b y %~. Recall t h a t t h e r e is no p.~ in these functions (2.2).

Lemma.

The ~-component of A'z.~ is

M O T I O N S OF R E L A T I V I S T I C H/kMILTONI2~:N I N T E R / k C T I O N S

40~

The ~-component is 9 ~H~

(Here there is no summation on g.) As we said, t h e proof would be as dull to read as it is easy to i n v e n t b y t h e reader.

Corollary. A generating ]unction ]or this vector ]ield in T~(R3N) is given by

(4.1)

To deduce this Corollary, one m u s t verify t h a t 8 / 8 ~ of t h e generating function is t h e ~ - e o m p o n e n t given in the L e m m a . One m u s t also show t h a t ~ / 8 ~ is the negative of the ~ - e o m p o n e n t . I f one could tolerate t h e a m b i g u i t y of using t h e x~, p ~ for the co-ordinates in T~(R3~), t h e n t h e symbols explained above would be merely t h e result of 4 differentiating (whenever derivatives are involved) and t h e n setting the m~ equal to 0. L e t us examine some of these generating functions. The vector field Z = = ~ / ~ represents an infinitesimal translation in t h e ~-direction. The generating function is t h e same for all co-ordinators, n a m e l y ~ § ~ § ... § ~ .

(4.~)

This looks more familiar when written as

Pii ~- P~i -~ "'" -~- P~i" L e t us n e x t consider Z----8/~$ 4. This is t h e infinitesimal t i m e translation in the forward direction. The generating function is (4.3)

-

~t~

-

o

~H~ --

... --

~.

0

These minus signs are no mistake. The reason for the difference in signs between (4.2) and (4.3) is built into the t h e o r y when the action was w r i t t e n in t h e f o r m (say for one particle) Pl dx ~~- P~ dx ~Jr P~ dx 3 -- H dt.

404

~. ~ s

A similar minus sign occurs even in the non-Hamiltonian theory, however. This is examined in ((~), (2.78)). The Hamiltonian of the N-particle system, for the co-ordinator x, is therefore the negative of (g.3), i.e.

E a c h t e r m of this sum m a y involve all 6hr co-ordinates.

5. - Lorentz-invariant systems o f genuinely interacting particles. Suppose each term of the Hamiltonian (4.4) is the same for all co-ordinators : 0 ~ Ho~ = yH~. Then the system is surely invariant under all space-time transformations because all the other dynamorphisms are expressible b y means of these xH~~ and other variables which are the same for all co-ordinators. Such systems we call completely autonomous. We want to show t h a t such systems do not necessarily consist of noninteracting particles. The notation presents difficulties, and so we will t r e a t only the case 3 r = 2. Moreover, we will specifically take space-time M to be R *. W e will not have to bother with co-ordinators x, y , ..., in this Section. Thus in this Section we will use x 1, x 2, x 3, x% yl, y2, y3, y4 not for the components of some co-ordinators, b u t as the Cartesian co-ordinates in M • M = Rs. F o r the associated (~mom e n t a )> we will use p~, p~, P3, P~, q~, q,, qs, q4. Thus the extra index ~ = 1, 2 can be avoided, at the cost of confusing readers who use q~ only for eon]igurational co-ordinates. E v e r y b o d y knows how the x's transform under the action of ~ , the inhomogeneous Lorentz group, or Poincar6 group, in Rs. The same goes for the y's. Let p~= g~pj, q~= r Then the p~, the q~ and the z~= x ~ - y ~ transform like (the components of) vectors. Define p . q (to take one example) b y g~p~qr g~r where the g~r are the Cartesian components of LorentzMinkowski metric -- (dxX)2 - (dx2) 2 - (dx3)~+ (dx4) ~. Define the following six variables, which are (except for one sign) the independent elements of the Gram matrix of p, q, z: (5.1)

e=p'p,

v=p'q,

a=q'q,

2=p'z,

~=z'z,

#=--q'z.

I f the functions 2~1, F~ of II-1 are functions of (only) these six invariants, then t h e dynamics will surely be Poincar6 invariant. W e must also ensure {El, F~} = 0. For t h i s reason we explore the c o m m u t a t i o n relations of the members of (5.1). For example, {@, ~} = O, (@, a} = O, (@, 2} = 2@.

2~OTIONS OF RELATIVISTICttAMILTONIA.NINTERACTIONS

405

T h e entire t a b l e is, in t h e order of (5.1), as follows: 0

0

0

0

0

0

0

0

0

- - 2@

2v

@ --

v

2@ v- @ -- 2v

-- 2a

--2(,~ ~- #)

- - 2v v -- a

4#

2a

0

2~

# --

A- - g

2~

0

2v

a -- v

4),

3.

I t was t o be e x p e c t e d t h a t each P o i s s o n b r a c k e t w o u l d be a Junction of t h e variables (5.1), b u t t h e y are a c t u a l l y linear combinations. T h u s t h e linear c o m b i n a t i o n s of t h e variables (5.1) f o r m a (six-dimensional) Lie algebra. W e w o u l d h a v e o b t a i n e d t h e s a m e Lie algebra if we h a d used t h e E u c l i d e a n g r o u p r a t h e r t h a n t h e P o i n c a r 6 group. N o r does t h e Lie algebra reflect t h e fourd i m e n s i o n a l i t y of M. W e do n o t explore or use these interesting facts here. W e use just t h e table.

Proposition.

The two quantities

and

~--

det

p'p

p.q

P'zl

p.q

q.q

q.z

p .z

q.z

z.z

I

each have vanishing Poisson bracket with every element o] (5.1). Conversely, every ]unction o] the invariants (5.1) which does commute with every element o/ (5.1) ij a Junction of these two quantities. T h e first s t a t e m e n t is easily verified. T h e m e t h o d for p r o v i n g t h e c o n v e r s e is as follows. F o r m a c o l u m n m a t r i x VJ whose entries are t h e p a r t i a l derivatives 8J/~@, 8 1 / ~ , ..., 8//8~. L e t M be t h e m a t r i x p r o v i d e d b y t h e table. T h e n 3/VJ = 0 is t h e condition of c o m m n t a t i v i t y . This leads e v e n t u a l l y t o our assertion. W e generalize this p r o p o s i t i o n as follows.

Proposition. A n y Junction of the invariants (5.1) which commutes with @ and a depends only on @, v, a and 1". T h e p r o o f is m u c h like t h a t of t h e p r e v i o u s p r o p o s i t i o n a n d will also be omitted.

406

~. ~ s

l~eturn t o t h e p r o b l e m of ( F ~ , / ~ } = 0. Using F ~ = @ d - const, F s = a d - const gives us t h e t w o n o n i n t e r a c t i n g particles. N o w suppose we t a k e a r b i t r a r y functions / g n d g, a n d let

(5.2)

F~ = ](@, v, ~, F ) ,

~ = g(@, ~,, (~, F ) .

T h e n we still h a v e {F~, F2}----0, or I L l . I n order t o o b t a i n also H - 2 a n d t t - 3 one m u s t solve t h e equations / ~ = 0, F~ = 0 for p ~ a n d Ps4, o b t a i n i n g a single-valued differentiable solution defined for all values of t h e o t h e r variables. W i t h t h e wide r u n g e of choice of ] a n d g, this is possible w i t h o u t h a v i n g t o fall b a c k t o d e p e n d e n c e on @ ~nd alone. A specific e x a m p l e leading t o a n o n z e r o i n t e r a c t i o n will be p r e s e n t e d in a f o r t h c o m i n g p g p e r (( Models of relutivistie H a m i l t o n i a n interactions )). W e p r o p o s e considering such s y s t e m s as c o m p l e t e l y H a m i l t o n i a n twoparticle systems. I f f a n d g do n o t d e p e n d on @, ~ a n d ~ glone, t h e n t h e fingl t t ~ m i l t o n i a n (see (4.4)) will definitely involve t h e interparticle displacement. H e n c e t h e i n t e r a c t i o n will n o t be trivial.

6. -

Are

these

really two-particle systems?

As is well k n o w n , CURI~IE, JOI~DAN a n d SUDARSHA~ (7) h a v e s h o w n t h e following. L e t t h e r e be a s y s t e m (*) (K, A) in w h i c h t h e s p a c e - t i m e g r o u p is t h e P o i n c a r 4 group. S u p p o s e t h e s y s t e m is c o m p l e t e l y H a m i l t o n i a n a n d inv a r i a n t . L e t t h e g e n e r a t o r s of t h e infinitesimal d y n a m o r p h i s m s be called H, P , , J~ a n d K~ as t h e y u s u a l l y are (~.s). S u p p o s e t h e r e are also defined on K some f u n c t i o n s (**) X 1, X 2, X 3, y~, y2, y3 such t h g t {x ~, x ~} ---- {y~, y~} = (x ~, y~} = = O, {x~, Jk} ---- -- s ~ x ~, {YS, J~} = -- s~k~Y~ a n d

(6.~)

{x ~, Kk} ---- x~{x ~, H } ,

{y~, Kk} = y~{yS, H } .

Then

(6.2)

{ix,, H}, It} = {iS H}, H} = 0.

(7) D. G. CURtCIE, T. ~. JORDAN and E. C. G. SUDAtCSHAN:Rev. Mod. Phys., 35, 350 (1963). (*) We reformulate the theorem in the terms of (s). (s) H. LEUTWYLlCR: ~Y~ovo Cimento, 37, 556 (1965). (**) These x's and y's correspond to the Cartesian co-ordinates ~ (~ = 1, 2; i = 1, 2, 3) in R% We feel that to call them ~i and V~ would cause some confusion with the V's we have earlier in this paper.

MOTIONS O~ R E L A T I V I S T I C ~-IA.MILTONIiN I X T : E R i C T I O N S

4{)7

The 2-particle systems we proposed above (Sect. 5) satisfy all these conditions except possibly (6._l) and (6.2). I n fact, (.6.2) is equivalent to f and g (see (5.2)) satisfying some second-order partial differential equations. Since ] a n d g do not h a v e to satisfy a n y other differential equations, t h e y can be chosen almost at r a n d o m in such a m a n n e r t h a t (6.2) does not hold, i.e. t h e particles do interact. Of course, t h e n (6.1) will not hold. W e h a v e therefore to justify a b a n d o n i n g (6.]). We first inquire into the basis for adopting (6.]). Although these conditions were obviously invented long before (8) we will base our discussion on (~). Suppose we t a k e the definition of 2-particle interaction as given in(9), reproduced in ((~), p. 157). Then((3), eq. (2.72)) tells w h a t the infinitesimal d y n a m o r p h i s m s should be if t h e space of states is T~(R6), t h e tangent bundle. There is no symplectic struct u r e on T~(R ~) as there is on the cotangent bundle T~(R~). W h e n there is a Lagrangian on 2~l(RS), one can use t h e Legendre trans]ormation ((~), p. 150) to transfer t h e dynamics f r o m TI(R ~) to TI(R~). Equivalently~ when there is a H a m i l t o n i a n given on Tl(Rs), one can m a p TI(R 6) onto Tl(Rs). W h e n this is done, the afore-mentioned infinitesimal d y n a m o r p h i s m s are transferred to TI(R6). I n particular, this is true for t h e Lorentz (, boosts. ~> T h e y t a k e the f o r m : x~8H

8

~SH 8

~p-~~x--;+ y ~ ~y~ +'"' where p, q are t h e <(m o m e n t a ~>associated with t h e x, y. The dots refer to the 8/8p and 8/8q terms. I t follows t h a t (6.3)

8H ~ ( x ~) ----x k . ~P,

I f the ~ are now supposed to h a v e generating functions K~, (6.1) results immediately. Sometimes the Legendre t r a n s f o r m a t i o n does not lead to the m o s t suitable generating functions. Consider a one-particles s y s t e m whose s t a t e space is the cotangent bundle TI(R3). Consider the infinitesimal space t r a n s l a t i o n ~/~x 1. This has t h e generating function Pl. This seems to be generally accepted, and is in agreement with t h e l e m m a of Sect. 4. See also ((~), Prop. 3.7), keeping in m i n d t h a t the Poisson b r a c k e t there is t h e negative of the present one. W e t u r n to the tangent bundle, TI(R~). C o m m o n sense, as well as ((~), eq. (2.72)), says t h a t the infinitesimal translation is 8/~x 1. However, in this partial deriva t i v e x ~, x ~, 21, ~2, ~3 are held constant, whereas in t h e formal" one, x 2, x ~, Pl, P~, P3,

(9) R. A~]~S: Journ. Math. Phys., 7, 1341 (1966).

408

n. A~zn~s

a r e h e l d c o n s t a n t . T h e s e t w o t h i n g s a r e n o t t h e same. W e c a n c o m p a r e t h e m , u s i n g a L a g r a n g i a n . A p e r f e c t l y r e a s o n a b l e one is ~,

A~)~ + I (&~ + A2)~ + 89(S:a+ A~) ~ ,

w h e r e A~, A s , As d e p e n d o n l y on x ~, x 2, x 3. T h e L e g e n d r e m a p g i v e s p~--~ = dL/d2~ ~ - ~ A~. N o w , in t h e / o r m e r sense, ~p~/~x ~ ~ O. I n t h e l a t t e r sense, ~ p ~ / ~ x ~ = ~ A ~ / ~ x ~. T h u s t h e s e t w o n ~ t u r a l i n f i n i t e s i m a l t r a n s l a t i o n o p e r a t o r s a r e not r e l a t e d b y t h e L e g e n d r e t r a n s f o r m a t i o n (*). H e n c e t h e L e g e n d r e t r a n s f o r m g t i o n s o m e t i m e s fails gs a g u i d e t o d e f i n i n g t h e d y n a m i c s on t h e c o t a n g e n t bundle. W e c o n c l u d e t h a t t h e L e g e n d r e t r a n s f o r m a t i o n s h o u l d n o t b e r e g g r d e d as n e c e s s a r i l y v a l i d r e i n on t h e choice of g e n e r a t i n g f u n c t i o n s . This s e e m s t o r e m o v e t h e b a s i s for c o n d i t i o n {6.1). T h e r e is no d e n y i n g t h a t a s y s t e m w i t h (6.1) is a 2 - p g r t i c l e s y s t e m i n a s t r o n g e r sense t h a n one w i t h o u t (6.1), g n d o n l y t h e wish t o a v o i d (6.2) l e a d s us t o q u e s t i o n (6.1). M a n y w r i t e r s (**) h ~ v e p r o p o s e d ~ l t e r i n g t h e C u r r i e - J o r d a n - S u d a r s h a n (7) d e f i n i t i o n for 3 / - p a r t i c l e H a m i l t o n i ~ n s y s t e m , in o r d e r t o a v o i d (6.2). T h e y h a v e , h o w e v e r , d u n g t o (6.1). (*) This m a t t e r discussed more completely in ((s), Theorem 3.3). I t is shown there, for the case of Einstein-Lorentz space-time structure, t h a t if a one-particle system is transferred to the cotangent bundle via the Legendre transformation and this new system is completely Hamiltonian, then the system is invariant (whence the particle is free). In the Gali]ei-Newton case, it turns out that the system must have a Lagrangian [(~i)~ + ... + ( ~ ) ~ ]

_

V(x 1, x~, x~).

(**) See L. B~L (~o) and the references given therein. (lo) L. B~J.: A n n . Inst. H. Poincard, 18, 57 (1973).

9

RIASSUNTO

(*)

Si presenta una classe di interazioni fra N particelle completamente hamiltoniane, invarianti rispetto al grnppo di Poincar6. Non si esprimono i movimenti con N-ple di linee di universo dello spazio-tempo, ma con certe sottomolteplicit~ N-dimensionali dello spazio delle fasi q~ per tt a~. L'insieme di qnesti movimenti ~ una moltephcit/~ simpletiea a 62V dimensioni ed i~ determinata dalla scelta di N funzioni in ~ che hanno la parentesi di Poisson nulla su q). Questa molteplicit~ a 6N dimensioni ~ in corrispondenza 1 a 1 con lo spazio delle fgsi del sistema a 6N dimensioni ogni volta ehe sia specifioato un osservatore. Lc transizioni dello spazio delle fasi cosl ottenute per ogni coppia di osservatori costituiscono la dinamica. (*)

Traduzione a cura della l~edazione.

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