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

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[a, b] =

I [a', b'

where the right-hand side is defined by the symplectic structure.

An automorphism S of .'(X) transforms each a e d into an operator

b = SaS-', defined by the condition

bSf = Saf

Vf c 9"(X).

b 0 0 if a 0 0. In general, b 0 d.

Definition 1.1.

G(1) is the group of continuous automorphisms S of 9"(X)

that transform sad into itself in the sense that

SaS ' e..4

Va e si.

G(1) is clearly a semigroup. If S E G(1),

a i--+ SaS

is clearly an automorphism of d. Therefore S-' E G(1), and G(l) is a group.

Under the natural isomorphism Z(1) - sad, the automorphism (1.5) of

d becomes an automorphism of the vector space Z(1):

s : a

F--' sa 1.

(1.6)

Since S commutes with the automorphisms of .9'(X) given by c e C, and

since [a, b] E C, we have

[SaS-', SbS-1] = [a, b],

or, considering (1.3) and the equivalence of (1.5) and (1.6),

[sa', sb'] = [a', b']

Therefore s is an automorphism of the symplectic space Z(1).

The group of automorphisms of the symplectic space Z(1) is called the

symplectic group and is denoted Sp(l):

S E SP (I).

By (1.1),

[sa', z] = [a', s-'z] = (a° o s-')(z)

In summary:

LEMMA 1. 1.

Under the natural isomorphisms of sat, Z(1), and a7°, the

automorphism

a -- SaS-'

of si, which is defined for all S E G(1), becomes

an automorphism s of Z(1), s : a' f--' sa', s e Sp(l),

an automorphism of sl' given by a° i--> a° o s-1.

The function S f--' s is a natural morphism

G(1) - Sp(1).

(1.7)

LEMMA 1.2.

The kernel of the morphism (1.7) is a subgroup of G(1) con-

sisting of automorphisms of 9'(X) of the form

f - cf, where f c .9'(X) and c e IC (complex plane minus the origin).

Remark.

This subgroup will be written as t.

Proof.

All c c t commute with all a e sad and thus belong to the kernel.

Conversely, let S be an element of the kernel. Therefore S is an auto-

morphism of 9"(X) commuting with all a E sad. Let p e X*. We have

-y<P.x)

vax+ple

=0.

Therefore, since S and (1/v)(0/8x) + p commute,

1 a

+ p)Se-'<P-'>

= 0.

vc?x

By integration of this system of differential equations,

Se_v<P,x> = c(p)e-'(P,'), where c:X* - C.

Taking the derivative with respect to p, we see that the gradient of c, cp,

exists and satisfies

- vS[xev<P.x>I =

-vxSe-''<P.x>

Cpe-v(P.x).

equivalently, since S and multiplication by x commute,

cp = 0.

c(p) is independent of p and will be denoted c. Let F be the Fourier trans-

form and let g = F-1 f e ."(X). By the definition of F,

(iL)'I2

f(x) =

2ni

e-'<P.X>g(p)d'p.

(1.8)

Since

Se-v<o.=>

= ce-v<P

X>,

we obtain

Sf = Cf Vf E .9'(X).

Now 9'(X) is dense in .9''(X). Therefore S = c e C. This proves the lemma.

Some other subgroups of G(l) will be needed in proving that the map

G(l) - Sp(l) is an epimorphism. They are

i. the finite group generated by the Fourier transforms in one of the

coordinates (some base of the vector space X having been fixed);

ii. the group consisting of automorphisms of .9''(X) of the form

f -. e°Qf,

where Q is a real quadratic form mapping X -. R;

iii. the group consisting of automorphisms of .9''(X) of the form

f' -* f, where f (x) =

det T f'(Tx), T an automorphism of X.

Each of these groups has a restriction to .9'(X) that gives a group of

automorphisms of .9'(X) and a restriction to .*'(X) that gives a group of

unitary (that is, isometric and invertible) transformations of .i*'(X). The

following definition uses these properties.

Definition 1.2. Let A be the collection of elements A each consisting of

1°) a quadratic form X Q+ X

R, whose value at (x, x') e X Q+ X is

A (x, x') = Z <Px, x> - <Lx, x'> + Z <Qx', x'), (1.9)

where, if `P denotes the transpose of P,

P = `P:X -. X*, L:X,-. X*, Q = `Q:X -+ X*,

det L A 0;

2°) a choice of arg det L = nm(A), m(A) e Z, which allows us to define

A(A) = det L

arg A(A) = (rz/2)m(A).

1,§ 1,1

Remark.

det L is calculated using coordinates in X* dual to the co-

ordinates in X and is independent of coordinates chosen such that

dx' n

. A dx' = d'x.

Remark.

m(A) will be identified with the Maslov index by 2,(2.15) and

§2,8,(8.6).

To each A we associate SA, an endomorphism of Y(X) defined by

rzi]`I2

A(A) I

(SAf)(x) [IV,,

where f' E ,9(X), arg[i]t'2 = nl/4.

(1.10)

Clearly SA is a product of elements belonging to the groups (i), (ii), and

(iii). Therefore SA is an automorphism of Y (X) that extends by continuity

to a unitary automorphism of A (X) and to an automorphism of 9"(X).

These three automorphisms will be denoted SA; SA e G(l).

The image sA of SA in Sp(l) is characterized as follows (where Ax is the

gradient of A with respect to x):

(x, p) = sA(x', p') is equivalent to

p = Ax(x, x'),

p' = -Ax.(x, x').

Proof of (1.11).

Let f' e Y (X). a(SAf')/(3x and SA(df'/dx) are calculated

by differentiation of (1.10) and integration by parts; the result of these

calculations gives the following relations among differential operators of

V 8x -

Px = - SA('Lx)SA',

SA (v

+ Qx}

SA' = Lx;

writing

(X, P) = SA(x', P'),

these relations mean

P - Px = -'Lx',

p' + Qx' = Lx

bx' E X, p' e X

This is proposition (1.11).

Definition 1.3.

We shall write Esp for the set of s e Sp(l) such that x and

x' are not independent on the 21-dimensional plane in Z(l) Q Z(l) de-

termined by the equation

(x, P) = s(x', p').

Let us recall the well-known theorem that the set of sA characterized by

(1.11) is Sp(l)\ESp.

Proof.

Clearly sA 0 Esp. Conversely, let s e Sp(l). On the 21-dimensional

plane in Z(l) Q Z(1) determined by the equation

(x, P) = s(x', p')

we have, since s is symplectic,

<p, dx> - <dp, x> = <p', dx'> - <dp', x'>.

Therefore

? d [<P, x> - <p', x'>] _ <p, dx> - <p', dx'>.

We assume s 0 Esp. Then x and x' are independent on the above 21-

dimensional plane. On this plane we define

A(x, x') = i <p, x> - 1 <p', x'>.

(1.12)

We therefore have

dA = <p, dx> - <p', dx'>,

that is,

p = As,

p = -As..

x and Ax have to be independent. Hence det;k(AX X ') i4 0. Therefore

s = 5A, which completes the proof.

The sA clearly generate Sp(1). Thus:

LEMMA 1.3.

The natural morphism G(l) - Sp(l) is an epimorphism.

By lemma 1.2, G(1) is a Lie group and

G(1)/t = Sp(l ).

(1.13)

[C is the center of G(l) because the center of Sp(l) is just the identity

element.]

Definition 1.4.

The metaplectic group Mp(l) is the subgroup of G(l)

I,§1,1-I,§1,2

consisting of those elements whose restriction to ff (X) is a unitary auto-

morphism of ,*°(X).

We have SA e Mp(l) VA. Now the SA generate Sp(l), so the natural

morphism

Mp(l) - Sp(1)

is an epimorphism. By (1.13), all elements of G(l) can be written uniquely

in the form

cS,

where S e Mp(l),

c > 0.

Writing R+ for the multiplicative group of real numbers > 0, we obtain

G(l) = R+ x Mp(l).

(1.14)

The study of G(l) therefore reduces to that of Mp(l), which has the follow-

ing properties:

THEOREM 1.

Mp(l) is a group of automorphisms of S°'(X) whose restric-

tions to.W'(X) are unitary automorphisms.

1°) Let S' be the multiplicative group of complex numbers of modulus 1.

Then

Mp(1)/S' = Sp(l). (1.15)

2°) Let EMp be the hypersurface of Mp(l) that projects onto Esp. Every

element of Mp(l)\EMp can be written as cSA, where c e S' and SA is given

by an expression of the form (1.10).

3°) The restriction of every S e Mp(l) to So(X) is an automorphism of

,°(X).

Proof of 1°): (1.13) and (1.14); S' is identified with a subgroup of Mp(l).

Proof of 2°). Let S e Mp(l)\EMp

. Then the image of S in Sp(l) is some

element sA, A e A; SSA' e S' by (1.15).

Proof of 3°).

By 2°), S = cSA, . .

. SAC

. Now the restrictions of c,

SA,

, ... ,

SA, to °(X) are automorphisms of , °(X ).

2. The Subgroup Sp2(l)

Of MP(l)

Definition 2.1.

We denote by Spz(l) the subgroup of Mp(l) that is

generated by the SA.

I,§1,2

The purpose of this section is to prove that Sp2(l) is a covering group

of Sp(l) of order 2.

In order to prove this, we calculate inverses and compositions of the

elements SA

Definition 2.2. - Given A E A, we define A* e A as follows:

A*(x, x') = -A(x', x), 1(A*) = i'A(A),

m(A*) = l - m(A).

LEMMA 2.1.

SA 1 = SA*; thus sA1 = SA+.

Proof.

This amounts to proving the equivalence of the following two

conditions for any f and f' EY(X):

f(x) - _

(±)"2A(A)

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

f (x) =

(Iv2rz

) 0(A)

Using the expression for A given by (1.9), this is the same as the equivalence

of the following two conditions:

f(x) = J

X x

f'(x') =

(jj)'detLI jev<1x'>f(x)dIx.

The equivalence is deduced

lemma follows.

from the Fourier inversion

formula; the

To compute compositions of the SA, we will find an explicit expression

for SA(e°`° ), where (p' is a second-degree polynomial. This is made possible

by the following definition.

Definition 2.3.

Choose linear

coordinates

in X such

that

d'x =

dx' A ... A dx' and choose the dual coordinates in X*. The following

notions are independent of this choice.

Let cp be a real function, twice differentiable:

cp : X -+ R.

1,§1,2

Hess,,((p) denotes the hessian of cp, the determinant of its second derivatives.

Alternatively this is the determinant of the quadratic form

X-3

Inertz((p) denotes the index of inertia of this form. It is

defined") when

Hess((p)

0. Clearly

Inert(-q') = I - Inert(q),

arg Hess((p) = n

Inert(g) mod 2n.

This formula makes possible the definition

arg Hess((p) = n Inert(q).

(2.1)

Thus, for example,

[Hess (9)]112 =

IHess((p)I112itnert(N).

(2.2)

If op is a real quadratic form,

rp: X a x i-- I<Rx, x>,

where R = 1R: X - X*,

then Hess((p) and Inert((p) will be denoted Hess(R) and Inert(R). Hess(R) is

the determinant of the symmetric matrix R. Inert(R) is the number of

negative eigenvalues of R. Clearly

Inert(R) = Inert(R-1),

[Hess(R)]1J2[Hess(-R-1)]112 = it.

(2.3)

LEMMA 2.2.

Let 9' be a real second-degree polynomial. Let A e A be such

that Hessx.(cp'(x') + A(x, x')) A 0. Denote by 9(x) the critical value of the

polynomial

Xax't--* A (x,x') + V (x');

rp is a second-degree polynomial. We have

SA(e'11)

= A(A)[Hessx.(cp' +

A)]-1l2e"°.

(2.4)

Remark 2.1.

This lemma assumes v/i > 0. Up to this point, it was

sufficient to assume v/i real and nonzero.

Proof. We know that

'It is the number of negative eigenvalues of the linear symmetric operator dx

F--. dcp_

exp [ - 2] dx =

2n.

Therefore if c e C and I arg p I< n1/2,

x +

c)2l

dx =

Iarg f <

JexP[-1(X

J J VP

We then have, for any p e C,

fex[_vx - 2(x + c)zl

exp

- 2u [vp + µ(x + c)] )} dx =

e (P

ev(O

I,§1,2

where (p is the critical value of the function

x i- p'(x) - px,

where gyp'

(x + c)2.

The Fourier transform F is the automorphism of 9 defined by

1/2

(Ff')(p)

2nli

Je<>f(x)dlxdf'

E.5(X)

(2.5)

We then have, for I = 1,

I arg p I < n1/2,

Fe°`° =

v I v I

evv;

= e"'4

VF,

11P 1/71-

Since F is a continuous automorphism of 5''(X), the preceding formula

remains valid for p = -Ev, r e $; then

f 1'/E

if r > 0

iJIEI

if r < 0.

In other words, when I = 1, the following result holds: Let p': X -' R be

a real second-degree polynomial such that Hess p'

0; let p(p) be the

critical value of the polynomial

xicp'(x)

- CP,x>,

we have

1,§1,2

Fe°`° =

[Hess p']-1ne"*.

(2.6)

Let us show that,

since relation (2.6) holds for I = 1, it holds for all I >, 1.

It suffices to choose

the coordinates x' in X such that

(P'(x) = i (pi(x').

j=1

Now using the definitions (1.9) of A, (1.10) of SA, and (2.5) of F, we have

in the case P = Q = 0,

(Snf') (x) = A(A) (Ff ') (Lx).

Then (2.6) establishes (2.4) in this case. From the definitions of A and SA,

the general case is clearly equivalent to this one.

Before taking compositions of the SA,we consider compositions of

the sA:

LEMMA 2.3.

1°) Let A and A' E A. The condition

$ASA' 0 E5p

(2.7)

is equivalent to the condition

Hess,, [A(x, x') + A'(x', x")] : 0 (the Hessian is constant).

(2.8)

2°) This condition is equivalent by lemma 2.1 to the existence of A" E A

such that

SASA.sA.. = e [identity element of Sp(l)].

(2.9)

A" is defined by the condition that the critical value of the polynomial

+ A(x, x') + A'(x', x") + A"(x", x)

be zero.

3°) Just as (1.9) defines A by P, Q, L, let A' and A" be defined by P, Q', L'

and P", Q", L". The condition (2.8) for the existence of A" is expressed as

1' + Q is invertible.

A" can be defined by the formulas

P" + Q' = L'(P' + Q)-"L',

P + Q = 1L(P' + Q)-1L,

(2.10)