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

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244

I V,§ 1,2

Q[x A p] = L001(cos 0 + ir3 sin (D) sin 0 + L0Q3 cos O

= Loule'm°3sin O + L0Q3 cos O

= Loe im°,/2(o1 sin 0 + Q3 cos O)eio";

Moreover, the functions of [L, M, t], 7., and µ, which are defined by

(2.10) of III,§1, satisfy

, =

- HL,

µt = - H

(2.14)

by (2.3) of 111,§2. Let

/3 =

e-i°3m;2+ir(1F+x)+iM(m+µ)y

(2.15)

Then the system (2.12) may be written

dy + i[(f - ZHM)Q3 + Log(Q1 sin 0 + r3 cos O)]ydt = 0.

(2.16)

It has now become evident that this system is completely integrable:

here y is independent of 0 and 'Y and is only a function of t.

Condition (2.10), which is that U be defined on V[L0, M0], is then

formulated as

evoL2+voLoY`+voMom-io3m/2+ii'(`Y+x)+im'(mtµ).

. V

C2.

(

)1j2

Now S2, A, and p only depend on t and, by (1.4) and (1.5) of III,§2, increase

AtS2 = 2tN[Lo, M0], A,.1 = 2rzNL[Lo, M0],

(2.17)

Ap = 2nNM[Lo, Mo]

when t increases by c[L0, M0N is defined by (2.9) of 111,§ 1.

Let vo = i/h (h > 0); then, by (2.11), condition (2.10) that U be la-

grangian on V may be formulated as follows, since e-"3 = -1: There

exists e e R such that

y(t + c[L0, M0]) =

e2" y(t),

(2.18)

N[L0,M0]+1'NL+m'NM+Z+e=Lo+l'-Z

= M0+m'-z=0mod 1.

(2.19)

IV,§1,2

245

The search for solutions y of (2.16) satisfying (2.18) is a classical problem:

(f - 4Hm)r3 + Log(Q1 sin 0 + o3 cos O)

is a self-adjoint matrix with zero trace; it is a periodic function of t; the

period is c[L0, Mo]; then the 2 x 2 matrix u(t) defined by

du + i[(f - IHM)Q3 + Log(Q1 sin 0 + o3 cos O)]u dt = 0,

u(0) = 1,

(2.20)

is a unitary matrix with determinant 1, that is, u c- SU(2); it satisfies

u(t + c[L0, Mo]) = u(t)u(c[L0, MO]);

the general solution of (2.16) is

y(t) = u(t)d, where 6 E C2;

y satisfies (2.18) if and only if 6 is one of the eigenvectors of the matrix

u(c[L0, Mo]) E SU(2). Let

e±2aie[Lo.Mo]

(0 -< e[Lo, Mo]

(2.21)

be the eigenvalues of u(c[L0, M0]). Then, in (2.18),

e = +e[L0, Mo].

Thus, there exists a lagrangian solution U of (2.2) defined on the torus

V[L0, Mo] if and only if there exist three integers 1, m - Z, and n such that

ILo+1'+1N[Lo, MO]+1'NL+m'NM=n+e[Lo,MO],

Lo + I' - 2 = 1,

Mo + m' = m, where IMo < Lo. (2.22)

This assertion does not assume that 1', m', (f -

HM)c[L, M], and

Lgc[L, M] have negligible squares on V[L0, Mo].

U and /l will be denoted by U± and #± depending on whether (2.22)

holds with

±e[L0, Mo]. By (2.15),

(2.17), and (2.18), where e =

+e[L0, Mo], fl± satisfies (2.8), where the c±k have the values (2.9).

Assume that the squares of 1' and m' and the squares of the derivatives

of he are negligible. Then (2.22) reduces to (2.5). The integers 1, m - Z,

and n must satisfy condition (2.5)3, which is independent of l' and m'.

Since 1 and n are integers, since a is near 0, and since N > 0, (2.5)3 implies

246

IV,§1,2

I < n. Since 1' and m' are near 0, (2.5)1 and (2.5)2 imply Iml < 1 + Z, which

completes the proof of (2.6).

It remains to prove that (2.4)

is an approximate expression for

e[Lo, Mo]. Since we have assumed that, on V[Lo, M0],

(f - 1HM)c[L, M] and Lgc[L, M]

are near zero and have negligible squares, the solution of (2.20), modulo

these squares, is

u = 1 - i[ r

(f + Z pt) dt a3 + L0

g dt (a, sin 0 + a3 cos O)

0 fo

by (2.14)2. In other words, choosing the right-hand side to be in SU(2),

like u,

u = e-i[fo(J+j,.

2)dta3+L0S0gdt(a,sin0+a,cos®)

Thus, by (2.17), definition (2.21) of e[Lo, Mo], and definition (2.4) of

fo and go, the eigenvalues +2rce[Lo, Mo] are approximately the eigen-

values of the self-adjoint matrix with zero trace given by

for3 + go(al sin 0 + a3 cos 0).

Now, by (1.7) and (2.13), its square is

(fo + go cos 0)2 + go sine 0 = f o + 2

fogo + go ;

hence the approximate expression (2.4)2 for e[Lo, Mo] follows.

Remark 2.2.

The-quantum numbers

± 1

become those used in the classical solutions of the Dirac equation by

imposing a condition

stricter than (2.6)

less strict than the condition that, on account of (2.7), would result from

the choice 1' = m' = 0, namely,

lmI<l - Z,

1 <n.

This condition is as follows.

(2.6)*

IV,§1,2

247

Definition 2.

In III,§1, we solved a system in one unknown, analogous

to the system (2.2) when 1' and m' are chosen to be zero. In III,§1, the

lagrangian amplitude of the unknown is necessarily constant. It is in

conformity with the nature of the amplitude a and of the phase v0q

in physics: a must vary more slowly than e"OQR. Let us make the present

situation as similar as possible: consider a quadruple

n, + 1

satisfying the condition (2.5)3 for the existence of a solution of (2.2) that is

lagrangian on V[L0, M0]; this condition is independent of (l', m'); require

that at (1', m') the function

(l',m')i_4Ic±1-112+Ict2-12+ c13+12ER

takes on values close to its minimum; in other words, by (2.9) and since

l'2 and m'2 are negligible in comparison with I' and m', at (l', m') the

function

(I', m') i--

l'2 + m 2 + [l'NL + m'N,f

E]2

takes on values close to its minimum; if such a choice, namely,

(1', m') near +

(NL, NM),

1+NL+NM

satisfies the condition (2.7), then and only then will we say that the

quadruple (l, m, n, + 1) is admissible.

In §2 (Dirac equation), the following case arises.

Example 2.

(NL, NM) is near (-1, 0); thus

(1', m') is near (+ e/2, 0).

By (2.7), the admissible quadruples (1, m, n, ± 1) are those that satisfy

condition (2.5)3 and the condition

0<I<n,

Iml±2

(2.23)

(the signs correspond).

It is customary in quantum mechanics to let

.j+1

and to use quadruples of quantum numbers

(1, m, n, j = I + 2).

248

IV,§1,2-IV,§2,1

They are admissible if they satisfy (2.5)3 and

0'< 1<n,

Im( _< j.

(2.24)

Then for each admissible choice of (1', m') there correspond solutions

U of (2.2) that are equal up to a proportionality factor. This factor is a

constant e C.

§2. The Dirac Equation

0. Summary

The Dirac equation is a system in 4 unknowns. Theorems 2.1 and 2.2

of II,§4 do not apply in this case because the zeros of det ao are double

zeros.

Theorem 1, whose statement resembles that of theorem 2.2 of II,§4,

reduces, mod(1/v2), this system in 4 unknowns to a self-adjoint system

in 2 unknowns.

By suppressing terms that are negligible in view of the order of magni-

tude of the magnetic field, this system is transformed in section 2 into

the reduced Dirac equation, which is a system of the form solved in §1.

In section 3, we observe that the energy levels defined by this reduced

Dirac equation in lagrangian analysis are those defined by the classical

resolution of the Dirac equation, even when the magnetic field is strong

enough to produce the Paschen-Back effect.

But the probability of the presence of an electron obtained in section 4

differs from that obtained in wave mechanics; it is connected with the first

quantum theory.

1. Reduction of the Dirac Equation in Lagrangian Analysis

Suppose we are given two infinitely differentiable mappings and a con-

stant:

A : E3 \{0} - R + ; B : E3 -. E3 ;

C E R

+ .

Let a' and a" be the two 2 x 2 matrices of lagrangian operators whose

expressions in Z(3) = E3 Q+ E3 are

A(x)+a[vex+B(x)l, a"=A(x)-a

rax+B(x)1'

IV,§2,1

249

where a- is the 2 x 2 matrix defined by (1.6) of §1; a' and a" are self-adjoint.

They are associated to the self-adjoint matrices

A(x) + a[p + B(x)] and A(x) - a[p + B(x)].

The Dirac equation is the system

a'U' = CU",

a"U" = CU', (1.1)

in which the unknowns U' and U" are vectors. Let us require that the two

components of each of these vectors be lagrangian functions, the four of

them defined on a single compact lagrangian manifold V in Z(3) = E3 Q E3.

The Dirac equation (1.1) is evidently equivalent to

i. the system

[C2

- a" o a'] U' = 0

in the unknown U',

ii. the system

[C2 - a'oa"]U" = 0

(1.3)

in the unknown U".

The calculation of C2 - a" o a' and C2 - a' o a" is easy and standard.

In order to state the result, let

H(x,p)=21 p+B(x)I2+

2 -2Az(x),

J'(x, P) = 2o[t

B(x)]

01A.],

J" = 2a[" A B - 2a [Ax

where

a n B = curl B;

Z[C2 - a" o a'] is the matrix associated to the 2 x 2 matrix

H+1J';

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IV,§2,1

z [C 2 - a' o a"] is the matrix associated to the 2 x 2 matrix

H +

1J".

Remark 1.1.

These matrices are not self-adjoint; but one is the adjoint

of the other.

Notation. Let W be the hypersurface in E3 Q E3 given by the equation

W: H(x, p) = 0. (1.5)

Let #3' and /3" be the lagrangian amplitudes of U' and U":

#':V-CZ

fl":V-'C2.

By theorem 4 of II,§3, equation (1.2) is equivalent, mod(l/v2), to the

conditions :

df +J'13'=0,

where d/dt is the Lie derivative 2' in the direction of the characteristic

vector x of V Similarly, (1.3) is equivalent, mod(1/v2), to the conditions:

VC W

dg-

+ J11 11 0 7

9 . )

( .

By (1.1), /3' and /3" satisfy the two equivalent relations

{A(x) + a[p + B(x)] }/i' = C$",

{A - a[p + B] } f3" = C/'

(1.8)

on V. The equivalence of (1.2) and (1.3) proves the equivalence of the

equations (1.6) and (1.7). Relation (1.8) evidently transforms the solutions

of one of these equations into the solutions of the other.

The definition (1.5) of W means that there exists a function

p:W -R+

such that

A = C cosh 2p, p + B = C sinh 2p.

IV,§2,1

251

Let

= 1

i C sink 2p

B],

so that

by (1.8) of §1. It follows that

Ce ±2pT = A ± a [ p + B].

Relation (1.8) connecting fl' and ji" is then written

#" = e2ptf'.

It means that there exists a function

#:V_+ C2

such that

fl'=e-PTY,

p"=e"fl.

(1.10)

The equivalent relations (1.6) and (1.7) may then be written

d +JfJ=0,

(1.11)

where the 2 x 2 matrix J is given by

J = epT

(e - PT) +

eptJ'e-pt =

e - pT d(epT) + e-PTJ"epT.

(1.12)

Let us show how this definition of J is equivalent to (1.14), which makes

the following theorem evident.

THEOREM 1.

Solving the systems (1.2) and (1.3) (which are equivalent to

the Dirac equation) mod(1/v2) on V is equivalent to solving the system

aU=0mod12

(1.13)

in two unknowns, where a is the matrix of lagrangian operators associated

to the 2 x 2 matrix

H+1J.

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IV,§2,1

H is defined by (1.4)1 and J by

J(x,P)=2Or[a

B]+2A+Ca[(p+ B)AAx]..

(1.14)

Remark 1.2. 1

J, and thus a, are self-adjoint, which shows by remark 1.1

the advantage of substituting (1.13) for (1.2)-(1.3).

Proof of (1.14).

By the definitions (1.4) and (1.12) of J', J", and J, it

suffices to prove the following formulas, where < , > is the scalar product

in E3 and curl B = a/ax A B:

° dt(e PT) + e

,dt(ePT)}

2(A

C)a[curlB

_ i

2A+Ca[(p+

B)AAx]

+ 2<p + B, curlB>ztanhp

(1.15)

4 {ePTQ[curlB]e-°T

+ e-ITQ[curlB]ePT}

2Au[curl B] - 2<p + B,curl B>itanhp,

(1.16)

{e°TQ[Ax]e-PT - e-PTa[Ax]ePT} = 2Q[(p + B) A A.J.

(1.17)

Remark 1.3.

The Pauli relations [(1.7) of §1] may be written

Q[x]Q[Y] = <x, Y> + iQ[x A y]

dx, y E E3.

(1.18)

It follows, in particular, that

Q[x]u[Y] + a[Y]a[x] = 2<x, y>,

(1.19)

Q[x]Q[Y] - a[Y]Q[x] = 2ia[x A y].

(1.20)

Proof of (1.15). Since T2 = 1 and p is a number,

e±°' = cosh p ± i sinh p.

Hence

IV,§2,1 253

e4:°' dt (e±P'`) = p ± t

dPcosh p ± dt

sinh p

= +t

+ e+Pi

sinh p,

therefore

[et-_te't

) + e PT

(e°`)1

= -t dt sinh 2 p.

(1.21)

In order to calculate t(dt/dt), differentiate the definition (1.10) oft:

dt sinh 2p

+ 2Ct

cosh 2p = a dt

(p +

B)1.

It follows by (1.9) and (1.10) that

CZtdtsinhz2p

-2dd

+ Q[p + B]u dt(P + B)

whence by (1.18) and considering that A

+ l p + B 12 on W,

C2td[sinh22p

iar(P

+ B) A

dt (p

B)1. (1.22)

By the definition of d/dt [(3.10) of II,§3],

= Hn(X, P),

= -HX(X, p);

dt dt

hence, by some easy calculations,

dt(p+B)= -(p+B) A curt B+AAx,

(p + B) A

d-(p + B) = p + B12curt B - <p + B, curlB>(p + B)

+ A(p + B) A A,,.

Therefore, since

2Ccosh 2 p = C[cosh2p + 1] = A + C,

l p + B12 = A2

- C2,

(1.22) may be written