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

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I,§3,3

Denoting

dp1 Adx2 A...Adx',

we may write (3.12) as

C I Ck m= d pk A dx 2

A ... A dx',

which is (3.9) for j = 1.

By (3.11), (3.13) implies

m i4 0on Vatz.

Using the expression

(3.13)

dx' = --- dx' mod (dx2, ... , dx', ... , dx')

in the right-hand side of (3.13), we obtain (3.9) for j > 1 and c; 0 0. If

j > I and c; = 0, then the two sides of (3.9) are obviously zero.

Proof of 2°). Suppose cl 0 0 at a point of ER. By (3.9), for z e V near

this point, we can choose the coordinates (pl, x z,

... , x') of Rz in Z(1)

as coordinates on V. Then on V, x1 and p are functions of (pl, x2, ... ,

x'). The equation of ER is

axl(P1, x2, ... , x')

aPl

Let t"'(z) be the vector in A(z) having the coordinates

(dpi = 1,dx2

=...=dx'=0).

Then in Z(1), RS'(z) has the coordinates

(dxi

dp; _

ax1(P1,x2, ...,x')

dx2 = 0, ...,dx' = 0,

aPl

z )

aP,

Let C(z) be the vector on R - 1 X * such that

RC(z) = dx = 0, dp =

aP(Pl, x2,

x')

aPl

I,§3,3-I,Conclusion

1'(z) is ayyfunctioyyn of z c V. Then

b(z) = b'(z) for Z E Y-R ;

'(z)] = [RC(z), RC'(z)] =

ax I

dpl _ P,

because, in view of (3.9), the obvious relation

dx' A ... A dx' =

Ox'(P),x2, ...,x')dpl

Opi

means

Ox'

d'x

apl

(3.14)

dx2 A...Adx`onV

Now by §2, theorem 6, for z r= V\ER,

mR(z) = m(R 1X

A,,(z)) =

for

C'(z)] < 0

1 + c for [Oz), Nz)] > 0,

c a constant c- Z. Considering (3.14), (3.10) follows.

4. q-Symplectic Geometries

If R is a q-frame, clearly the group

R-' Spq(l)R = Spq(Z)

acts on Z and its lagrangian manifolds while preserving their q-orienta-

tions; this group is independent of R by (3.1).

F. Klein defined a geometry by specifying a manifold and a Lie group

acting on that manifold.

Each of the groups Spq(Z), where q c (1, 2, ... , cc

}, thus defines a

geometry on Z which it is convenient to call the q-symplectic geometry.

Conclusion

This chapter in §1 defined a unitary representation of the group Sp2(!)

in the Hilbert space K(X), where dim X = 1. As a result of §2, in §3 the

study of this group was subsituted by that of the isomorphic group Sp2(Z)

that defines the 2-symplectic geometry of Z, where dim Z = 21. Earlier §2

I,Conclusion

and §3 detailed the properties of the index of inertia and the Maslov index,

already introduced in §1. The interest of these various properties is to make

possible the definition and study of a new structure, lagrangian analysis,

which is the subject of the next chapter.

I I

Lagrangian Functions; Lagrangian Differential Operators

Introduction

Summary.

In Chapter I we studied only differential operators with

polynomial coefficients and functions defined on all of R' = X.

The aim of chapter II is to extend those results. In §2, we consider a

symplectic space Z of dimension 21 provided with a 2-symplectic geometry.

We define

lagrangian functions and lagrangian distributions, on which Sp2(Z) acts

locally;

lagrangian operators, more general than differential operators, which

are transformed by Sp2(Z) like differential operators with polynomial

coefficients (see I,§l,theorem 3.1).

Each lagrangian function U is defined on a lagrangian manifold V in

Z; in each 2-frame R, U has an expression UR, a function with formal

values defined on V\ER. In each frame R, a lagrangian operator has an

expression

1 a 1

V, X, I -lJ

V ax

that is a formal differential operator of order < oc, acting locally on the

UR.

A change of frame

S = RR"' e Sp2(l) (see 1,§3,3)

is a local operator that transforms

UR, into UR =

aR' into aR = SQR'S-1,

and hence transforms

aR, UR' into aR UR = S(aR. UR.).

It follows that lagrangian operators act on lagrangian functions and

lagrangian distributions.

All the expressions aR of a single lagrangian operator a are readily

obtained from a single formal function a° defined on Z.

In geometry, a change of frame leaves invariant (or changes linearly)

the value of a scalar (or vector) function at a point; but an essential

II,Introduction-11,§1, 1

characteristic of lagrangian analysis is the following: at each point z

of V, the group Sp2(l) of changes of frame acts on the germs of expres-

sions UR of a lagrangian function U and not on the values at z of these

expressions. U. = SUR, may have singularities on ER, but has none on

ER.\ER, n ER, whereas the singularities of UR, are on ER.. Thus the singu-

larities of the expressions UR of U are not singularities of U; they may be

described as apparent singularities. Their nature can be made explicit by

making use of the Maslov index.

Historical account. V. P. Maslov [10] elucidated this essential charac-

teristic of lagrangian analysis, even though he only studied the projections

of lagrangian functions onto X without defining either lagrangian oper-

ators or lagrangian functions explicitly. He only used a subgroup of

Sp 2(1), depending on a choice of coordinates of X : the subgroup generated

by Fourier transforms in a single coordinate.

§1. Formal Analysis

0. Summary

In section 1 we define and study asymptotic equivalence classes of functions.

In section 2 we define formal functions; formal functions defined on X

with compact support are asymptotic equivalence classes. Then we may

integrate them (section 3) and transform them by Sp2(l) and by differential

operators, whose definition can be generalized (sections 4 and 6). We

deduce (sections 4 and 6) that Sp2(l) and formal differential operators act

locally on formal functions defined on lagrangian manifolds V or their

covering spaces V; these functions may be compactly supported or not.

In section 5 we study their scalar product.

Formal functions on V, which are no longer asymptotic equivalence

classes, enable us to define lagrangian functions in §2.

1. The Algebra ' (X) of Asymptotic Equivalence Classes

The algebra -4(X)

Let I be the purely imaginary half-line

I = ill, 301EC.

I(X) denotes the algebra of mappings

f:I x X-3 (v,x)F-- f(v,x)EC

II,§1,1

all of whose x-derivatives are continuous in (v, x) and satisfy

Sup

(v,x)EI X X

8 1

f(v, x)

8x J

< oc

dq,rEN`;

Sp2(l) and the differential operators with polynomial coefficients in

(1/v, x),

1 al a_

a =

- Ox/I=-

vOX

act on V (X ).

If 1 = dim X = 0, 24(X) is denoted :#?: thus.1 is the algebra of bounded

continuous mappings I - C.

The algebra (X) Let N E R. Let -IN(X) be the set of f e .4(X) such

that the mapping

(v, x) - vNf(v, x)

still belongs to ,4(X). Clearly 24N(X) is an ideal in 2(X) that shrinks as

N grows; hence -4,)(X) =

is an ideal in 24(X). We define

the algebra 16(X) as follows:

W(X) = 24(X)i2x(X)

(1.2)

l'(X) is an algebra over C; its elements are called asymptotic equivalence

classes. The condition that the elements f and f of .(X) belong to the

same class is

f - f'E2N(X)

dNeR+.

Clearly,

"eN(X) = -qN(X)114.(X)

is an ideal in 16(X) that shrinks as N grows:

f w,(X) = to}.

NeR,

Let f and f e :4(X) with classes f and f' e''(X ). To express the equiva-

lent relations

f - f e 24N(X),

f -

' e'6,(X), where N 5 cc,

(1.5)

II,§1,1

we shall write one of the following:

f = f modvN,

f = f' modf'E f modvN.

(1.6)

If I = dim X = 0, '(X) is denoted W. Clearly

'(X) is a subalgebra of

the algebra of functions X

Thus an element f of '(X) has a restriction to any subset of X and a

support

Supp(f)

X. (1.7)

Remark 1.

Let F be an entire holomorphic function

F:Ci - C such that F(O) = 0;

for every fi c fi, the F(fl , ... ,

f,) are in the same class, denoted

F(.r,,...,

j). Thus d.

eW(X)(j = 1, ..., J), F(f,,...,.Tj) Ew(X).

Let F be a Holder function

F : C' -p C such that F(O) = 0.

We similarly define

F(.Ti , ... , .fi) E W

bfi E W, j = 1, ... , J.

For example, if f e

', then j f1, Re f, and Im f are defined. f c W is

real if Imf = 0; then f+ (the positive part) and f_ are defined. f = 0

is written 0 1 and defines a partial ordering on Re W, the set of real

elements of 1W: let f and E Re 16; f > 0 and g > 0 imply f + j > 0

and f g > 0; 12 0; f 0 (that is, -.r > 0) excludes f > 0 (that is,

0 < f

0). However, f' may satisfy neither f < 0 nor f > 0.

Let F be a Holder (and increasing) function

F : R - R such that F(0) = 0;

df E Re W we have F(f) E Re q' (the order relation on Re q' being

preserved).

In particular we have the Schwarz inequality

llz r

z]l/z

dri, 9i e 1K.

Irilz]gi

II,§1,1

The differential operators with polynomial coefficients in (1/v, x) given

l al

l a

a=a lv,x,vax/-a

vex'x

are clearly endomorphisms of -4(X), -4N(X ), 'K(X ), and 'N(X ). They act

locally: the restriction of of to an open set S in X only depends on the

restriction off to this open set:

Supp(a.T) c Supp(.T)

As in §1,3, the differential operator

a =

a+(v,

1 al

(v, 1

x)--) =

- ,

vax vax

is associated to the polynomial a° in (1/v, x, p) given by

a°(v, x, P) = lexp

Cax' 6)] a+ (v, x, P)

8 a

= [exp

Cax'

]

a- (v, p, x).

Integration is defined on '(X): if f c .J.A(X ), the function

v i-- f

f(v, x)d:x

belongs to

S. Its asymptotic equivalence class, which depends only on

the class f of f, is denoted

J(v, x) d'x c 16;

f is called the asymptotic integral.

The scalar product (I) is a sesquilinear form on '(X) x W(X) with

values in W defined by

(T, 9) =

ff(v,x)g(v,x)dlx,

where f e f c'(X), g c 4 c'(X);

(1.9)

II,§1,1

g(v, x) is the complex conjugate of g(v, x); v = -v. Thus

(J,f) _ (f, 9)

The seminorm

is the function W(X) - 16 with values >, 0 defined by

IIfll2

llf l + llg

I(f,g)l -<

g11 in 16.

LEMMA 1.1.

Sp2(1) forms a group of automorphisms of c(X) that are

unitary, that is, preserve seminorms and scalar products.

Proof.

Let SA E Sp2(1)\Esp2. Its definition in I,§1 by the integral (1.10)

shows that SA transforms the asymptotic equivalence class f' into the

class SA,' defined by the asymptotic integral

(SAf')(x) =

(27riJL2A(A)

evA(x.x'f(x,)d'x'.

This formula makes sense because multiplication by e"A, where vA is

purely imaginary, clearly preserves asymptotic equivalence. The SA act

'(X), composition being the same as when they act on Y(X ). Now

they generate the group Sp2(1); thus this group acts on W(X).

Since Sp2(l) is unitary on ,'(X), it is unitary on W(X).

Theorems 3.1 and 3.2 of i,§1 and their corollaries give the following.

LEMMA 1.2.

The transform

SaS-'

of the differential operator a by

S e Sp2(l) is the differential operator associated to a° o s-1; s denotes the

image of S in Sp(1); s acts on the space Z(1) of vectors (x, p).

LEMMA 1.3. A necessary and sufficient condition for the differential

operators a and b to be adjoint is that

b°(v,x,p)=a (( P

In particular, a is self-adjoint when a°(v, x, p) has real values for

V E I, (X, P) E Z(1).

S transforms adjoint operators into adjoint operators.

II,§1,2

2. Formal Numbers; Formal Functions

Theorem 2.2 connects the following definitions with the previous ones.

A formal number is a formal series

u = u(v) = Y- E

1',e°`0',

(2.1)

jE rEN

where J is a finite set, aj, e C, qj e R, and cpj -A cpk if j -A k.

The expression (2.1) for a formal number is unique, by definition.

The set of formal numbers is a commutative algebra F for which the

addition and multiplication rules are obvious.

We give

the topology defined by the following neighborhood system

.NV(N, e) of the origin:

NE N, eeR+, uE,N'(N,e)means Ylajrj <e

dr4N.

jEJ

A formal function on X is a mapping u : X -+ .

that can be put into

the form

u = u(v) =

a-'

(2.2)

jEJ rEN

where J is a finite set, ajr: X -+ C, cpj: X - R, and ajr and cpj are infinitely

differentiable.

The mapping of u may be put in the form (2.2) in many ways at a point

where two of the cpj are equal in a neighborhood of that point. The value

of u(v) at x is denoted

)

E E

aJr(

u(v vNj(x), x e

(2.3)

jEJ

Moreover, u(v) has a support

Suppu(v) c USuppaj,r.

(2.4)

j.r

The set .F(X) of formal functions on X is an algebra over .f, which

is commutative. The set of formal functions on X with

compact support

is a subalgebra .FO(X) of .F (X).

The cpj are called phases, the aj = ErEN(xjr/vr) amplitudes.