Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

I,§2,8

define functions m. They are again called Maslov indices. From theorems

5, 7.1, and 7.2 and lemmas 5.3 and 7, they evidently have the following

properties.

THEOREM 8.

1°) The Maslov index defined by (8.1)1 is the only function

defined on transverse pairs of elements of Aq(l) that is locally constant on

its domain and that makes the following decomposition of the index of

inertia possible:

Inert(.lq, 'q, n,9) = .19) -

.1q) + m(),9, )9) mod q.

(8.2)

2°) The Maslov index defined by (8.1), is the only function

m: Spq(l)\EsPq H Z2q

that is locally constant on its domain and that makes the following decom-

position of the index of inertia possible:

Inert(s, s', s") = m(S) - m(Sr-1) + m(S") mod 2q.

3°) These functions have the following properties:

(8.3)

m(.,q, :t9) + m(ti9, ).q) = I mod q,

m(S) + m(S 1) = I mod 2q,

(8.4)

Inert(s, A, ).') = m(S) - m(A2q, XZq) + m().Zq, XZq) mod 2q

if S E Spq, ),Zq = S.'Iq. We recall that Spq(l) acts on A2q(l).

(8.5)

By §1,theorem 2,3°), assuming q = 2, theorem 8,2°) identifies:

the Maslov index that formula (2.15) of §1 defines on Sp2(l) mod 4,

the Maslov index that (7.2) and (8.1) define on Spq(l) mod 2q.

Definition 8.

Let A E A (§1,definition 1.2). Let us give a definition of

SA E Spq(l)\EsPq that coincides with that of §1 for q = 2.

Recall that m(sA) is defined mod 2. By formula (2.15) of §1, m(sA) = m(A)

mod 2. Then by (7.5), there exists a unique element of Spq\EsPq, denoted

S.4. such that

its projection onto Sp(l) is sA ;

m(SA) = m(A) mod 2q.

(8.6)

I,§2,8-I,§2,9

The mapping

A-3

is clearly surjective and, if q = oc, bijective. If q -A co, the condition

S,, = Se.

is equivalent to the following:

A(x, x') = A'(x, x')

Vx, x' e X,

m(A) = m(A') mod 2q.

9. Lagrangian Manifolds

An isotropic manifold in Z(l) is a manifold on which

d<p, dx> = 0; i.e., >dp; A dx1 = 0; i.e., d[z, dz] = 0.

(9.1)

Its tangent plane is isotropic, and hence it has dimension < 1.

A lagrangian manifold V is an isotropic manifold such that

dimV=1.

Let V be the universal covering space of V (see Steenrod [17], 14.7).

Evidently (9.1) means that there exist functions

cp : V R,

1k: l

- R,

defined up to additive constants, such that

dcp = <p, dx>,

VI(x, p) = (0(x, p) - i<p, x>,

dtji = z[z, dz]

cp is the phase of V and i(i its lagrangian phase.

The I-plane A(z) tangent to V at z = (x, p) e V is obviously lagrangian.

The apparent contour Ev of V is the set of z E V such that %(z) is not

transverse to X*. On V\E,, and on its tangent l-planes, x may serve as a

local coordinate. By (9.2), the equations of V and its tangent I-planes are

then

V: p = (p.(x),

.l(z):dpi

= i cps,x.

dxk.

(9.3)

k=1

Each element of Sp(l),

S: Z(l) -3 Z' f-+ sz' = z E Z(I),

1,§2,9-1,§2,10

maps every lagrangian manifold V' in Z(1) into a lagrangian manifold V

in Z(l), with lagrangian phase

O(z) = i/i'(z') + const. for z = sz'.

If z = (x, p) and z' = (x', p'), the phases o and cp' of V and V' are then

related by

(p (x) -

< p, x> = cp'(x') - 1 < p', x'> + const. (9.4)

If s = sA 0 Esp(,), then p = Ax, p' _ -Ax, by §1,(1.11), and so

tp(x) = cp'(x') + A(x, x') + const,

(9.5)

p = (px = Ax,

p' = (px, = -Ax..

LEMMA 9.

Let SA e Sp(l)\Esp; let z' _ (x', p') e V'\E,,, be such that

sAZ' = z = (x, p) e V\E,,, where V = sAV'.

Evidently these relations define a diffeomorphism x' -+ x of the local

coordinates of V and V'. Its jacobian takes the form

d'x _ Hessx.[A(x, x') + cp'(x')]

d'x'

A2(A)

(9.6)

(In the calculation of Hess,,, x and x' are viewed as independent.)

Proof.

By (9.5), the diffeomorphism x' f-- x satisfies

cp', (x') + As (x, x') = 0.

Then (9.6) follows, because, from §1,1,

A2(A) = det(-Ax:x,,).

10. q-Orientation (q = 1 , 2, 3, ... , co)

The notions defined in this section enable us to supply a complement to

formula (9.6). This is theorem 10, which will be crucial in the sequel.

By a q-orientation of A E A(l) we mean a choice of a it2q E A2q(l) with

natural projection A.

A q-oriented lagrangian manifold, denoted Vq, is a lagrangian manifold

V together with a continuous mapping

V -3 Z H A2q(Z) E A2q(1)

I,§2,10

that, when composed with the natural mapping

A2q(1) - A(1),

gives the mapping

V -3 z f--' A(z) e A(1), where ).(z) is the tangent 1-plane to V at z.

Each element of Spq(1) maps a q-oriented lagrangian manifold Vq into

another.

A q-orientation of V is characterized by the values taken by the locally

constant function

V\Ey

z f

m(Xzq, 2q) E Z2,-

By (5.12), a change in this q-orientation is equivalent to the addition of

a constant E Z2q to the function m.

The statement of theorem 10 will be simplified by the following

definition.

Definition 10.

The argument of d'x at a point of Vq, where the tangent

1-plane is A2q e A2q(l), is defined by the formula

arg d`x = rcm(XZq, ).Zq) mod 2q7[. (10.1)

For example, by (5.13),

argd'x = 0 on X2y. (10.2)

Let

SA E Spq(1)\Espq

(definition 8),

and let Vq be a q-oriented lagrangian manifold in Z(1); denote

Vq = SA Vq .

Let ) 2q and 2q be the tangent planes to V9 and Vq at z' and z = sAZ'. In

formula (8.5), taking S = SA and s =

SA, we have, by (8.6) and §1,

definition 1.2,

m(S) = m(A) =

arg(t (A)).

By theorem 7.3 and equation (9.3) for ).'we obtain that Inert(s,

is the

inertia of the symmetric matrix

I,§2,10-I,§3,1

x,; x

[A(x, x') + T'(x')]

By §1,definition 2.3 of arg Hess, this is

arg Hessx, [A(-x, x') + p'(x')].

Thus, by (8.4), formula (8.5) may be written

m(Xzq, 2q) - m(XZq, A'2q) _ argHessx.[A + lp]

- 1

arg A2 (A) mod 2q,

whence, by definition 10, we have the following theorem.

THEOREM 10. If Vq and Vy are two q-oriented lagrangian manifolds such

that

Vq = SAV4, where SA e Spq(l)\Espq,

then not only are the two terms of (9.6) equal, but so are their arguments

mod 2gir.

It is the special case q = 2 of this theorem that will be used (see §3,

corollary 3).

§3. Symplectic Spaces

0. Introduction

Symplectic geometry makes it possible to state the preceding results in

the following form, which will be used in chapter II.

Z(l) has been provided with a symplectic structure, and moreover with

a particular frame consisting of a pair (X, X*) of transverse lagrangian

I-planes. It is important to state conclusions that are independent of this

choice of a particular frame.

1. Symplectic Space Z

A symplectic space Z consists of R21 together with a symplectic form, that

is, a form

I,§3,1

x Rz1a(z,z')"[z,z']ER

that is bilinear, alternating, real valued, and nondegenerate. We then have

[z, z'] = - [z', z];

[z, z'] = 0 for a given z and all z'

R21 only if z = 0.

Z(l) is provided with the symplectic structure defined in §1,1; the

isomorphism of the symplectic structures of Z and Z(l) is obvious and

has the following consequences.

The subspaces of Z on which [ ,

] vanishes identically are called

isotropic ; their dimension is <, 1.

The isotropic subspaces of dimension 1 are called lagrangian subspaces.

The collection of lagrangian subspaces A(Z) is called the lagrangian

grassmannian of Z; A(Z) is homeomorphic to A(l) and therefore has a

unique covering space Aq(Z) of order q c- {1, 2, ..., oo}.

The projection of Aq E Aq(Z) onto A(Z) is denoted A. The following

definition (see §2,4) makes sense on Z.

Definition 1.1. Let A, A', A" e A(Z) be pairwise transverse.

Inert(A, A', A") is the index of inertia of the nondegenerate quadratic form

z H [z, z'] = [Z', Z,] = [z", z],

(1.1)

where

z E A,

Z C -A"

Z" E

z + Z' + Z" = 0.

Clearly its values belong to {0, 1, ... ,

1), and

Inert(A, A', A") = Inert(.', A", A) = 1 - Inert(A, A", A').

(1.2)

Let Y.^9 denote the set in A9(Z) consisting of pairs of nontransverse

elements of Aq(Z). By theorem 8 and (8.1), we have the following theorem.

THEOREM 1.

For each q c {1, 2, ... ,

oo}, there is a unique function

m : A9 (Z)\1^, -,

Zq,

called the Maslou index, that is locally constant on its domain and that

satisfies

Inert(A, A', A") = m(A.q, A) - m(Aq, A') + m(AQ, Aq) mod q.

(1.3)

It has the following properties:

60 I,§3,1-I,§3,2

Aq) = I modq,

(1.4)

m(Aq, Aq) = m(A, Ate,) mod q.

(1.5)

Theorem 6 of §2 applies to the jump of this Maslov index across Ego.

Definition 1.2. - A lagrangian manifold V in Z is a manifold of dimension

I on which

d[z, dz] = 0; (1.6)

its tangent 1-plane is lagrangian. In other words, on the universal covering

space V there exists a function

0: R such that d =

[z, dz], (1.7)

called the lagrangian phase, which is defined up to an additive constant.

2. The Frames of Z

A framet3t of Z is an isomorphism

R:Z-+Z(1)

respecting [ , ]. If R and R' are two such frames, then RR'-t is called the

change of frame:

RR'-t E Sp(l).

(2.2)

Clearly, if A, A', A" E A(Z), then RA, R1.', RE' E A(1) and

Inert(A, A', A') = Inert(RA, RA', RA"). (2.3)

If V is a lagrangian manifold in Z, then RV is a lagrangian manifold in

Z(1) with phase tpR given by

(pR(z) = O(z) + '< p, x>, where Rz = (x, p) E X ®X Z(1). (2.4)

The apparent contour of V relative to the frame R is the set ER of points

in V at which the tangent l-plane is not transverse to R X *.

On V\ER, x may serve as a local coordinate.

By lemma 9 of §2 we have the following theorem.

'The use of the letter R to denote a frame comes from the initial of the French word repere.

[Translator's note]

1,§3,2-I,§3,3

THEOREM 2.

Let x and x' be the local coordinates defined on V\(ER U ER,)

by two frames R and R' such that RR'-' a Sp(l)\Esp. Then there exists

A (§1,2) such that SA = RR'-'; let rp'(x') = cpR,(z) be the phase of R'V.

Then the local diffeomorphism X 3 x' r+ x e X has the jacobian

d`x

_ Hessx.[A(x, x') + cp'(x')]

d'x'

A2(A)

(In the calculation of Hess,., x and x' are viewed as independent.)

Remark 2.

The frames we have just defined do not allow us to fix the

q-orientations if q > 1.

3. The q-Frames of Z

The q-frames of Z do allow this: each q-frame (q E [1, 2,

... ,

oe}) consists

i. an isomorphism jf:Z -+ Z(l) respecting

ii. a homeomorphism hR: A2q(Z) -+ A2,(1) whose natural image is the

homeomorphism A(Z) - A(1) induced by jR.

If R and R' are two such frames, then the change of frame RR'-'

consists of

1. S = .1RIR'1 E SP(1),

ii. H = hRhR., a homeomorphism of A2,(1) whose natural image is the

homeomorphism of A2(l) induced by s (cf. §2, example 3.1).

To define H knowing s it suffices to give HX Zq E AZq(l) with the projection

sXz E A2().

There are q elements of Spq(l) with image s e Sp(l). By §2,(3.12), they

map X *4 into the q elements of A2q(l) with image sX*2 e A2(1).

The unique element S E Sp,(1) with images E Sp(l) and such that SX Zq

HX 2q thus induces the homeomorphism H of A2q(l). It characterizes

RR'-', which we denote S :

RR'-1 E SPq(l)

(3.1)

R denotes either jR or hR; we write

(x,p)EX O+ X* = Z(1),

R : A2q(Z) 3 %.2q t--+ R22q E AZq(l).

I,§3,3

Evidently

m(A2q, )'2q) = m(RA2q, RA'2q) mod 2q VA 1,, A'2q E A2q(Z). (3.2)

R maps a q-oriented lagrangian manifold V in Z into another in Z(1).

If A2q E A2q(Z) is the tangent plane to V at z, we define

m,e(z) = m(R-'Xzqi A2q)E Z2q.

(3.3)

If x is the local coordinate of V\ER (q-oriented) defined by the q-frame

R, we define

argd'x = nmR(z) mod 2gir.

(3.4)

Example 3.

On R -1X24, arg d`x = 0 mod 2gir by §2,(10.2).

Theorem 10 of §2 evidently allows us to supplement theorem 2 as

follows.

THEOREM 3.1.

In the statement of theorem 2, suppose that V = Vq is

q-oriented and that R and R' are q -frames. Then

RR' -' = SA E SP q(l) (3.5)

and the arguments of the two sides of (2.5) are equal mod 2g7r.

Let us recall that §1 gave the definitions 1.2 and 2.3 of arg A(A) and

arg Hess. By §2,(8.6),

m(A) = m(SA) mod 2q.

The following special case of this theorem will be used in part (iii) of

the proof of theorems 4.1-4.3 of II,§1.

COROLLARY 3.

If q = 2, the half-measure [d'x]'/2 on the lagrangian

2-oriented manifold V is defined by

arg[d`x]1/2 =

mR(z) mod 2n.

We have

[dIx]1/2 = ,A) {Hess..[A(x, x') + 9'(x/)]11/2[d'x']1/2.

(3.7)

In a neighborhood of a point of ER, where dim). n R-1X* = 1, mR(z)

has the following expression, which follows from §2, theorem 6.

I,§3,3

THEOREM 3.2.

Let ).(z) be the tangent 1-plane to the lagrangian manifold

Vatz;zEERmeans

dim(.nR-'X*)> 0.

We stay in a neighborhood of a point of ER at which

dim(ti n R-'X*) = 1;

then for z e ER, the projection of R)i(z) onto X parallel to X* is a

hyperplane:

icjdxj=0.

j=1

1°) There exists a regular measure w on V such that for z E ER, for all

and all k

dx' A

A dxj-' A dpK A dxj+' A . A dx' = cj cKrJ.

(3.9)

2°) If V = V9 is q-oriented, there exists a constant c E Z2, such that

mR(z) = c mod 2q for d'x/m < 0,

(3.10)

mR(z) = 1 + c mod 2q for d'x/ra > 0,

provided that d'xl'm vanishes to the first order where it vanishes (namely

on ER).

Remark. A regular measure to on V is a differential form of maximal

degree 1 on V with everywhere nonvanishing coefficient.

Proof of 1°). The cj entering into (3.8) are not all zero. Say c, # 0;

then, on V at z,

dxZ A

.. A dx' * 0; 3k such that dpk A dxz A .

. A dx'

(3.11)

Since Ejdpj A dxj = 0 on V, we have, for all k,

dp1 A dx' A dxZ A dxk A dx'

+dpkAdxkAdx2A .

(3.12)

where the cap suppresses the term it covers. By (3.8)

dx' = -Ckdxk mod(dx2,

... ,

dzk, ... ,

dx').

Leray J. Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index

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