Analyticity and quantization

L]~TTER~ AL -NUOVO C~M]~N~O VOL. 3, N. 1 1 Gennaio 1972 Analyticity and Quantization. E. ONO~RI and M. PAURI Istit~to di di ]?isica dell' Universith - Parma Istit~to Nazionale di Fisica N~cleare - Sezione di Milano (rieevuto il 15 Novembre 1971) A general technique of <~geometrical ~) or <~canonical ~) quantization has been firstly exploited by VAN HOVE (~) for Euclidean phase spaces /22.,~ and, successively, genera- lized b y SOURIAU(:") to a wide class of symplectic manifolds. This theory is intrinsic (i.e. co-ordinate-independent) and satis]actory/rom the beginning in that the operators ~, which it associates to the classical dynamical variables/(q, p), are really self-adjoint (not simply symmetric as a priori they are in the ordinary algebraic approach); moreover, the operators ~ satisfy the commutation rules required by standard Dirae formulation of the quantization procedure. However, already for the Euclidean ease, the theory does not provide a solution o] the full Dirac problem. Actually, it contains no prescription for finding, if any, a suitable subspaee ~ of ~cP2[~22,] where the Heisenberg group (generated b y ~, ~, "i) is irre- ducibly represented and the Hamilton operator / t is defined: exp [ i t l ~ ] ~ c ~ . The existence of this i n v a r i a n t irreducible subspace must be verified case b y ease. STRE- AT~n (3) has shown how to define it in the ease of the one-dimensional harmonic oscil- lator. However, already in the simplest cases of Hamiltonians which are not linear or quadratic in the canonical variables, ~ cannot be constructed. For these reasons the theory of canonical quantization is better called a <~prequantization ~) theory. I n the case of non-Euclidean phase spaces, the Iteisenberg algebra itself has no longer an intrinsic meaning because the canonical variables loose their global character; then the full Dirae problem is not even defined. We have proposed in ref. (4) a new approach to this matter which amounts to a reformulation of the full Dirae problem and renders the quantization a system-dependent procedure. Essentially, it consists in abandoning the Heisenberg group as a universal quantization group substituting for it a quantization group c~ which is characteristic of the particular classical system to be quantized; precisely, the phase space ~22, and the Hamiltonian flow are entirely defined in terms of it: .Q~ is a homogeneous space (~) /~. VAN HOVE: M~em. Acad. Roy. Belg. CI. Sci., 6, 26 (1951). (2) J . M . SOCRIAC: Comm. Ma~h. Phys., 1, 374 (1966); Structure des sys~dmes dy~amiques (Paris, 1970). (3) R . F . S t r e a t e r : Comm. Math. Phys., 2, 354 (1966); 4, 217 (1967); C. ]:TZYKSON: ComY~. Math. Phys., 4, 92 (1967). (4) E. ONOF~I a n d M:. PAURI: Dynamical quan~ization, I s t i t u t o di Fisica dell'Universith, P a r m a , pre- p r i n t ( J u l y 1971). 35 36 E. O/~OFRI and M. PAURI of ~ , i.e. the action Y ~ of ~ in D2~ is canonical and transitive and the Hamiltonian flow ~ is defined as a one-parameter subgroup S ~ of Y ~ . Finally, the maximal dynami- cal-symmetry group go of the Hamiltonian is contained in ~ in the form g0 | D. Then, a Hamiltonian system is dynamically q,t~ntizable if ~ can be prequantized (~ la Souriau ~). The prequantization brings to a ~'educible unitary representation ~2 ~ ) of ~ within the Hilbert space of Lebesgue square-integrabIe functions over the so-cal- led (, quantum manifold ~> ~ § constructed over the phase space. The true quanti- zation is accomplished by the selection of an irreducible constituent :~ of ~(llh), which we called the quantal representation. This selection amounts to requiring that all pos- sible observables of the quantum system are contained in the enveloping algebra of and corresponds to the choice of an irreducible representation of the Heisenberg algebra, in the standard procedure. The usual Schr5dinger aspect of quantization, whenever it exists, is possibly recovered a postcriori through the definition of ~,canonical operators )>Q, P {which do not come from an intrinsic quantization, in general) in terms of which the Hamiltonian operator m a y assume a ((correspondential ~> structure. In ref. (4) all these features of dynamical quantization have been evidenced b5~ studying the n-dimensional harmonic isotropic oscillator and the n-dimensional hydrogen atom. Some points were not completely clarified, however, and relevant questions needed more satisfactory answers. In particular, the existence and uniqueness of ~ for a given classical system were not completely clear; since the automorphisms of a symplectic manifold define an infinite Lie pseudogroup, the choice of a finite Lie algebra containing H has a high degree of arbitrariness a priori. Furthermore, the basic quautization condition, i.e. the selection of ~ , though stated in terms of very natural requirements, was not free from ambiguities. The purpose of this note is to give a first account of new results which i m p r o v e sub- stantially the previous approach. The basic conceptual point is the explicit recogni- tion of the rote that a'~alyticity plays in the abstract process of quantization. AetuM]y, the ambiguities in the definition of ~ and in the selection of ~ are reduced to the choice of a suitable complex structure in the phase space; ~ is defined as a group of holomorphie transformations and ~ is the unique holomorphic component of ~(~jt~) (in a sense to be specified in the following). At the same time we obtain a reformulation of the whole approach which widens the class of dy~amically quantizable systems. Without entering into details here we can say that this class can be roughly identified with the class of stable Hanfiltonian systems, a fact that, we feel, has a significant epistemological appeal. We give now in some detail the general reformulation of the theory and discuss some examples. In particular, for the harmonic oscillator we generalize the result of STnEATER. Special emphasis deserves the connection with Hilbert spaces of holomor- phic functions. The fundamental new requirement is the following: there exists a complex structure j (5) in (-O-2,~, co) where o) is the fundamental two-form, such that l) g(X, Y) ~ o ( J X , Y) is a Hermitian form; 2) o equipped with g is a homogeneous Kaehler mani/old, i.e. the group of holomorphic automorphisms ~h is transitive; 3) the Hamiltonian flow is described by a one-parameter subgroup of ~ . We choose (r as the dynamical quanti- zation group, i.e. ~ ~ (~j~. The relevant implications of these requirements are the following: 1) ~ is a Lie group of dimension no~ greater than n ~ + 2n and -Q,z,,~ ~ / K with K compact; the simplest examples of such manifolds are the homogeneous bounded domains in ~'* for which the group of all nonsingular holomorphic transformations coincides with the group of isometrics. 2) K has a noudiscrete center, i.e. there is a compact generator in the (5) S. I(OBAYAS~I a n 4 ]i:. NOM:[ZU: Foundations o] Dit]erential Geometry, Vol. 2 (New Y o r k , 1963). ANALYTICITY ANJ) QUANTIZATION ~7 algebra, of K w]fich c o m m u t e s w i t h all the g e n e r a t o r s of K ; if, in particular, K If0(~ SO.,_, t h e n ~ / K o is "l preq'*tan.tization, of (.(2:~, ~). 3) The complex s t r u c t u r e J p r o v i d e s a n a t u r a l solution to the p r o b l e m of the reduction of the rcl)rescntation ~ / ~ ) of ~ : to o b i a i n the r e d u c t i o n it is sufficient t h a t t h e action of ~ , lifted to the qua,ntuln manifohi - Q 2 ~ , be h(,h)morphic, i.e. lhc t r a n s f o r m a t i o n s in each fibre s' s ~. at(z, z*) be a.ctm~.lly h o l o m o r p h i c : s r = s 4- ~;,(z); if this is the case, thc m u l t i p l i e r exp [(i/h)..'ry(z)] in :~(~/~) is h o l o m o r p h i c as well; consequently, the subspace of holom.orphie ].~tnetions is an i~waria~t irred.u, eible su.b.r for ~ according to a theorem by K o ~ a Y a s m (~) a d a p t e d to our case. This holom.orphie .represe-ntalio.n is defined as the (( q u a n t a l ,) rep- resentation ,~. 4) The t [ a m i l l o n i a n systems for which it will be possible to c a r r y t:hrough this p r o c e d u r e have a priori a f n n d a m e n t a l p r o p e r t y : since the. I I a m i l t o n i a n flow is a group of i.~om,etries, the flow is s t r i c t l y stable (t31R~:m)FF) with respect to t h e met- ric g (:). Sine(, the q u a n t i z a t i o n procedure is extensiblc w i t h o u t ambiguities toc las- sieal generatin~ functions of the form ll = . ~ ( ~ ceeXe) , where X o define ~ basis in the L i e algebra of ,~" and t" is such t h a t X~r is complete, Birkhoff st~tbtlity is tt()f mMn~ained for lhis e x t e n d e d class; however, all these d y n a m i c a l systems arc stable with respect to the orbits. E:ramples. We shall discuss here the one-dimensional oscillator t o g e t h e r w i t h its generalizations to ~ dimensions and the one-dimensional n o n l i n e a r system H = -~ ,~ (pO+ x".~-~/x~) which has been r e c e n t l y reconsidered as it.. leads ~o an e x a c t l y solvable N - b o d y i n t e r a c t i o n (s). l) l,ct .(2~-- R", t o = d.cAdy ( x , y f'.artcsian canonical co-ordinates). L e t t i : = ~(x ~" -b y:); then X n = x(~/~y) - y ( 5 / ~ x ) . According to the a b o v e prescription we m u s t look for the most general complex s t r u c t u r e such t h a t 1 ) t o ( J X , J Y ) = co(X, Y), 2) exp[tX/,] is an holomorI)hic isomciry, 3) the group of "mtomorphisms ()f (-Q2, g) is t r m l s i t i v e in R ~. The result, is the following: the most general g in locM polar co-ordin~tes takes the form (dr) ~ (l) g =i (e/4)r z i- 'r2( 1 -(e/4)r'~)(dq@' , where c is the negative consi~ant c u r v a t u r e of the I I c r m i t i a n form g (also c-- 0 is allowed). The complex struclm'c J defines a m a p p i n g z : R ~- ~(~ given by (2) z = ( 1 -- (e/4) r'~)-~r cxp [iqq , which is an i s o m e t r y of (~2o, g) onto the bounded d o m a i n Dx(e):{z~(~]!z] < (--4/c)-~}. (*) ,% KomtYxsHt: Proc...liner. Math. Soc., 12, 359 (1961). (') ]~. ABltAHA.~t:I~'mzmlations o] Mechanics, Chapt. V (New York, 1967). (s) J. ~'RI~, "~r )~AN'DRASOV, wl~A. V. ~.~{OH()I)Z.~'SKY, ) g I~IILIR ~tnd ]). ~VZNTE~N'ITZ: P h y s . L e f t . , 16, 354 {1965); I.'. C:kLO(~ERO: J o u r u . . ' ~ [ a t h . l>hys., 10, 2191, (1969); 12, 419 (1971); 1'. CAM]Z, A. GERARDI, C. MARCIITORO, E. ]~RESUT~[ ~ n d E. SCACCIXTELLI: Exact solution ol a time-dependent quardum harnumie oscillator w i t h a singular perturbation, Istituto di l.'isica dell'Universith, Rom~, Nora intern~ No. 301 (1970). 3~ E. ONOFRI a n d ~. ~AURI I t holds dz* dz dz*A dz (3) g= 2 ( . 1 + (c/4)z*z) ~' t~ T h e t r a n s i t i v e group of h o l o m o r p h i e (nonsingular) t r a n s f o r m a t i o n s of D~(c) onto itself is well k n o w n to be ~z(- 4/c)~ + (4) c < O: SU~: y(z) = ( - 4 1 c ) 8 9 1 8 9 + o~* (lal z - IZT= 1), (5) c = 0: E~ : y(z) = exp [ir + ~ (r e (0 ... 2~), , e ~ ) . S~1 contracts to E , w h e n c ~+ 0 according to fl/~* = T(-- c/4)89 a/~* = exp [ir The pre- q u a n t i z a t i o n is g i v e n b y (6) c < o: G = {%'~,er ~- I~1 ~ = - 2/c}, 0 = iGd%- ~]d~). SU1. ~ acts t r a n s i t i v e l y in ~3 l e a v i n g 0 i n v a r i a n t : y(Zo, Z~) = (fl*zl + a*Zo, azl + flzo). The canonical p r o j e c t i o n of [J~ onto .Q~ is g i v e n b y z = (--4/c) 89 If we p u t = (-- 2/c)89 0 a n d the t r a n s f o r m a t i o n s of SUez t a k e t h e following f o r m : z* dz (7) 0 = -- 2/(ic)d(log ~) -~ 2i(1 + (c/4)z*z) ' (s) ~,(~) = r 4/c)~ + ~*)-~ (z t r a n s f o r m s as in (4)). The i n t e g r a l i t y c o n d i t i o n ~0 = -- 4n/c = Nh (*) selects o n l y c = - 2/57h. W e r e c o v e r V a n H o v e ' s expression for 0 p u t t i n g s = - i N h l o g $; the functions (9) ~(z, z*, s) = e x p [is/T~] yz(z) e x p [ i N a r g ~](1 -- z * z / 2 N h ) ~12 ~f(z) constitute an irreducible i n v a r i a n t subspace for a u n i t a r y r e p r e s e n t a t i o n of S~I,1, namely (lO) (G~o)(z) = (r ~*)-~ ~+ 9 the i n v a r i a n t scalar p r o d u c t b e i n g (11) (~1, ~2) = J V J exp [/(s - s*)lh] ~l(z)* ~,).(z) 21 -- 1 / " dS*Ad5 = l i m ~ J ' " -" , , ' t ----z"*"z ~ 2,~. ,-,"-" 2,% f z ~ * w 2 ~ z ~ - - : -->--,v / "-' z 2i /)1 (*) Note that Nh is exactly the fixed value of the canonical invariant in the irreducible canonical realization 3ff~1.1(~2). ANALYTICITY AND QUANTIZATION 39 W e rediscover the well-known r e p r e s e n t a t i o n s D+/2 given b y BARGMANN (9): (12) e = o: ~=~• zE~2{Im(s) = z*z/~}. E~ acts holomorphi[ea]ly i n ~a: y(z) == exp [ir + v, ~(s) = s + i(v* exp [ir + ]v]2/2). The functions exp [is/hi ~(z), w i t h ~ holomorphic, form a n i n v a r i a n t irreducible sub- space: explicitly we have (13) (U~yJ)(z) =: exp [-- (3* exp [ir + [z{2/2)/2~] ~(V(z)), 1 (14) (91, ~ ) =:-- [exp[--i(s*- s)/l~]~l(z)*~fo(z)r z3 lfc dz*Adz = xp [-- z*z/2t~] YJl(z)* yJ~(z) 2i F o r N -> c~ we verify t h a t D+/2 contracts to the r e p r e s e n t a t i o n of E~ except for a phase factor exp [ilVr i n fact, D+I~ is a t r u e representation, whereas the r e p r e s e n t a t i o n of E 2 is a projective one: (15) (a* 4- fl*z/(2Nli)89 -zz = (a*)-~(1 + 3" exp [ir -zz = = exp [i5Yr 1 - ]vp2Nh)~( 1 4- 3" exp [ir -~ , (16) (1 -- z*z/2Nt~) N-~ ~ exp [ - z*z/2h] . The result of S T ~ A T E a has been derived here i n a straightforward m a n n e r . I t is easy to see t h a t also the quantization through S U~ 1 is equivalent to the q~sual one : i n the Hilbert space of the q u a n t u m oscillator In> = (n!)-~(a~)~[0>, we i n t r o d u c e the overcomplete base of eigenveetors of t h e operator (*) (17) A = (1 4- a C a / N ) - 8 9 A{z> = z*/(2~)]]z>. W e find t h a t <zI~ > ~ (1 -- z*z/2Ni~)~y~(z) with ~ holomorphic i n z*z < 2Nh;Iz> are the ~ principal vectors ~>in the representation D+~ ; i n p a r t i c u l a r for s c~ t h e y reduce to B a r g m a n n ' s coherent states (lo). 2) I n the n - d i m e n s i o n a l ease, ~2~ ~ R ~ , we i n t r o d u c e the complex s t r u c t u r e as follows : (18) z~: .(22,~-> D,~(c)c~'~:z~ 1 -- (c/4) ~ (x~ 4- y~) (x~ 4- iy~) . 1 (') V. BAI~GMANI~: Ann. o] Math., 48, 568 (1947). (*) Note that A :~V89 +Hs)-IH , Hs, H+, H_ being the generators of SUI,,. Qo) V. B ~ G ~ C N : Comm. Pure Appl. Math., 14, 187 (1961). 40 ~. ONOFRI and .M. P A U R I T h e n t h e g r o u p of h o l o n m r p h i e t r a n s f o r m a t i o n s is ,q[,,.~ for c < 0 a n d c o n t r a c t s to IU,, for c - + 0 . T h e prcquantiz, a t i o u is giv(,n b y (,9, {.o.o, ..... 1 :',: -'4. ,SU,.: acts l i n e a r l y on the a~'s l e a v i n g t h e o n c - f o r n t 0 i n v a , ' i a n t : 0 - - i a*da o -- a~(ta k ; t h e c a n o n i c a l l)rojection ~z,+:-~-(22,, is g i v e n by z ~ = - ( - - 4 / c ) ~ a ~ / a o a n d t h e (<c o r r e c t >> p a r a m e t r i z a t i o n of t h e fihrcs is g i v e n b y ~ . (-- 2/c)89 o. W(; lind (20) 0 = 2i/cd(log:-) + )= z~.dzk / 1 t (4/c)~... T h e s u b s p a e c of f u n c t i o n s of t h e f o r m (--v~/.,(z].... , z , ) constitute an inwtriant irredu- cible s u b s p a c c fi)r "t u n i t a r y r e p r e s c n t a l i o n of ,S'U,,,,: r1 ~ k,j-- 0, 1 .... , n , ?'(~M = ~-'~aj, zk = ~'~./ao, (21) ( U~,r)(:, . . . . . &) = U~-;j) ')- ~ (i (N>n) . t [ c r c :also t h e c o n n e c t i o n w i t h t h e u s u a l f o r m u l a t i o n is o b t a i n e d b y i n t r o d u c i n g t h e o v c r c o m p l c t c base of c i g e n v e c t o r s of t h e c o m m u t i n g o p e r a t o r s Ak : (9 1 4- \ 2o, aJ2r )' 1 ak (k = 1 .... , n ) . T h e l i m i t N - , oo is ~ c o n t r a c t i o n SU,,a-->IU, a n d we lind t h e u s u a l f o r m u l a t i o n ill [trills of c o h e r e n t states. We stress t h a ! it is p o s s i b l e to q u a n t i z c in t h i s way c v c r y t i a m i l t o n i a n v e c t o r field w h i c h b e l o n g s to t h e a l g e b r a of S U , : for a n y v'due, of :V. i n p a r t i c u l a r , t h e Ha~Mtonian el the an.isot.ropic harm~)~ic oscillator is g i v e n b y H=- ehH~.t.- ... 4- w~H~ (Hi b e i n g t h e C a r t a n b a s i s of U~c SU,,.:): 3) L e t -% = {x, v e n ~ l x > o } , • :(p"-+x~+,~/x~), ..>o. T h e c o m p l c x s t r u c t u r e is fir:cd b y z:-Q 2 -+ D ~ C ~ : p =- i(x - x/x) dz*Adz (m ~" v - - i(.,, ~ T i . ; i ' ~ . . . . ~ (7-- z * ~ ) ' T h e a c t i o n of SU~I i n Dj is t h e u s u a l one, ciz. cq. (4) w i l h ,; = -- 4. W e find t h a t SU~.~ can be p r c q u a n t i z e d o n l y for i n t e g e r w d u c s of ~, since in t h i s case u is e x a c t l y t h e ('anonical i n v a r i a n t N; h o w e v e r , t h e g e n e r a t o r s of SU~I l i f t e d io ~3 a r e complete aild t h i s i m p l i e s t h a t t h e r e e x i s t s a g l o b a l a(;tion of t h e u n i v e r s a l c o v e r i n g g r o u p SU~.~ o n ~ a (sac P:tr.ArS, rcf. (n)). I n t h i s way, for (;very n > 0, we. o b t a i n % r e p r e s c n t a t i o n a- of SU~.~, 1)~/2 w h i c h is a r a y r e p r e s e n t a t i o n of SU, i (see rcf. (.%4)); t h e r e is n o r e a s o n to discar(1 t h e s e r e p r e s e n t a t i o n s , of course. (n) I f . S, PALArS: 3fern. _4mer. 3 I a t h . S e e . , 22, C h a p t . XV, T h . I I I (1957). ANALYTICITY AND QUANTIZATION 41 W e n o w s h o w t h a t t h e r e e x i s t s e l f - u d j o i n t o p e r a t o r s Xo, , Po, i n D~+~ s u c h t h a t t h e H a m i l t o n i a n a s s u m e s a n (( a l m o s t e o r r e s p o n d e n t i a l ~) s t r u c t u r e . L e t us define (23) (Xo~,)~ = 2 ( H a -- Ha) = -- (z - 1)2d/dz d- fl(1 -- z ) . T h e (generalized) e i g e n f u n c t i o n s of Xo, a r e f o u n d t o be (24) (z]x} ~ JiZ(x)(1 - z) -u cxp [ - x2/(1 - z ) ] . To n o r m a l i z e t h e m i n s u c h a w a y t h a t ( x t x r } = 8(x-x') we e x p a n d o n t h e e n e r g y e i g e n f u n c t i o n s ~0,(z) ---- [ ( z ) , / n !]89 (*) : co co (25) (z[x) = H ( x ) ~. k ! ((-1 ~x~)~ ~)~+~ -- JV'(x) ~ %(z)[(~)r/r !]-89e x p [ - x ~ ] / ~ - ~ ( x s ) . k=O k=O I n t r o d u c i n g a ~ cut-off ~ ~ - ~ 1, we h a v e co (26) <xlx'} = ~AQx)H(x')~o "~" r(z+~'~F(z)r!L,-1, ( L'-~. (x':) exp[- x ~ - x '2] = , r(~) , ~ 9~ . _ . ( 2 x ~ ' v'~l(1 - ~)) ~-~7~ H ( x ) H ( x ) ~-(xx )- +. e x p y- xx']~(~ - x'). In conclusion rr(.)]. 9 r 1 l+z] (27) <~lx>= (1-~ -- ~ X 2 - 1-.J - I t is easy to v e r i f y t h a t t h e H a m i l t o n i a n t a k e s t h e f o l l o w i n g f o r m : with <,,.,(-5 (29) 11~1I' ~dxl<-l,p)L' 9 0 W e see t h a t we c a n o b t a i n H t h r o u g h S c h r S d i n g e r ' s p r e s c r i p t i o n , b u t t h e i n t e r a c t i o n c o n s t a n t ~ m u s t b e p r e v i o u s l y r e n o r m a l i z e d . L e t us n o t e t h a t t h e s t a t e s 1 l+z* (*) (a). =-r(a +n)/r(a). ~2 E. ONOt~RI and M. P A U R I have very simple transformation properties under SUb.l: (31) ;./(8.* + In particular, their time evolution is periodic. We are examining the other classes of irreducible tIermitian symmetric spaces to produce further examples. The hydrogen atom seems to fit with SOn J(SOn S02); note that S 0 , 2 does not include the maximal symmetry group SOn+l; on the other hand, S0~+1~ does not admit a I-Iermitian 2n-dimensional homogeneous space, although it admits a sympleetie one (see ref. (4)). Mathematical details and a discussion about the significance of analytieity from a general point of view will be given in a forthcom- ing paper. We are indebted to Prof. F. ])UIMIO for his kind interest in our work.