On the Leray-Maslov quantization of Lagrangian submanifolds

JOURNAL OF

Journal of Geometry and Physics 13 (1994) 158—168 North-Holland

GEOMETRYAND

PHYSICS

On the Leray—Maslov quantization of Lagrangian submanifolds Maurice A. de Gosson Residence du Parc 32, allée des Vanniers, F-54600 Villers les Nancy, France Received 28 July 1992 (Revised 1 February 1993)

We define a generalized Maslov index on A 4 (n), the covering of order 4 of the Lagrangian Grassmannian A (n). That generalized Maslov index, defined on all pairs of elements of A4( n), allows us to state Leray—Maslov’s quantization rule independently ofthe choice of frame.

Keywords: Maslov index, Lagrangian submanifolds 1991 MSC: 58 F 06

1. Introduction In his treatise “LagrangianAnalysis and Quantum Mechanics” [4] Leray gives a quantization condition for Lagrangian submanifolds VofZ=OR~xl~,equipped with the symplectic form w=dq A dp. That condition [4, ch. II, § 3.6, definition 6.21 can be stated as follows: (1) For every two-frame R the function ER3imR(i)+çoR(~)eR/47L

(1.1)

is defined modulo 1 on V\ER, where J~ is the universal covering manifold of phase V, ER ofRVand the apparent of J~’~ 0R the m~contour is the Maslov relativeto the frame R, ER its projection, ( index on V relative to R and i/v 0 is Planck’s constant (see ref. [4, ch. I, §~2.5, 3.2, 3.3] for these notions). Condition (1) is independent of the choice of the frame R: if it holds in one frame it holds in every frame; however, its statement requires a choice offrame. We show in this paper that it is possible to give a quantization condition [condition (2) below] which is equivalent to (1), but which does not require a pre0393-0440/1 994/$7.00 © 1994 Elsevier Science B.V. All rights reserved 0390-0440(93)E0033-L

SSDI

M.A. de Gosson /Leray—Mas/ov quantization

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liminary choice offrame. This is made possible by the use of a Maslov index, free of any transversality assumption, and which is a slig1~tvariant of the Maslov index we defined in our papers [2,3]. Condition (2) thus obtained can be written iF

1_L

4~m~j 2mi’~’~~’yeir1 where C~,=~ dq, m~is the jump of the Maslov index along y. That condition when applied to the classical trajectories of the harmonic oscillator very simply yields the energy levels predicted by quantum mechanics; it has in fact a great similarity with the classical Bohr—Sommerfeld quantization condition; we therefore call condition (2) the “generalized Bohr—Sommerfeld quantization condition”. Our notations are essentially those of ref. [3]. For xc P we denote by ~ (~) the class of x modulo 4 (modulo 8). Let Z = P “ x IP” be equipped with the symplectic form w(z, z’) =x’~y—x~y’,z= (x, y), z’= (x’, y’). We denote by A the Lagrangian Grassmannian of (Z, w): leA if and only if 1 is a n-dimensional subspace of Z, and w=0 on 1. Sp is the symplectic group of (Z, a), i.e., the group of all automorphisms ofZ leaving invariant; Mp is the metaplectic group, i.e., the 2(P”)w of Sp unitary representation in L 2, the double cover of Sp. Mp is generated by generalized Fourier transforms SA given by

(~) J n/2

SAf(x)=

A(A)

dx,

e~f(x)

with A(x, x’)= 1Pxx—Lx~x’+ p....pT QQT = mldet(L) ~2, m~r=Arg(det L) ~Qx’~x’, mod 2~r(see ref. [4, ch. I]),det(L) vci[ #0, 1, oc)A(A) being i parameter. Finally, for q=2, 3, we write 7Lq =7L/qZ. a ...

2. Extended Arnold—Leray—Maslov indices We denote as in ref. [3] the signature of a triple (1, 1’, 1”) of Lagrangian planes by a(l, 1’, 1”) and its class modulo 8 by à(l, 1’, 1”); recall [1—3]that a is a Spinvariant, antisymmetric, 7L-valued cocycle on (A (n) ) locally constant on each set ~,

Ak,kk’ ={(l, 1’, 1”) :dim(lnl’) =k, dim(/’r~/”)=k’, dim(/nl”) =k”} for 0~k,k’,k”~n. Furthermore a(l, 1’, 1”) ~n+dim(lr~l’)

+dim(l’n/”) +dim(lrml”) mod 2

As Dazord [5] we define the modified signature d by

.

(2.1)

MA. de Gosson /Leray—Maslov quantization

160

d(l, 1’, 1”) = ~( —n—dim(lnl’) —dim(l’nl”) +dim(lnl”) +a(l, 1’, 1”)).

(2.2)

It is immediately clear by (2.1) and the properties of a that Proposition 2.1. The Dazord signature d is a Sp-invariant 7L-valued cocycle, locally constant on each Ak k ‘,k’~ Observe that due to the presence of the minus sign before dim(lnl”), dno longer is antisymmetric; however, d(l, 1’, 1”) =d(l”, 1’, 1). For fixed leA we define d1(S,S’)=d1(s,s’)=a(l,sl,ss’l)eZ4, 5,

51

e Sp

(2.3)

being the projections of S, S’; it is clear from proposition 2.1 that (2.4)

i.e., d1 is a Z4-valued cocycle on the group Mp. Define now m,(S)=~(~1(S)+ñ+dim(slnl))eP/47L,

(2.5)

where ~ is the Leray—Maslov index on Mp relative to 1 [3, § 3, p. 2681; it immediately follows from the properties of~1[3, thm. 3.21 that m1 (S) e 74 and that Theorem 2.2. (a) The function m1is the onlyfunction Mpi-+7L4 having thefolowing properties. (1) m1(SS’) rn1(S) m1(S’) =d1(S, S’) (i.e., d1 is a coboundary ofm1); (2) the mapping (S, 1’) rn1 (S) d(sl, 1, 1’) e7L4 is locally constant for s1r~l’=lnl’=0,hence m1 is locally constantfor slnl={0}. (b) Furthermore m1 has thefollowing properties: (3) m1(S)+m1(S~)=ñ+dim(lnsl), m1(I)=ñ, (4) m1(—S)=rn1(S)+2, (5) m1(S) —rn1 (5) =d(5l, 1,1’) —d(sl, si’, 1’) +d&i(slr~l)—di~i(sl’nl’). 1 and S=SA; then /.t1(SA)= It is instructive to compute m1(S) when l={0}xR’ ~(SA)=2th(A)—h [3,formula (3.12) thm. 3.2(i)];thus —

—

—*

—

(2.6) hence m 10—_ m0 is the Arnold—Maslov index modulo 4 [4, ch. I, §2,81. This motivates following definition:

Definition 2.3. We call m1 : Mp—JL4 the extended Arnold—Leray—Maslov (ALM) index on Mp, relative to 1. Let now 1, 1’ be two elements of A4, the covering space of order 4 ofA=A(n); there exists SEMp such that T’=Sland if S’eMp is such that 1’=s’t, then S’=SH,

M.A. de Gosson /Leray—Maslov quantization

161

H being in the stabilizer St (1) of! in Mp; from theorem 2.2(1) and the definition (2.3) of a, it follows that m1(S’)=m1(S)+m1(H)+a(l,sl,sl) hence, using (2.2) and the antisymmetry of a, m1(S’)=m1(S)+m,(H)=m1(S)+m1(H). Now, m1(H)=~~i,(H)=O in view of lemma 4.1 of ref. [3]; hence m1(S) is independent ofthe choice of S in Mp such that T’=Sland only depends on the pair (!~1’); we therefore denote it m (1, 1’); the properties of m (1, 1’) follow from theorem 2.2 (compare with ref. [3, thms. 4.1 and 4.2]): Theorem 2.4. (a) Thefunction m :A~(1,1’) F-*m(1, 1’)~7L4 is the only function A~—+7L4which is locally constant on {(1, T’):lnl’={O}} and such that (1) m(T, T’)—m(l 1”)+m(I’, T”)=n+a(l”, 1’, 1). (b) m has furthermore thefollowing properties: (2) m(T, 1’)+m(I’, 1)=ñ—c n(lnl’), rn(1, l~=O, (3) m(S1, S1’)=rn(I, 1’), VSeMp. Using the same argument as in ref. [3, thm. (4.2)(iii)], one readily proves that the following identity holds, where i=,~/Ji,ke7L: (2.7) note that (2.7) then appears as a particular case of property (3) of theorem 2.4 when k=k’. For a triple of pairwise transverse lagrangian planes (1, 1’, 1”), one defines [4, ch. I, §2,4] the index of inertia i(!, 1’, 1”) as being the index of inertia of the quadratic form on 1 (or 1’, or 1”) defined by Q(z)=w(z, z’) (or Q’(z’) =a(z’, z”), or Q”(z”)=w(z”, z)) for (z, z’, z”)elx!’Xl”, z+z’+z”=O. That index is related to the signature a(l, 1’, 1”) by a(!, 1’, 1”) =2i(!, 1’, 1”) —n if!nl’=!’nl”=lnl”={O} from that relation and theorem 2.4(a) it follows that Corollary 2.5. The restriction m’ of m to (A ~ )‘ = { (1, 1’) :1 n 1’ = {0} } is the only locallyconstantfunction (A~)’—~7L4 such that: m’(T, I’)—m’(I, T”)+m’(I’, 1”)=i(l, 1’, 1”)

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MA. de Gosson /Leray—Maslov quantization

for In!’ = 1’ n 1” = 1” n 1= {0}.

Proof Immediate in view of theorem 2.4(a) since we have ~÷d(l”, 1’, 1) =n+ ~ ( —ñ+à(l”, 1’, 1)) =1(1, 1’, 1”) whenlnl’=l’nl”=l”nl={O}. Corollary 2.5 identifies m’ with the index onA4 defined by Leray [4, ch. I, § 2,5, § 2,8, thm. 8]; Leray’s index was an extension of Maslov’s index, whose correct definition had been given by Arnold [61,thus justifying Definition 2.6. We call m :A~—~l4 the extended ALM index on A4. For fixed l0eA consider the multiplication on Sp x 74 defined by (s,th)(s’,th’)=(ss’,th+th’+d10(s,s’))

(2.8)

;

in view of the cocycle property of d~0it is clear that Sp x 14 is a group G10 for that multiplication; (I, ñ) is the identity of that group and (s, th) = (s~, —

ñ—th+dim(slnl)).

Let leA, SeMp; in view of theorem 2.2(4) the image of m1(S) in 12 only depends on land on the projections of S; we denote that image m1 . Theorem 2.7. For every l0eA the mapping MpoSi.-~(s,m10(S))eG10 is an isomorphism ofMp onto the subgroup 10={(s, th); m em10} of G,; the restriction ofthat mapping to {S : sl0 n l~ = {0} } is a homeomorphism. Proof Absolutely similar to the proof of theorem 5.1 in ref. [3].

E

We may thus, for given 1, identify Mp with
(*)

is a bijection. The restriction of that bijection to A(O)={T:lnlØ={0}} is a homeomorphism.

M.A. de Gosson /Leray—Maslov quantization

163

Proof The mapping (*) is injective for, if 1=1’, then [= jkt for some k ci, and m (1 ~) = m (1’, T~) = m (1, 1~) + k~ then implies k 0 mod 4, hence 1= T’. The mapping (*) is obviously surjective. The second statement follows from the fact that m is locally constant on A(0) [theorem 2.4(a)] and hence continuous. We may thus identify A4 with AX 14; transporting the topology ofA4 on that set the identification also becomes topological; we denote by 10 the topological space thus defined. Using the identification A4 = 10 it is easy to calculate explicitly the ALM index: Corollary 2.9. For T=(l, S), 1= (1’, X’) in 10 we have: (1) m(T, 1’) =,~—,~“+ñ+d(lo, 1, 1’); hence in particular (2) m(T, 1~)=X—X0+dim(1n10) for To= (lo, X0). Proof Obvious in view oftheorem 2.8 and formula (1) in theorem 2.4. Sp acts transitively and continuously on A; that action is covered by an action of Mp on A4 (more generallythe covering group SPq of order q acts on the covering space A24 of order 2q, see ref. [4, ch. I, § 2,3, thm. 3, 3°].The corresponding action of (SpX14>10 on 10 is easily described using the Maslov index. Theorem 2.10. The group 10 acts continuously and transitively on 10 via (s,~2)(l,X)=(sl,~2+X+d(!0,s!o,sl))

Proof Obvious in view of the cocycle property of d, the continuity of the action being a consequence of proposition 2.1 and the definition of the topologies of 10, 10.

3. A quantization condition for Lagrangian manifolds in Z Let V be a connected lagrangian manifold in Z, J~ its universal covering space. We assume that V (~)is two-oriented, i.e., that there exists a continuous mapping Vn z i.-~I(z) eA4 (J.~oI~-~ 1(1) eA4) (a “two-orientation”) which composed with the natural projection A4—~A gives the mapping z—~l(z)= T~V (f—*l(i)=T~J~’) (thetangentspaceto V(J~)atz(~)). For each T3eA4 we define rn10: V—~l4by:

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MA. de Gosson /Leray—Maslov quantization

3.1. m~0(f)=m(1(i),10) [=X(f)—Xo+d1~i(lnl0)

if one identifies A4 with

,0,1(f) with (l(f),X(f)),10 with (10,X0)]. The index m/0 has the following properties: m~0(i)—m11(f)=n+a(l(z),l0,l1)—m(T0,11) .

Proof Formula (1) in theorem 2.4(a).

(3.1) n

3.2. m10() is constant on each connected component of the set J~\E10,where E10={z :1(z) nlo={0}} is the “apparent contour” of Vrelative to 1~. Proof Obvious again by theorem 2.4(a), since the mapping i—41(i) is continuous. The first homotopy group ir1 ( V) acts on V: if yer~( V), Ic J~then yle Vand has the same projection z e V as I. The following result is essential: Proposition 3.3. The difference mî0(yI) therefore denoted thy; clearly th~e Z4.

—

rn~0(I)only depends on yeir1 ( V); it is

Proof In view of formula (1) in theorem 2.4 we have —m(I(yI), Io)+rn(1(f), 10)=ñ+d(10, 1(1), l(yI))—m(I(yI), 1(1)) Now, both mappings z i.-~1( yf) and I T( I) are continuous, and cover the mapping Ii.-~l( I) e T~ since I and yl have the same projection z e V; it follows that —~

~,

—rn(I(yI), 1~)+m(T(I),1~)=—m(I(yI), 1(1)) noting that d( l~,1(1), l(yf) ) = n, a being antisymmetric. The Lagrangian manifold J~ is connected since Vis, hence (1( yf), 1(1)) describes a connected subset 0fA4XA4 this implies that m(1(yf), 1(1)) has a constant value in view of theorem 2.4(a). —

The symplectic form a = dp A dq vanishes on every Lagrangian manifold V; hence there exists a function ~: c’—~~, the phase of V, such that dç~= (p, dx> (sensu stricto, ça is only determined up to an additive constant since V is connected). Let v0 e i [1, cc) be apurely imaginary number (in quantum mechanics v0 would be 2iri/h, h Planck’s constant), and consider the function F(f)=~m10(I)+~~~:~(f)

.

(3.2)

Replacing I by yi, ye m1 ( V), the phase q, (I) increases by a constant value, de-

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165

noted C~,and depending only on y; in fact C~=ço(yI)—ço(I)= ;

(3.3)

hence, taking into account proposition 3.3, F(yI)—F(I)=~th~,+

C~.

(3.4)

Definition 3.4. The connected Lagrangian manifold V satisfies the generalized Bohr—Sommerfeld quantization condition if ~th~+

-~

2iri

~,ei4

(3.5)

for every yem1(V). Example. Consider the one-dimensional harmonic oscillator whose Hamiltonian function is given by 2+m2w2q2) (rn, w>0). (3.6)

H(q,p)=

(p

The classical trajectories in phase space are solutions of Hamilton’s equations q(t) =ÔH/äp, p(t) = —aH,/aq and are thus the ellipses Va,a:

q(t)=acos(wt+a),

The universal cover ~

p(t)=—mawsin(wt+a),

a,areal.

of Va~is parametrized by

~q(t)=acos(wt+a), ~a,ap(t)=—m0sin(0jt+1~T5), (teP) (u(t) =wt Setting z= (z, u) = (q, p, u), the differential dço=p dq is given by dço(I)=~ma2w2(l—cos2oit) that is, ~(I)=~(ma2w2u+pq). Let y: [0, 2ir/w]~ti.-*(a cos(wt+a), —maw sin(wt+a)) be the generator of it 2wir,and, using 1 ( V) =ir~ (Z, +);2.9 one formula (1)(S’)= in corollary weimmediately get as well checks that C~=ma th~=m

2 10(q,p,u+2ir/w)—m10(q,p, u)=

(intuitively, one has to rotate a two-oriented line twice to get it back to its initial

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MA. de Gosson / Leray—Maslov quantization

position); choosing v0=i/h, h=h/2ir, Vaa is thus quantized if and only if 2wme7z rna for all n ci, which condition is equivalent to +

a2=(2n+l)h/m2w,

n=0,1,2,...,

(3.7)

hence the quantization of the trajectories of the harmonic oscillatorin phase space; reporting the values of a2 given by (3.7) in the classical formula E= ~ma2w2 giving the total energy ofthe oscillator (3.6), we get the usual energy levels E~=(n+~)hw, n=0,1,2,...

(3.8)

predicted by Quantum Mechanics, and which are the eigenvalues E~of the Schrödinger operator

H(q,p)=—

~

+~mwq

associated to the Hamiltonian (3.6). We do not explain here the agreement of the values given by (3.8) with these eigenvalues; instead we relate our quantization condition (3.5) to the quantization condition given by Leray [4, ch. II, § 3,6, p. 144]. Let V be a connected Lagrangian manifold (in Z), equipped with a two-orientation denoted I~—~1(I). Let R eMp; we define RVas being the image of Vby r, the projection of R onto Sp; clearly RV is a Lagrangian manifold, and using the identification of Mp with /0 (theorem 2.7)), together with theorem 2.2, it is also clear that RVis two-oriented; we will, in this situation, call R a two-frame ofZ (it is not exactly the definition Leray gives of a two-frame [4, ch. I, § 3,3]; it is, however, equivalent). Let now w be the Lagrangian phase of V; it is the function w: J)’_~p,uniquely determined up to an additive constant, such that d~(I)= ~w(z, dz); the phase of Vrelative to the two-frame R is then

(x,p)=Rz, ztheprojectionofl.

(3.9)

Let ~*EA4 have projection X*={0}XP~~, ref. [4, ch. I, § 3,3] defines mR(z)=m(RX,1(z))

,

(3.10)

where rn’ is the restriction ofrn to the set { (1, t) : ln l’={O}} (see corollary 12.11); thus m~ is not defined on V, but only on V\ER, where ER = {Ie J~:rl(z) nX*={0}} is the apparent contour of Vrelative to the two-frameR. Now, Leray’s quantization condition (called “Maslov’s quantum condition” by Leray) can be stated as follows: 3.5. For every two-frame R, the function

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167

ER3Ii-+—~mR(I)+~WR(1)EP/47L is defined modulo 1 on V\ER (ER the projection of ER on V). Remark. If condition 3.5 holds for one two-frame R, it holds for all two-frames [4, def. 6.2]. Proposition 3.6. The Bohr—Sommerfeld quantization condition (eq. 3.5) and the Leray—Maslov quantization condition 3.5 areequivalent; they thus define the same quantizedmanifolds. Proof In view of (3.9) the Leray—Maslov quantization condition is equivalent to 3.6. GR(I)=—~mR(I)+~-~fr(I),

IeJ~\ER,

is defined modulo 1 on V\ER. Let ye it1 ( V); yl and I have the same projection z, hence in particular yE~= ER. In view of formulas (2k) and (3) in theorem (2.4) we have m~(yl) —rn~(I) =mR~.(z) —m~~*(yI)

for Ic 1~~\ER; by proposition 3.3 the r.h.s. of this equality depends neither on R, nor on I, hence mR (yl)

—

mR (I) =

~i(yI)—yi(I)= hence ~i(yI)—w(I)=C~

—

thy;

on the~otherhand it is obvious that

~w(z,dz)=

~pdq,

it thus follows from condition 3.6 that +~thy+~

hence Leray—Maslov quantization implies Bohr—Sommerfeld quantization. Suppose conversely ~(m~0(yI)—m~0(I))+ ~

(~(yI)—~(I))ei4,

that is, ~(mio(yI)—mi0(I))+ ~ (i,u/(yI)—~/.I(I))ei4

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MA. de Gosson /Leray—Maslov quantization

for all ye ir~( V); choosing an element R in Mp such that Rl0=X*, which is possible since Mp acts transitively on A4, we can rewrite that relation as ~(m(I(yI), R~*)_m(1(I), R~*))+ ~ (~(yI)—~(I))ei4, 7\ER again in view of formula (2~)in propowhich is equivalent to 3.6 for IE J sition 2.4.

References [1] M. Demazure, Classe de Maslov II, Exposé no. 10, Séminaire sur le fibre cotangent (Orsay, 1975/ 76). [2] M. de Gosson, La definition de l’indice the Maslov sans hypothèse de transversalité, C.R. Acad. Sci. Paris Ser. I, 310 (1990) 279—282. [3] M. de Gosson, Cocycles de Demazure—Kashiwara et gbometrie métaplectique, J. Geom. Phys. 9 (1992) 255—280. [4] J. Leray, Lagrangian Analysis and Quantum Mechanics (MIT Press, Cambridge, MA, 1981); Analyse Lagrangienne, R.C.P. 25 (Strasbourg 1978; College de France 1976—77). [5] P. Dazord, Invariants homotopiques attaches aux fibres symplectiques, Ann. Inst. Fourier Grenoble 29(2) (1979) 25, 78. [6] V.1. Arnold, On a characteristic class intervening in quantum mechanics, Funct. Anal. Appl. (1967) 1—14.