On the Interpretation of Diracs alpha -Matrices

PHYSICS: G. BREIT

10

PROC. N.A. S

1 W. D. Coolidge, J. Frank. Inst., 202, 693 (1926). C. C. Lauritsen and R. D. Bennett, Phys. Rev., 32, 850 (1928). 3 M. A. Tuve, L. R. Hafstad and 0. Dahl, Phys. Rev., 36, No. 7, 1262 (1930). ' A. Brasch and F. Lange, Naturwissenschaften, 35, 765 (Aug. 29, 1930). 6 R. Wideroe, Arch. Elektrot., 21, 387 (1929). 6 E. 0. Lawrence and N. E. Edlefsen, Science, 72, No. 1867, 376 (1930). 2

ON THE INTERPRETA TION OF DIRA C'S a MATRICES By G. BREIT DEPARTMENT OF PHrsics, NEW YORK UNIVERSITY

Communicated December 5, 1930

It is known that in a certain sense the matrices a,, a2, a3 used by Dirac in his relativistic equation have the significance of velocities. It has been recently shown by Schroedingerl that in a certain sense the motion of a free electron can be decomposed into two parts. The part XK represents the usual uniform motion of translation while {K is a superposed motion of a vibratory character. Although Schroedinger's paper makes it clear just how the aK may be thought of as instantaneous velocities it seems appropriate to point out some of the characteristic features of this interpretation. We wish to bring out that: (1) it is possible to have solutions of the general relativistic one electron problem which correspond to one of the aK having definitely a given value say + 1 or (-1) at one particular time and which imply that the velocity is -c (or +c) in the direction XK, (2) such solutions give a definite value of a asK only at one time, (3) such solutions for a free electron involve in general a finite probability of states of negative energy, (4) in the special case of the momentum PK becoming infinite the probability of states of negative energy may become zero. In the notation of Dirac dx = -cC aK and in order that al = 1 we must

have V1 = 'I'4, in terms of X1i =

42

=

43. The four Dirac equations are easily expressed

1-464,

X2 =

42-

x3

= 11

+ k4,

X4 = 12

+ {3 (1)

and are

where

(Po - PI)Xl + (mc - iP2)X3 + P3x4 = 0

(2.1)

(Po + PI)X3 + (mc + iP2)Xl - P3x2 = 0 (PO + PI)X4 + (mc - ip2)X2 + P3X1 = 0 (Po- PI)X2 + (mc + iP2)X4 -P3x3 = 0

(2.3) (2.4)

e PO = - 2iri cbt + C Ao'

(2.2)

e

K (2) PK - +XA PK ri~ XK + CK(25

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PHYSICS: G. BREIT

and the A,, form the four potential. On account of (1) 4

4

(2.6)

Z x,* x;. = 2 E , 1

1

For a, = 1 we must have xi = X2 = 0 for all Then equations (2.2),. (2.3) become

(xI, x2, X3) and given t

= to.

(3) (Po + P1)X3 = (PO + P1)X4 = 0. Using (2.5) we have on considering the complex conjugate of (3) and eliminating Ao, A1

5a- (xx3*x) =

)(X4*X4) =

0.

= cat - bx1,)X2 Xa

a--xl / Since X = X2 =0, we also have (Xi\CJ

0,

even though -a-l #d 0, hence using (2.6)

(catdx.)

\6D

*4

= 0.

(4)

Equation (4) is true at the particular t = to and all (xl, x2, X3). Combined with (2.6) it shows that for t = to the charge density moves with the velocity c in the direction - xi. The rotational invariance of Dirac's equation assures us that this result is independent of the choice of axes. We see, therefore, that it is possible to have solutions of Dirac's equation which correspond to the velocity being c at a given instant in a given direction (1l, 12, 13). These solutions mean formally that hla, + 12a2 + l3a3 is on a principal axis at that instant. We have used so far equations (2.2) and (2.3). Equations (2.1) and (2.4) determine the rates of change of X1 and x2 at the time to. This rate of change may be shown to be distinct from zero so that it is impossible to have one of the aK on its principal axes permanently. In fact, if = 0, and Xi = X2 = 0, equations (2.1), (2.4) become on adding tl =

6Xt2

them to each other and also on subtracting: mCvol + (ip2 + P3)'02 = 0 = 0 mc'p2 + (ip2 - P3)

(5)

where P2 = X4-X3, l = X4 + X3@ The vector potentials may be eliminated from equations (5) and their complex conjugates. It then follows that

PHYSICS: G. BREIT

72

(27rmc/h)(spi*spi + (P2*SP2) = (- a

PROC. N. A. S.

+ i aX) (S01 P2)

(\~

AX2ix)(l

2

(6)

Integrating both sides over the X2, x3 plane we obtain zero on the right, whenever (pl, j2 vanish at infinity to a sufficiently high order. In order that the integral of the left side should have a finite contribution from the infinite region i*(Pl + (2 *2 must vanish at least as fast as i/p2 where p2 - X2 + X2. Under these conditions the integral of the right side vanishes and hence (7) (P1 = 2 = O

everywhere. The above proof is not quite rigorous because a function may vanish as J/p2 and faster and yet its derivative may not vanish so that the integrals of the right side of (6) may diverge. This has been pointed out to the writer by Prof. T. H. Gronwall who has been also kind enough to supply a mathematically rigorous proof applying with great generality to the case of vanishing vector potential. Since physically we cannot admit solutions implying divergence of the integrals of the right side of (6) we may say that no admissible solutions of Dirac's equations exist which correspond to one ak being permanently on one of its principal axes.2 We thus conclude that from the purely formal point of view, i.e., taking Dirac's equation and using it according to the rules of quantum mechanics the matrices CaK have a general meaning of instantaneous velocities in the sense of equation (4), that it is possible to have solutions for which one aK iS instantaneously on a principal axis, but that it is impossible for the aK to remain on a principal axis. The only definite value of a velocity component is therefore c but this definite value exists only instantaneously. We now analyze the solutions xl = x2 = 0 (t = to) for a free electron. In order that such solutions should exist we must have a special type of Fourier analysis of the solutions. Every Fourier component has the form -

x, = a, (p) exp [(27r i/h)(-Et + We substitute this into (2) and obtain

(E/c + pi)a3 = (E/c + pl)a4 = In order that at t = 0, XI = X2 al(p) + b1(p)

p1x1 + p2x2 + p3X3)].

-(mc + ip2)al + P3a2 -p3al - (mc - ip2)a2. 0 we must have = a2(p) + b2(p) = 0

(8) (8')

(9)

PHYSICS: G. BREIT

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73

where b,, refer to coefficients of exponentials with opposite signs of the energy E. The probability of an energy E is proportional to 4

aa,,

=

(a a, + aa2). 2 1 E 1/(I E I +cPi)

(10)

1

as becomes clear on using (8'), and the probability of the energy similarly 4 b, b, = (b b1+ b* b2). 2 1E 1/(I E -cl).

-

E is

(11)

1

The probability P(+) of the energy +j E is related to the probability by P(-) of the energy

-|El,

p(+)

IE

cpl pp(_)

(12)

The energy E is connected with the momenta by E =+VM2 C2 + C2

(p2 + p2 + p2).

(13)

If pi is large and negative P(+)/P(-) is small on account of (12). For moderate pi this ratio is of the order of 1. Thus if we wish to have a, = 1 even instantaneously we must in general use states involving appreciable probabilities of negative energies. Only if the momentum is made infinite in the direction -xi can these states give a vanishingly small probability P( -). In this case component xl of Schroedinger has a velocity - c. We see therefore that even though formally the -CaK are velocities, the absence of negative energies in the physical world and the apparently false prediction of these energies by Dirac's equation necessitate considering only those experimental conditions for the measurement of the aK which correspond also to a definitely infinite momentum in the same direction as the velocity. 1E. Schroedinger, Zitzungsberichte preuss. Akad., 24, p. 418, 1930.

2 Purely formal solutions becoming infinite exponentially at infinity may be constructed, however. Such solutions have no place in the general structure of quantum theory. If desired, one could interpret them as meaning that an electron at infinity (in the y direction) moves permanently with a velocity c in the perpendicular x direction.