conformal symmetry of the hamilton-jacobi equation and quantization

IL NUOV0 CIMENT0 VoL. 31 B, N. 2 11 Febbraio 1976

Conformal Symmetry of the Hamilton-Jacobi Equation and Quantization.

C. P. BOu

Centro de Inves t igaci6n en Matemdt icas Ap l i cadas y e n S i s temas Univers idad Nac iona l A u t d n o m a de Mdxico - Mdxico 20, D.F. , Mdxico

M. PENAFIEL l~. (*)

Ins t i t u to de Fis ica , Univers idad Nac iona l A u t d n o m a de Mdxico - M~xico 20, D. F . , Mdxico

(ricevuto 1'8 Settembre 1975)

Summary. - - We discuss the symmetry group of the time-dependent Hamilton-Ja~obi equation for a nonrelativistie free particle in n spatial dimensions. I t is found to be isomorphic with the conformal group of an ( n + 2)-dimensional Minkowski space. A prescription is given whereby quantization is equivalent to selecting a unitary irreducible representation of the subgroup of linear representations over the space-time manifold. The generalization to arbitrary manifolds and the inclusion of spin is discussed.

1 . - I n t r o d u c t i o n .

In the s tudy of par t ia l differential equat ions occurring in physics or applied mathemat ics , one often gains informat ion and insight b y s tudying the symmetr ies of the equat ion bo th f rom the point of view of hav ing physical inter- p re ta t ions of the symmetr ies and f rom being able to generate solutions f rom the symmetr ies . I t also seems tha t in the pas t this approach has m a n y t imes been overlooked. For example , while the symmetr ies of the classical Poisson brackets , i .e. the pseudogroup of canonical t ransformat ions has been studied

(*) Present address: Instituto de Fisica, Universidad de Bolivia, La Paz, Bolivia.

195

196 e . P . B O ~ R an d ~ . PENAFI~L ~ .

extensively (1.2), the closely re la ted Hami l t on - Jacob i equat ion (3) has not been

extens ively studied f rom this poin t of view. I t is the purpose of the present pape r to a t least par t ia l ly fill this gap, and in doing so we will gain some

interes t ing insight into the relat ionship between classical and q u a n t u m

mechanics . We presen t a detai led s tudy of the s y m m e t r y group of the t ime-dependen t

t I ami l t on - Jacob i equat ion for a free nonrelat ivis t ic particle. By s y m m e t r y group we m e a n the unex tended group, t ha t is s y m m e t r y t r ans format ions

are not pe rmi t t ed to depend on the spat ia l t empora l der ivat ives of the Hami l t on pr incipal function. This restr ic t ion is made only for reasons of convenience and tha t the unex tended group has enough interest ing s t ruc ture in itself.

Indeed , in the more general case when potent ia ls are included it would seem not only desirable bu t also necessary to include the more general t r ans formations. We only ment ion here t ha t the ex tended symmet r ies fo rm a Lie pseudo-

group which can be expressed as the semi-direct p roduc t of the pseudogroup of canonical t rans format ions ex tended by dilatat ions with the diffeomorphisms of the real line (4).

I n contradis t inct ion to the ex tended group, the unex tended group for the free part icle is a f inite-dimensional Lie group which for n spat ia l and one temporal dimensions is isomorphic to the conformal group of an (n ~-2)-dimensional

Minkowski space. Thus for n----3 we have a factor group of the group 05,8. I t is easily seen t h a t 05.2 contains the SchrSdinger group (5) as a subgroup and, hence, the centrally extended Galilei group of quantum mechanics (s). I t is the fac t t ha t an equat ion of mot ion of classical mechanics a l ready contains in its s y m m e t r y group the symmet r ies of q u a n t u m mechanics t h a t leads us to a quant iza t ion prescript ion in t e rms of the symmet r ies of the Hami l t on - Jacob i equat ion t h a t avoids some of the usual pitfalls of the canonical quant izat ion. Fo r example , it is well known t h a t the canonical quant iza t ion makes no sense

on compac t spaces, e.g. spheres. On the other hand, our quant iza t ion prescrip-

t ion makes sense on spheres, and gives what one expects in t e rms of un i t a ry representa t ions of the ro ta t ion group, while agreeing with the canonical quantiz- a t ion in Eucl idean space. Our procedure m a y ~lso provide some insight into

(1) L . V . ~ Hove: Mere. Aead. Roy. Belg., 26, (1951); Bull. C1. Sci. Aead. Roy. Belg., 37, 610 (1951); V. UHLHORN: Ark. Fys., 11, 87 (1956); K. B. WOLF: Group Theory and its Applications, Vol. 3, edited by E. M. LO~B~ (New York, N. u 1971) ; R. ABRAHAm: Foundations o] Mechanics (New York, N.Y. , 1967). (2) J .M. SOURIAU: Comm. Math. Phys., 1, 374 (1966); R. F. STREAT]~R: Comm. Math. Phys., 2, 354 (1966); E. 0NOFRX and M. PAURI: Journ. Math. Phys., 13, 533 (1972). (a) H. GOLDSTEIN: Classical Mechanics (Reading, Mass., 1950). (4) This is the case for any first-order partial differential equation. We are indebted to Prof. I. KUPK~_ for showing us this. (5) U. NIEI)]~I~ER: Helv. Phys. Acta, 45, 802 (1972). (6) J.-M. L]~vY-L~BLOND: Journ. Math. Phys., 4, 776 (1963); Group Theory and its Applications, Vol. 2, edited by E. M. LO~BL (New York, N.Y. , 1971).

C O N F O R M A L S Y M M E T R Y OF T H E H A M I L T O N - J A C O B I E Q U A T I O N E T C . 197

the quantizat ion problem for nonlinear equations. I t is clear t ha t our approach applies as well to the relativistic Hamil ton-Jacobi equation.

Amusingly enough it is also known tha t 0s.~ contains not only the Poincar6 group of relativistic mechanics, bu t also the invariance group 0~,2 of electro- dynamics. However , we have been unable to a t t r ibu te any meaningful physical significance to this fact. This would presuppose being able to in terpre t Hami l ton ' s principal funct ion as ~ generalized co-ordin~te.

In sect. 2 the Lie algebra of the symmet ry group of the Hamil ton-Jacobi equat ion is obta ined using Lie's me thod (7). Then in sect. 3 we make a (( light- cone ~ t y p e t ransformat ion to arrive at a covar iant formulation. The geo- metr ical s t ructure of the problem is elucidated by lifting to the cotangent bundle. Final ly in sect. 4 we s tudy the role of the SchrSdinger subgroup and the quant izat ion process. This procedure is then recast in the language of fibre bundles to allow the generalization to a rb i t ra ry manifolds and inclusion of spin.

2. - The de terminat ion o f the s y m m e t r y group.

We consider a general par t ia l differential equat ion given implicitly by

(2.1) F(u,,, ut, u; x, t) = 0 ,

where the funct ion u is at least once differentiable and its derivatives, denoted by u~, ut , are functions of the under lying (n + 1)-dimensional Eucl idean space spanned by (x, t ) = (x 1, ..., x", t). In general the function F of (2.1) can in- volve derivat ives of arbi t rar i ly high order, however the case we will t rea t here is of first order. Following LIE (7) we consider infinitesimal t ransformations of the form

(2.2)

u' = u + sV(x, t, u) + 0(e2) ,

x" = x~ + s~i(x, t, u) + 0(e2) ,

t' = t + e~(x, t, u) + ~ ( ~ ) .

The t ransformations (2.2) are said to be a sym m et ry of (2.1) if u'--~ S '(x , t, u) is a solution of

! ! ! (2.1') F ( u : , , u~,, u ; x , t') = o

wherever u = S(x, t) is a solution of (2.1). Geometrically, one can view the t ransformations (2.2) as t ransformations in R "+2 given by the local co-ordinates

(7) S. LIE: Theorie des Trans]ormationsgru~Ten, reprint (New York, N. Y., 1970).

198 c .P . BOYER and M. PENAFIEL lg.

(x, t, u) which m a p the surface defined b y a solution of (2.1) into another such surface.

To proceed fur ther we compute how the der ivat ives t ransform. The procedure is s t andard and can be found in ref. (7.8). The results are

(2.3)

a x i t

~x,- ~ = ~ - ~(8~, + ~8S~,) + 0(~ ~) ,

at ax" ~(Tx, + ~S~,) + 0( .~) ,

at = 1--~(~, + *~S,) + 0(~ ~) ,

ax ~ - - i S , at' = - - ~ ( ~ + is , ) + 0(~ 2)

Using (2.2) and (2.3) we calculate to order

(2.4)

aS' Sx,-4-e[V,,+ J a x t i - -

aS'

Now let S(x, t) be a solution to the H a m i l t o n - J a c o b i equat ion

(2.5) 1

2m (Sx)2 -~ S t ---- O .

Then the condit ion for the invar iance of (2.5) under (2.2) is

(2.5') 2m (S:,)'- + S;, = 0 ,

which upon using (2.5) and (2.4) to first order in e yields

(2.6) 4m ~ ~s(S.) 4 + ~ (v~,-- m~s) S~(S.) 2 + ~m [(Tz ~- %) 6, ~-

1 D

m

Since S(x, t) is an a rb i t r a ry solution of (2.5), each coefficient of the powers of the der ivat ives of S in (2.6) mus t vanish and we obtain the set of first-

(s) G. W. BLUMAI~ and J. D. COLE: Similarity Methods ]or Di]]erential Equations (New York, N.Y. , 1974).

CONFOI%MAL SYMMETRY OF THE HAMILTON-JACOBI EQUATION :ETC. 1 9 9

order coupled part ial differential equations

~ = O, v , = m~:~, r h = O,

~, + ~, = o (i # j ) , (2.7)

V~, = m ~ ,

~/~ + r t = 2~, (no sum over i ) .

These equations can be integrated in a s traightforward manner to give the infinitesimal t ransformations

m v(x, t, S) = mC~x~ t § -~ Cox~ x' § mC~ox~ § C3t~ § C2t § C~ ,

~'(x, t, S ) = C~ (5~tS m xZ 5~ § mxi x ~) - - ~ ! + C~x~S + (2.8) \

§ C~ox'S § Catx~§ C2x'/2 § C7x'/2 § R~x~§ C~t § C~,

~(x , t, S) = mC~x~S + m ~ C3x'x ~ + mC~x § C~S 2 § C,S + C~ .

From (2.8) we immediate ly compute the vector fields

X1 = ~ L i~ = x ~ , - - x i~ , ,

X~ = t~ ~ + ~ x ~ ~, ,

X 3 =

X ~ =

(2.9) X6 =

X 7 =

X ~ =

X 9 =

Thus it is seen tha t the Lie 1 (n -4- 3)(n + 4).

m 2

1

mx~t~t § mx ix '~ . , § ( t S - - 2 x ~ ) ~ . , § mxiS~, ,

m 2

algebra defined by (2.9) has dimension

200 c . P . BOreR and M. PF, N A F I E L 1~.

I t is easily seen t ha t the vec tor fields X~, . . . ,X~ and Z " fo rm a basis for

the Lie a lgebra of the Schr6dinger group (5) 5 : ~ [SL2,~Q 0. ] ~ W,, where W is the (2n + 1)-dimensional Weyl group. This i m p o r t a n t subgroup will be

discussed more th roughly in sect. 4 in connection with quant izat ion. Looking closely at (2.9) we notice an interest ing au tomorph i sm of the vec tor

fields given b y in terchanging t and S. Expl ic i t ly defining the t r ans fo rmat ion

P• = (t ---> S, S --> t, x ---> 4- x) , we have

(2.1o) { X~ ~ X . , X~ ~+ X~, X~ ~ X~, X~ -> • X~,

+ , + , z . z . .

Thus if we denote the Sehr6dinger subgroup jus t discussed b y 5#(, t) (2.10)

gives us another Schr6dinger subgroup 5:~ ) . We will see t h a t the au tomorph i sm (2.10) is ac tual ly an inner au tomorph i sm under the group action and thus 5:(~ t)

and 5f~ ) are conjugate subgroups. The s y m m e t r y (2.10) will give us a clue to the s t ruc ture of the algebra (2.9) which we will exploi t in the nex t section. This

s y m m e t r y also tells us tha t , if S(x , t) is a solution of (2.5), then apply ing the t r ans fo rmat ion P~ and using the implicit funct ion theorem gives us ano the r solution.

3 . - T h e c o v a r i a n t f o r m u l a t i o n a n d t h e g r o u p s t r u c t u r e .

The s y m m e t r y discussed a t the end of the previous section suggests t and S p lay a symmet r i ca l role. Accordingly we introduce the var iables

(3.1) x ~ t"~-- y X n + I - - v / ~ t - - ,

and the covar ian t no ta t ion /~ = 0, ..., n + 1, with the metr ic go0 = - - g , -----

---- - - g,+~,,+~ = 1, g,~ --~ 0, /x :/: v. Then upon introducing (~)

(3.2)

Ms, ---- x ~ . - - x ~ , ,

D = x ' ~ - ~ x ~ . ,

K~ = 2x~x " ~ -- x 2 ~ , ,

(9) G. MACK and A. SALAd: Ann. o] Phys., 53, 174 (1969).

COI~FORMAL SYMMETI~Y OF THE H.AMILTON-JACOBI ~,QUATIOI~ ETC. 2 0 ~

where x 2 = x"xu, we find tha t the generators

(3.3)

Mio -- ~/2 X~ + m

M.+~.o = - - ( X ~ - - X ~ ) ,

M~+L~ = - - ~

M . = Li j ,

D = X , §

/), = X~,

1 Po = - ~ (X~ + mX~) ,

1 P.+I = ~/--~ ( X l - - reX. ) ,

2 K , = - - - X ~ ,

m

satisfy the commuta t ion rules

[M~, M ~ ] = (g~ M~o + gvo M++ - - g.~ M~Q - - g~ M ~ ) ,

(3.4)

[M#~, Po]

[P . , P.]

[D, P . ]

[M.., K.]

[K., P.]

= (g~oP~,- g~,,P~) ,

= [g~, K.] = [M.~, D] = 0 ,

= -- P . , [D, K~] = K . ,

= -- 2(M.~ + g.~D).

Thus in this formulat ion we immediately recognize the Lie algebra of the

conformal group 0~+~,~ in an (n-t-2)-dimensional Minkowski space M "+~. In

the physically interesting case n----3 the group is 05,~. I t is seen t h a t the

Hamil ton-Jacobi formulat ion appears in the light-cone or inf ini te-momentum

variables (10) obtained by the inverse t ransformat ion of (3.1).

(lo) K. BAI~DAKCI and M. B. HALP]~R~: Phys. Rev., 176, 1686 (1968); H. BACRV and N. P. C~tA~G: An~. o] Phys., 47, 407 (1968); G. BURDET, M. PEI~RII~ and P. SORBA: Comm. Math. Phys., 34, 85 (1973).

202 C.P. BOYER and M. PENAFIEL N.

Now the group action can be obtained from the well-known action of the r group:

1) Lorentz t ransformat ions: genera ted by M,,

x , = A t x~ with A t e 0n+ia ;

2) t ranslat ions: generated by P ,

!

x~ = x~ + a t , a t e R;

3) di latat ion: generated by D

r =

x~ Qx., ~ e R +;

4) special conformal t ransformat ions: genera ted by K .

x~,-- C~x ~ C~ e R. !

x , = 1 - - 2 C ' x ~ C~x 2 '

Writing these t ransformat ions in terms of the light-cone variables t and S we obtain the symmet ry group for the Hamil ton-Jacobi equat ion:

x ,i : A i j x ~ -~- Ai+t ~- m - l A i_ S

1) t' = A++ t § m - I A + S -4- A+i x~

S ' = m A + t § A - - S + m A - i x i ;

�9 t' S ' = S - 5 a - ; 2) x ' i = x i + a ' , = t + a + , (3.5)

= = S' = QS; 3) x 'i ~x i , t ' ~t ,

x '~= a-l(x~ t, S ) [x ~ - C~x 2 - - 2 m - l C ~ t S ] ,

4) t ' = a--X(x ~, t, S ) [ t - - C+(2m-~ tS - x~)],

S' = o~-l(x ~, t, S ) [ S - - C - ( 2 t S - - mx~)] ,

where

and the parameters indicated above with + and -- can easily be expressed in terms of their covariant counterparts .

The new solution of (2.5) is given by (3.5) implicitly. In order to find the explicit solution we have to solve the system of equations (3.5). This m a y

(30/qFORMAL SYMMETRY OF THE I-IAbIILTON-JACOBI E Q U A T I O ~ ETC. 2 0 ~

not be possible at certain points of R~+2; however such points form a set of

measure zero (Sard's theorem (H.r..)).

We now discuss how the geometric s t ructure can be elucidated and the

s y m m e t r y group found in a simpler way by introducing the concept of the

cotangent bundle (~3). Generally let M be a C ~ manifold with (X a) aS local co-

ordinates at p r M, then the cotangent space T*(M) at p is spanned by (dx~),.

The cotangent bundle can be defined (~) by the disjoint union

(3.6) T * ( M ) = (.J T * ( M ) . ~EM

:Now a topology can be defined on T*(M) which makes it a C ~ manifold.

Wi th the space T*(M) there is associated the projection map~:T*(M)--> M with zt(~) = p, ~ c T*(M). The fibre z-~(p) is the cotangent space T*(M) at p.

:Now let a Lie group G act on M g:p-->g(p), then this induces a mapping in

T*(M) dg:T*r by the usual chain rule for derivatives

~(g'x) ~ ~x ~ ~x ~ ~x'~

:Now dg is a bundle isomorphism which provides a representation of G on T*(M). I n our case we have M----R '~+2 and the trivial bundle T * ( M ) = Rn+2x R ~+2. The conformal s t ructure of the group G becomes elucidated by introducing

the graph W(x, t, S ) ~ 0 of the solutions of (2.5). Then W ~ - - W z S ~ and

W~ ~----WsS t and the Hamil ton-Jacobi equation (2.5) becomes

3 (3.7) 2m (W")~-- Ws Wt = 0 .

Then by introducing the t ransformat ion (3.1) we have

(3.8) wows=0 .

:Now any section of T*(M) can be writ ten as d W = W~,dx ~. Thus (3.8) is a cone in the cotangent bundle T*(M) and by the above isomorphism the

symmet ry group of the I tami l ton-Jacobi equation is isomorphic to the group

(11) S. STERNBERG: Lectures on Di]]erential Geometry (Englewood Cliffs, N.J . , 1964). (12) Actually since not all solutions are submanifolds (e.g. cones) one must use a slight generalization of Sard's theorem. Moreover, we must stay away from the point where a(x, t, s) vanishes, or we should compactify R ~+2. (13) N. STEENROD: The Topology o/ Fibre Bundles (Princeton, N.J . , 1951). (14) F. W. W~NER: Foundations o] Di/ferential Ma~,i/olds and Lie Groups (Glenview, Ill., 1971). The definition of tangent and cotangent bundles given differs from the usual one given in ref. (11,13), but it is more convenient for out purposes.

204 c.P. BOYER and ~. PENAFIEL N.

which leaves the cone (3.8) invariant. I t is well known tha t this group is 0~+~.~,

and the group t ransformations can be wri t ten down as previously. This ap-

proach has also the advantage of providing us with a global description of

the symmetry . The group 0,+2,~ does not act effectively on M ~+~ however.

This can be seen by linearizing the group action. The group 0~+2,~ acts l inearly

on the cone defined by {~" ~ 2 2 = 0}, where O, n + 3 , �9 ~ o - - "'" - - ~ + 2 A - ~ + a a . . . . ,

and we extend the metric tensor g,~ to g,b by adding g~ = ~ g b b = - 1 while

the off-diagonal components vanish. Then the co-ordinates x~ of M ~+2 can be

defined by homogeneous variables

( 3 . 9 ) x ~ = ~/~

and the group action obtained is tha t given previously. However, we see t h a t

the t ransformat ion ~ , - - > - - ~ in 0.+2,~ is mapped by (3.9) to the ident i ty

act ing on M -+~. Therefore, the conformal group C -+z,~ on the Minkowski space

M "+~ is the factor group

C "+~'~ ~ O(n A- 2, 2) /Z~,

where Z~ = {~a--~ =t= ~a}.

We now discuss some of the global symmetries in the form of discrete t rans-

formations. I t was seen previously tha t the t ransformations P ~ = { t - + S ,

S --> t, x --> • x} are automorphisms of the algebra O.+m. In fact if we use the

covariant formulation it is easy to see tha t these transformations are members

of 0.+2.3. I f we write S'--~ t and t ' = S and use the covariant co-ordinates

(3.1), it is seen tha t this corresponds to the t ransformation

(3.20) x ~ = coshflx ~ sinh f lx , ,+l,

x ''+~ = sinh flx ~ -- coshfl x n+~ ,

where coshfl = (m + 1/m)/2 and sinhfl = (m -- 1/m)/2, thus P~ are members of the subgroup 0,+1,1 of rotations of M-+~. Notice fl does not connect (3.10)

to the ident i ty so P+ is not in the connected component of On+l,~; however, / ) - is in the connected component if n is odd and is not if n is even. The sym-

metries P• are nontrivial symmetries of the Hamil ton-gaeobi equation, i.e.

they cannot be obtained merely by inspection. Another such symmet ry is

the covariant t ime reversal

T = (x ~ -+ -- x ~ x,+l -+ x-+l, x -+ x)}

CONFORMAL SYMMETRY OF THE HAMILTON-JACOBI EQUATION ETC. 2 0 5

or in terms of S and t

T~--{t--> - S , S-->--mt, x---~x I .

Clearly T e 0~+,,~ but it is not connected to the identi ty. Final ly we ment ion tha t the inversion symm et ry given by

is a member of 0~+2,~ which is not in the connected component . Notice tha t I is not defined on the cone with its centre at the origin in R ~+2. ~ o w the group C ~+2.2 acts t ransi t ively on R ~+2, in fact we have

(3.11) R ~+~ ~ r -~ 1, 1) ~) D] ~ A~+2,

where (~ and ~ denote direct and semi-direct product respectively, D is the one-parameter group genera ted by the covariant dilatations, and An+~ is an (n-~ 2)-dimensional Abelian group generated by the special conformal transformations. Previously we had lifted the group action to the cotangent bundle and obta ined a cone. Indeed cones play a special role in the base space R ~+2 a l so - - the nonsingular t ransformations of C "+~.2 map cones into cones. Such a special solution is the cone at the origin given by S ~ mx~/2t.

4. - Linear representation and quantum mechanics.

I t is well known tha t the classical t t ami l ton-Jacobi equat ion affords a close connection with quan tum mechanics. More precisely, the I Iamil ton- Jacobi equat ion is the geometrical-optics limit of the SehrSdinger equation. Indeed it was the original approach of SCHRSDINGER himself. Now, given a si tuat ion as above where one equation is the limit of the other, we m a y ask what happens to their invariance groups or how does kinematics behave uuder taking a limit. In general, the limit group must be of the same dimension or bigger. Usually two situations arise: i) the limit group is u contract ion of the original group, ii) the limit group contains the original group as a subgroup - - t h e process of embedding. We have already seen tha t the invariance group of the Schr5dinger equat ion . ~ is a subgroup of the invariance group of the t t ami l ton-Jacobi equat ion 0n+:,2, thus we have an embedding in the limit.

The inverse of the above procedure is cer tainly of interest, i.e. given an equat ion and its invariance group, classify all generalizations which give back the original equat ion and group in a certain limit. For situations of type

2 0 6 c . e . BOYeR a n d ~ . P.E~'A.FI~L 1~.

i) above this is the problem of deformations of algebras and groups, which is still an open problem al though some results ~re known (see ref. (~5) for a review). For type ii) the situation is somewhat s imple r - - tha t of restr ict ion to subgroup. Although all subgroups of 0 .+m or even 0~,2 are still not known, the procedure for obtaining them is (le). We do not wish to dwell fur ther on these questions here, bu t only to ment ion tha t subgroup reductions other than the one considere4 might also prove to be of interest physically.

A remarkable feature of the s y m m e t r y group 0~+~,2 is tha t it contains the SchrSdinger group ~ as a subgroup. This is best seen f rom the realization of the algebra given by (2.9). Indeed an impor tan t subgroup of ~ . is the (2n -~ 1)- dimensional Weyl group genera ted by the subalgebra X~, X~ and X s. Thus we see tha t the s y m m e t r y group of the Hamil ton-Jacobi equat ion contains the seeds of quan tum m e c h a n i c s ~ t h e Weyl group or algebraically the Iteisen- berg commuta t ion relations. P u t another way the sy m m et ry group of the t tami l ton-Jacobi equat ion contains the centrally extended Galilei group of quan tum mechanics, not the unex tended Galilei group of classical mechanics (e). I t is emphasized tha t this is not the case for other classical equations such as Newton's equations.

We consider the Hamil ton-Jacobi equat ion in its homogeneous form

1 (4.1) 2-~ (~.)2 + ~wt = 0 ,

where ~(x, t) ~- exp [S(x, t)] and now ~ is allowed to be complex. Equat ion (4.1) is similar to eq. (3.7); however now we are not embedding the problem in R n+2. Making the above change of variables, i.e. S~- In ~, we can wri te the group t ransformations (3.5) as well as the Lie algebra (2.9) in terms of the new co- ordinates. The group t ransformations (3.5) can then be viewed as a nonlinear representat ion of the eonformal group On+m acting on functions over R "+I. Moreover the subgroup of linear representations is precisely the SchrSdinger group ~fn. We can then derive the SchrSdinger equat ion by choosing a a irreducible un i ta ry representa t ion of ~n. Similar in spirit to the work of KOSTANT (17) we then have:

Quantization and hence the SchrSdinger equation is equivalent to selecting a unitary irreducible representation o] the subgroup o] linear representations of the symmetry group o/the Hamilton-Jacobi equation.

(15) C. P. BOYER: Rev. Mex. Fis., 23, 99 (1974). (is) j . 1)ATERA, I ). WINTERNITZ and H. ZASSF, NHAUS: Journ. Math. Phys. 16, 1597, 1615 (1975). (17) B. KOSTANT: Lectures in Modern Analysis and Applications, Vol. 3, LNM 170, edited by C. T. TAA~ (Heidelberg, 1970).

CONFORMAL SYMMETRY OF THE HAMILTON-JACOBI EQUATION ETC. 207

The only catch is in the word linear, which is co-ordinate dependent . We must use the homogeneous form (4.1), which then gives rise to the linear uni- t a ry representat ion (5) of 5~

(4.2a) [T(g-~)~p](x, t) =- #(g; x, t)v/(g. (x, t))

with the action given by

t+Z R_.~+__,,t+_~ (4.2b) g . ( x , t ) = 7~l + oc(t+ fl) , ~ l + a ( t + fl) ] '

where R ~ O(n), a, v e R ~, o~, f ie R, 7 ~ R+ and multiplier

(4.2c)

[ - - imh~(x , t) "] ~(q, x, t) = r-o,~c~ + ~(t + ~)],~,~ e~p L 2 ( i u '

h~(x, t) = ax 2 @ 2R" x" Iota-- (1 @ aft)v] @ a~ot -

- - (1 + otfl)tv ~ + 2 a t a ' v .

For now we take yJ to be C ~ and of compact support in x and C I in t, so yJ e C~ (R'). However this representat ion is not irreducible since the t ransformations (4.2) leave invar iant the SchrSdinger equation, restr ict ing ourselves to the subspace of C~ (R n) which contains solutions of the Schr6dinger equations, i.e.

1 (4.3) i~t -~ ~mm y~x.= = 0 ,

we obtain an irreducible representat ion of 5~n. The representa t ion can be made un i ta ry by the completion of Co(R n) with respect to the norm induced by the inner product

(4.4) (w~, ~) =fd~x V,~(x----7 t) v,~(x, t) .

So now we have yJe ~f~(Rn). I t is noticed tha t in the above procedure there nowhere appears the funda-

menta l constant of quan tum mechanics h. This can be t raced back to the choice of ~v made in (4.1) and can be unders tood in terms of symmetries as follows. I t is easily seen tha t the generator X7 in (2.9) generates a one-parameter group of outer automorphisms of the Schr6dinger group Sf~. In fact on R ~+2 this one-parameter group acts as

(4.5) x'=- ~ - l x , t '= t , S ' = h-~S ,

2 0 8 c . P . ~0YF.:R a n d M. P~.~AI+IEL lq.

where the choice ?~-~ is suggestive. Correspondingly we have (4.1) given in

t e rms of the p r imed var iables with ~ ' ~ - e x p [S'] . This then gives rise to an irreducible subspace condit ion (4.3) in t e rms of the p r imed variables, which

upon wri t ing in t e rms of x and t becomes

~2

(4.3') i~, + ~m ~o=.= = o .

Thus it is seen t h a t h appears as a p a r a m e t e r of a di la tat ion of C "+~,~ which produces conjugate Schr6diager subgroups St(, a~. The classical l imit ~ - + 0 of 5p(n~ is singular~ bu t the Schr6dinger equat ion passes over to the Hami l ton - n

J a c o b i equat ion as is well known.

We are now interes ted in how this procedure migh t generalize to a rb i t r a ry

manifolds and the inclusion of spin. To see this we briefly discuss our procedure in the language of fibre bundles (~a). F i rs t we a lways deal with the homogeneous equat ion (4.]) and allow ~ to be complex. Then G acts on M x C and leaves (4.1) invar iant .

~ o w M X C can be viewed t r iva l ly as a complex line bundle wi th M as the base and C as a fibre. We search for a m a x i m a l subgroup H act ing on M of bundle-preserving maps (~3) such t h a t the d iagram

(4.6)

M • C(h'r(h))) - M • C

M - h ' > M

is commuta t ive , and T(h) acts l inearly on each fibre. The cross-sections of the bundle are the functions ~0(x, t) and the act ion

of H on the bundle induces ~n act ion of the fo rm (4.2a) on the cross-sections.

We then look for subspaces of the space of cross-sections such t h a t the restr ic t ion of the representa t ion T(h) to the subspace is irreducible. To imp lemen t uni- t a r i t y we int roduce a bil inear I t e rmi t i an fo rm on the fibres [~1, ~02]-->~lYJ2 and an inner p roduc t on the cross-sections b y means of a quas i - invar iant measu re dr(x).

We should keep in mind here t h a t ordinari ly in q u a n t u m mechanics the inner p roduc t is built as an integral over the spat ia l co-ordinates only. T h a t this can be done in a t ime- independent fashion is a weU-known consequence of (4.3). Of course, this is not a necessi ty in the general si tuation. Suffice it to say t h a t we have an inner p roduc t on the cross-sections given b y (4.4). T h e condit ion for un i ta r i ty is then gua ran teed by the choice of # in (4.2c).

The point of this discussion is to see t h a t this procedure is readi ly genera- l izable to a rb i t r a ry manifolds M and bundles ~ . For functions over an arbi- t r a r y manifold which car ry spin s degrees of f reedom, we mus t replace the t r iv-

4~0NFORMAL SYMMETRY OF THE tIAMILTO~-JACOBI EQUATION ETC. 20~

ial product bundle M • C in (4.6) by a nontr ivial bundle ~ whose fibres are (2s ~-1)-dimensional vector spaces r over C. The bundle-preserving map (4.6) easily generalizes and again we demand tha t T(h) acts l inearly on each fibre through ~--~#~jv~j. The cross-sections are now vector-valued functions ~(x , t) over M and the irreducible subspace condition (4.3) suitably generalizes (an example with M---- R ~+~ would be the Levy-Leblond equations (18)). The bilinear Hermi t ian form on the fibres then generalizes to [ ~ , ~2] --~ <~1, ~2>, where <-,. > is an inner product on ~/F. The inner product on the cross-sections then generalizes to

(4.7) (w,, =fd t), t)>,

and uni tar i ty is guaranteed by the choice

dr(g" x) (4.8) I#(g; x, t)] 2 - dr(x)

We now give a ve ry brief discussion of how our quant izat ion prescription should go for a case where the canonical quantizat ion fails, namely when the spatial universe is compact , e.g. a sphere S n. In such a case the symmetries are more restr icted and the temporal and spatial symmetries untangle completely. We will no longer get a Schr6dinger-type equat ion (e.g. a rigid ro ta tor (19)) f rom specifying irreducibil i ty but should add it as an auxil iary condition. We will, however, obtain un i ta ry irreducible representat ions of the group SOn+l, thus circumventing the difficulties with canonical quantization. These tech- niques should be of interest in the quant izat ion problem for ex tended objects (,0).

Finally, we can add potentials, in which case we probably have to s tudy the ex tended group except perhaps for quadz'atic Hamil tonians (21), i.e. the a t t rac t ive and repulsive harmonic oscillator and linear potentials. Our approach is similar in spirit to t ha t of others (2.17) who s tudy quant izat ion over sympletic manifolds. Indeed the fully ex tended sy m m et ry group of the I tami l ton-Jacobi equations contains the pseudogroup of canonical t ransformations, however whether the quaut izat ion schemes are equivalent or not is not known at this time.

Qs) J . -M. LEvY-LEBLO~D: C o m m . M a t h . P h y s . , 6, 286 (1967). (19) J. D. SMIT~: NUOVO Cimento, 22 B, 337 (1974). (2o) A. O. B~UT: contribution at the International ConJerence on Mathematical Physics (Warsaw, 1974). (21) U. •IEDERER: Helv. Phys. Acta, 46, 191 (1973); Proceedings o] the I I International Colloquium on Group Theoretical .Methods in Physics (Nijmegen, 1973); C. P. BORER: Helv. Phys. Acta, 47, 589 (1974); C. P. BOYER and K. B. WOLF: Journ. Math. Phys., 16, 1493 (1975).

14 - 1l Nuovo Cimento B.

210 c . P . BOY'ER and ~ . PEN&FIEL ~ .

We are g rea t ly i n d e b t e d to I. KuPKA for n u m e r o u s discussions especia l ly

on the geomet r i c aspects of the work. We also t h a n k E. G. KALI~INS, M. I~O-

SIt:[NSKY~ ~. I:)LEBANSKu a n d P. WINTER1NITZ for the i r c o m m e n t s a n d in te res t .

�9 R I A S S U N T O (*)

Si discute il gruppo di simmetria delI'equazione di Hamilton-Jacobi dipendente dal tempo per una particella libera non relativistica in n dimensioni spaziali. Si trova che esso ~ isomorfo al gruppo conforme di uno spazio di Minkowski ad n ~ 2 dimensioni. Si ds una prescrizione per cui la quantizzazione ~ equivalente alla scelta di una rappre- sentazione irriducibile unitaria del sottogruppo delle rappresentazioni lineari sulla molteplicits spaziotemporale. Si discutono la generalizzazione a molteplicit~ arbitrarie e l 'inclusione dello spin.

(*) Traduzione a cura della Redazione.

I~OHt[~OpMHaSI CHMMeTplL~ ypaBHeHlm FaMH.m~THlla-~IgO6H H tmauTonaHne.

PealoMe (*). - - M~I o6cym~aeM rpymiy CHMMeTpI414 HC 3aBacamero OT BpeMeHH ypaBneHI4~ BaMHn~TOHa-J;IKO6H ~JI~ HepeYI~IT~nCTCKOfi ~IaCTI~I~I B IIpOCTpaHCTBe n-n3MepeHHI~. IIony~aeTc~, qTO 3Ta rpymm H3OMOpqbHa C KOHC~OpMHO~ rpym10~ (~-2)-MepHoro npOCT- paHCTBa MI~KOBCKOFO. I-[pe~JIaraeTc~I pet~eI~, B KOTOpOM KBaHTOBaHHe 3KBI~BaYleHTHO BbISopy ym~TapHoro HenpHBO~tMOrO npe~cTaBnem~ no~rpyiIn~i nmiefiHBiX npe~cTa- BJIeHHI~ Ha BCA~M npOCTpaHCT~H~o-BpeMeHHOM MHo~KeCTBe, O 6 c y m ~ a l o T c ~ oSo6me~tHe

Ha IIpOH3BOYIBHBIe MHOhYeCTBa H BKYIIOqeHHe cIn~Ia .

(*) llepese3eno peOamlue~.

conformal symmetry of the hamilton-jacobi equation and quantization

Documents