Tus theory we have been building up so far is essentially a non-
relativistic one. We have been working all the time with one par-
ticular Lorentz frame of reference and have set up the theory as an
analogue of the classical non-relativistic dynamics. Let us now try
to make the theory invariant under Lorentz transformations, so that
it conforms to the special principle of relativity. This is necessary in
order that the theory may apply to high-speed particles. There is no
need to make the theory conform to general relativity, since general
relativity is required only when one is dealing with gravitation, and
gravitational forces are quite unimportant in atomic phenomena.
Let us see how the basic ideas of quantum theory can be adapted
to the relativistic point of view that the four dimensions of space-
time should be treated on the same footing. The general principle
of superposition of states, as given in Chapter [, is a relativistic
principle, since it applies to ‘states’ with the relativistic space-time
meaning. However, the general concept of an observable does not fit
in, since an observable may involve physical things at widely separated
points at one instant of time. In consequence, if one works with a
general representation referring to any complete set of commuting
observables, the theory cannot display the symmetry between space
and time required by relativity. In relativistic quantum mechanics
one must be content with having one representation which displays
this symmetry. One then has the freedom to transform to another
representation referring to a special Lorentz frame of reference if it
is useful for a particular calculation. ,
For the problem of a single particle, in order to display the sym-
metry between space and time we must use the Schrodinger repre-
sentation. Let us put 21, 2%, 2; for x, y, 2, and x for ct. The time-
dependent wave function then appears as (7,2, 2,23) and provides
us with a basis for treating the four x’s on the same footing.
We shall use relativistic notation, writing the four 2’s as x,
(u = 0, 1, 2,3). Any space-time vector with four components which
transform under Lorentz transformations like the four elements dz,,
will be written like a, with a lower Greek suffix. We may raise the
254 RELATIVISTIC THEORY OF THE ELECTRON § 66
suffix according to the rules
a = G, a = —d, a = —a,, a = —as. (1)
The a,, are called the contravariant components of the vector a, and
the a# the covariant components. Two vectors a, and 6, have a
Lorentz-invariant sealar product
Ag by — Ay by — Gy by — ig bg = ab, = G4, bY,
a summation being implied over a repeated letter suffix. The funda-
mental tensor g#” is defined by
g® = 1 git = g? = g3 = —]
gg’? = 0 forp -v.
With its help the rules (1) connecting covariant and contravariant
components may be written
Qe = gta,
In the Schrédinger representation the momentum, whose com-
ponents will now be written p,, pp, ps instead of p,, Py, Pz, 18 equal
to the operator
p, = —haléx, (r= 1, 2, 8). (3)
Now the four operators 6/éx,, form the covariant components of a
4-vector whose contravariant components are written @/éx#. So to
bring (3) into a relativistic theory, we must first write it with its
suffixes balanced, p, = th alex’,
and then extend it to the complete 4-vector equation
Dy, = th 8/800". (4)
We thus have to introduce a new dynamical variable p,, equal to
the operator 7% 6/éay. Since it forms a 4-vector when combined with the
momenta p,, it must have the physical meaning of the energy of the
particle divided by c. We can proceed to develop the theory treating
the four p’s on the same footing, like the four x’s. ,
In the theory of the electron that will be developed here we shall
have to introduce a further degree of freedom describing an internal
motion of the electron. The wave function will thus have to involve
a further variable besides the four z’s.
67. The wave equation for the electron
Let us consider first the case of the motion of an electron in the
absence of an electromagnetic field, so that the problem is simply
§ 67 THE WAVE EQUATION FOR THE ELECTRON 255
that of the free particle, as dealt with in § 30, with the possible
addition, of internal degrees of freedom. The relativistic Hamiltonian
provided by classical mechanics for this system is given by equation
(23) of § 30, and leads to the wave equation
{po (mc? + pit p3+-p3)}p = 0, (5)
‘where the p’s are interpreted as operators in accordance with
(4). Equation (5), although it takes into account the relation between
energy and momentum required by relativity, is yet unsatisfactory
from the point of view of relativistic theory, because it is very un-
symmetrical between p, and the other p’s, so much so that one cannot
generalize it in a relativistic way to the case when there is a field
present. We must therefore look for a new wave equation.
If we multiply the wave equation (5) on the left by the operator
{pot (m?c?+ pi+ p2+p3)!}, we obtain the equation
{pj— me? — pi—pi— PP = 9, (6)
which is of a relativistically invariant form and may therefore more
conveniently be taken as the basis of a relativistic theory. Equation
(6) is not completely equivalent to equation (5) since, although every
solution of (5) is also a solution of (6), the converse is not true. Only
those solutions of (6) belonging to positive values for p, are also
solutions of (5).
The wave equation (6) is not of the form required by the general
laws of the quantum theory on account of its being quadratic in pp.
In § 27 we deduced from quite general arguments that the wave
equation must be linear in the operator 6/ét or po, like equation (7)
of that section. We therefore seek a wave equation that is linear
in p, and that is roughly equivalent to (6). In order that this wave
equation shall transform in a simple way under a Lorentz transforma-
tion, we try to arrange that it shall be rational and linear in p,, po,
and p, as well as in py, and thus-of the form
{Po— 1 Pi 2 P2— %3 Pg— BJ = O, (7)
where the «’s and 8 are independent of the p’s. Since we are consider-
ing the case of no field, all points in space-time must be equivalent,
so that the operator in the wave equation must not involve the z’s.
Thus the «’s and B must also be independent of the x’s, so that they
must commute with the p’s and the z’s. They therefore describe
some new degree of freedom, belonging to some internal motion in
256 RELATIVISTIC THEORY OF THE ELECTRON § 67
the electron. We shall see later that they bring in the spin of the
electron.
Multiplying (7) by the operator {py-+ a, p1+o2Po+%3 P3+f} on the
left, we obtain
(Bi 3 lod t+ (or ao-+on a) Pr Pa + (a B+Boy)p] Bly = 0,
where > refers to cyclic permutations of the suffixes 1,2, 3. This is
123
the same as (6) if the «’s and f satisfy the relations
of = 1, Oly Ny + Oy oy == 0,
B= mec? B+ Ba, = 0,
together with the relations obtained from these by permuting the
suffixes 1, 2, 3. If we write
B = a, Me,
these relations may be summed up in the single one,
gq Ap toy Xy = 26,, (a,b = 1, 2, 3, orm). (8)
The four «’s all anticommute with one another and the square of
each is unity.
Thus by giving suitable properties to the a’s and 8 we can make
the wave equation (7) equivalent to (6), in so far as the motion of
the electron as a whole is concerned. We may now assume (7) is the
correct relativistic wave equation for the motion of an electron in
the absence of a field. This gives rise to one difficulty, however,
owing to the fact that (7), like (6), is not exactly equivalent to (5),
but allows solutions corresponding to negative as well as positive
values of p). The former do not, of course, correspond to any actually
observable motion of an electron. For the present we shall consider
only the positive-energy solutions and shall leave the discussion of
the negative-energy ones to § 73.
We can easily obtain a representation of the four a’s. They have
similar algebraic properties to the o’s introduced in § 37, which o’s
can be represented by matrices with two rows and columns. So long
as we keep to matrices with two rows and columns we cannot get a
representation of more than three anticommuting quantities, and we
have to go to four rows and columns to get a representation of the
four anticommuting a’s. It is convenient first to express the a’s in
terms of the o’s and also of a second similar set of three anticom-
muting variables whose squares are unity, p;, po, pg Say, that are
§ 67 THE WAVE EQUATION FOR THE ELECTRON 257
independent of and commute with the o’s. We may take, amongst
other possibilities,
Oy == PL Fis Og —= Pi Fas 3 —= P1583, Om = Par (9)
and the «’s will then satisfy all the relations (8), as may easily be
verified. If we now take a representation with p, and o, diagonal,
we shall get the following scheme of matrices:
= /0 1 0 0\ = /0O-i 0 0\ o=/l 0 0 0
10 0 a0 0 0 0-1 0 0
00 01 0 0 0-1 0 0 1 0
0 0 1 0 0 0 4 0 0 0 O-!1
p= /0 0 1 0\ pp= /0 0-%.0\ pp=/l1 0 0 0
000 1 0 0 0-1 0 1 06 #0
10 0 0 +0 0 0 0 O-—1 0
0 1 0 0 0 i 0 0 0 0 O-T/.
It should be noted that the p’s and o’s are all Hermitian, which makes
the «’s also Hermitian.
Corresponding to the four rows and columns, the wave function 4
must contain a variable that takes on four values, in order that the
matrices shall be capable of being multiplied into it. Alternatively,
we may look upon the wave function as having four components, each
a function only of the four x’s. We saw in § 37 that the spin of the
electron requires the wave function to have two components. The
fact that our present theory gives four is due to our wave equation
(7) having twice as many solutions as it ought to have, half of them
corresponding to states of negative energy.
With the help of (9), the wave equation (7) may be written with
three-dimensional vector notation
{Po— pr(3, P)—pamepp = 0. (10)
To generalize this equation to the case when there is an electro-
magnetic field present, we follow the classical rule of replacing p, and
p by pte/c.A, and p+e/c.A, Ay and A being the scalar and vector
potentials of the field at the place where the electron is. This gives
us the equation
{p,+£4.—o,(o. p+5A)—pomel = 0, (11)
which is the fundamental wave equation of the relativistic theory of
the electron.
258 RELATIVISTIC THEORY OF THE ELECTRON § 67
The four components of ¢ in (10) or (11) should be pictured as written
one below another, so as to form a single-column matrix. The square
matrices p and o then get multiplied into the single-column matrix
according to matrix multiplication, the product being in each case
another single-column matrix. The conjugate imaginary wave func-
tion that represents a bra should be pictured as having its four com-
ponents written one beside another, so as to form a single-row matrix,
which can be multiplied from the right by a square matrix p or a to give
another single-row matrix. We denote this conjugate imaginary wave
function pictured as a single-row matrix by J‘, using the symbol * to
denote the transpose of any matrix, i.e. the result of interchanging
the rows and columns, Then the conjugate imaginary of equation (11)
reads ve e e
B [pote do—n(eP+2A)—pame]|=0, 02)
in which the operators p operate to the left. An operator of differentia-
tion operating to the left must be interpreted according to (24) of § 22.
68. Invariance under a Lorentz transformation
Before proceeding to discuss the physical consequences of the wave
equation (11) or (12), we shall first verify that our theory really is
invariant under a Lorentz transformation, or, stated more accurately,
that the physical results the theory leads to are independent of the
Lorentz frame of reference used. This is not by any means obvious
from the form of the wave equation (11). We have to verify that, if
we write down the wave equation in a different Lorentz frame, the
solutions of the new wave equation may be put into one-one corre-
spondence with those of the original one in such a way that corre-
sponding solutions may be assumed to represent the same state. For
either Lorentz frame, the square of the modulus of the wave function,
summed over the four components, should give the probability per
unit volume of the electron being at a certain place in that Lorentz
frame. We may call this the probability density. Its values, calculated
in different Lorentz frames for wave functions representing the same
state, should be connected like the time components in these frames
.of some 4-vector. Further, the 4-dimensional divergence of this 4-
vector should vanish, signifying conservation of the electron, or that
the electron cannot appear or disappear in any volume without passing
through the boundary.
§68 INVARIANCE UNDER A LORENTZ TRANSFORMATION 259
For brevity it is convenient to introduce the symbol a) = 1 and to
suppose that the suffixes of the four a, (4 = 0, 1, 2, 3) can be raised
in accordance with the rules (1), even though these four «’s do not
form the components of a 4-vector. We can now write the wave
equation (11) {ou"(p,-+e/e.A,)—a, mo} = 9. (13)
The four « satisfy
oor, 0 are, a == Qgh a, (14)
with g«” defined by (2), as one can verify by taking separately the cases
when p and v are both 0, when one of them is 0, and when neither of
them is 0.
Let us apply an infinitesimal Lorentz transformation and distinguish
quantities referring to the new frame of reference by a star. The com-
ponents of the 4-vector p, will transform according to equations of
th
e type pe = py +a,"D,, (15)
where the a,” are small numbers of the first order. We shall neglect
quantities that are quadratic in the a’s and thus of the second order.
The condition for a Lorentz transformation is that :
DL pe* = p, ph,
which gives a,"p, pe+p,, ap, = 0,
leading to ghee Lage == 0. (16)
The components of A,, will transform according to the same law, so
we have
Pytele.A, = pte. Ai—a,"(pyt efe. A*),
Thus the wave equation (13) becomes
((at—ahayt)( pis 6/0 Af) — cy, me}h = 0. (17)
Define M = ha, ofa, a7. (18)
Then from (14)
CO gy ME — Mi csp, ot = FU yey (LM Oly, AP + OP Og OH) Oty, CE —
po
= HP Ob (ott py Ff LO, Ox)
BA pg (GhPou? — chy?)
I
—4 oP
with the help of (16), and hence
oH (1 +-om M) = (1+ Mery) (oi —a,F0). (19)
Thus, multiplying (17) by (1+ M«,,) on the left, we get
{oe(1 tot, M) (pi + e/c. A) — (om -+ Mme} = 0.
260 RELATIVISTIC THEORY OF THE ELECTRON § 68
So if we put +o, Mb = o*, (20)
we get {a(pi-te/c. AZ) —o,, mepp* = 0. (21)
- This is of the same form as (13) with the starred variables pi, AZ, $*,
and shows that (13) is invariant under an infinitesimal Lorentz trans-
formation, provided % is subjected to the right transformation, given
by (20). A finite Lorentz transformation can be built up from infinite-
simal ones, so under a finite Lorentz transformation the wave equation
(13)is also invariant. Note that the matrices «+ do not get altered at all.
The invariance proved above means that the solutions } of the
original wave equation (13) are in one-one correspondence with the
solutions %* of the new wave equation (21), corresponding solutions
being connected by (20). We assume that corresponding solutions
represent the same physical state. We must now verify that the
physical interpretations of corresponding solutions, referred to their
respective Lorentz frames of reference, are in agreement. This requires
that J's should give the probability density referred to the original
frame and J*'%* the probability density referred to the new frame.
Let us examine the relationship between these quantities. pty is the
same as ys'a% and forms one of the four quantities Jfas, which should
be treated together.
Equations (18) and (16) show that M is pure imaginary. Thus the
conjugate imaginary of equation (20) is
prt oe = pl Mee)
PRang* = FL —Moy, ot poy Mp
= £t(1—Ma,,)(1+ Ma,,)(o*—a,to” yb
from (19). This reduces to
Dean = Pata, yb
= Porpam, plary
with the help of (16}. If we lower the suffix u here, we get an equation
of the same form as (15), which shows that the four quantities J'c,, y
transform like the contravariant components of a 4-vector. Thus pty
transforms like the time component of a 4-vector, which is the correct
transformation law for a probability density. The space components
of the 4-vector, namely ¢*u, 1, if multiplied by c, give the probability
current, or the probability of the electron crossing unit area per
unit time.
Hence
§68 INVARIANCE UNDER A LORENTZ TRANSFORMATION 261
It should be noted that #*a,,) is invariant, since
Day P= Pt — Mey, )og(L-f0tp, ME)
= Pom op.
We must verify finally the conservation law, that the divergence
ao;
—— (ft 99
ae, (Pow) (22)
vanishes. To prove this, multiply equation (13) by #1 on the left.
The result is
Brar(in +24, ‘| — fin, mop = 0.
The conjugate imaginary equation is
(inh grea Jat mors = 0.
axe ef ™
Subtracting and dividing by i#, we get
7, Ob , Gt
tye 4 Qty ==
Pra dae © Bch b= 0,
which just expresses the vanishing of (22). In this way we complete
the proof that our theory gives consistent results in whichever frame
of reference it is applied.
69. The motion of a free electron
It is of interest to consider the motion of a free electron in the
Heisenberg picture according to the above theory and to study the
Heisenberg equations of motion. These equations of motion can be
integrated exactly, as was first done by Schrédinger.{ For brevity
we shall omit the suffix ¢ which the notation of § 28 requires to be
inserted in dynamical variables that vary with time in the Heisen-
berg picture. ,
As Hamiltonian we must take the expression which we get as equal
to cp) when we put the operator on in (10) equal to zero, ie.
H = cp,(o, p)+pymec? = c(a, p)+pgme’. (23)
We see at once that the momentum commutes with H and is thus a
constant of the motion. Further, the z,-component of the velocity is
dy = [2,,H] = cay. (24)
This result is rather surprising, as it means an altogether different
t Schrédinger, Sitzungsd. d. Berlin. Akad., 1930, p. 418.
3595.57 iS)
262 RELATIVISTIC THEORY OF THE ELECTRON § 69
relation between velocity and momentum from what one has in
classical mechanics. It is connected, however, with the expression
bea, for a component of the probability current. The #, given by (24)
has as eigenvalues --c, corresponding to the eigenvalues -+-1 of o,.
As % and @, are similar, we can conclude that a measurement of a com-
ponent of the velocity of a free electron is certain to lead to the result -ke.
This conclusion is easily seen to hold also when there is a field present.
Since electrons are observed in practice to have velocities con-
siderably less than that of light, it would seem that we have here a
contradiction with experiment. The contradiction is not real, though,
since the theoretical velocity in the above conclusion is the velocity
at one instant of time while observed velocities are always average
velocities through appreciable time intervals. We shall find upon
further examination of the equations of motion that the velocity is
not at all constant, but oscillates rapidly about a mean value which
agrees with the observed value.
It may easily be verified that a measurement of a component of the
velocity must lead to the result +-c in a relativistic theory, simply
from an elementary application of the principle of uncertainty of
§ 24. To measure the velocity we must measure the position at two
slightly different times and then divide the change of position by the
time interval. (It will not do to measure the momentum and apply
a formula, as the ordinary connexion between velocity and momen-
tum is not valid.) In order that our measured velocity may approxi-
mate to the instantaneous velocity, the time interval between the
two measurements of position must be very short and hence these
measurements must be very accurate. The great accuracy with
which the position of the electron is known during the time-interval
must give rise, according to the principle of uncertainty, to an almost
complete indeterminacy in its momentum. This means that almost
all values of the momentum are equally probable, so that the momen-
tum is almost certain to be infinite. An infinite value for a component
of momentum corresponds to the value +c for the corresponding
component of velocity.
Let us now examine how the velocity of the electron varies with
time. We have tidy <= oy H— Hon.
Now since «, anticommutes with all the terms in H except ca, D,,
0 AE Hoy = on Coy Py + Coy Py oy = 2H,
§ 69 THE MOTION OF A FREE ELECTRON 263
and hence
they = 2a, 1—2cp,, (25)
= —2Ho,+2cp,.
Since H and , are constants, it follows from the first of equations
(25) that ihe, = Qe H. (26)
This differential equation in &, can be integrated immediately, the
result being (27)
where a? is a constant, equal to the value of &, when ¢= 0. The
factor e~2/#"% must be put to the right of the factor a? in (27) on
account of the H occurring to the right of the &, in (26). The second
of equations (25) leads in the same way to the result
dy == ORIG D,
eee BO p—Bi Hh
ay = aye’ tr
We can now easily complete the integration of the equation of motion
for x,. From (27) and the first of equations (25)
oy = fthad eA on, HO, (28)
and hence the time-integral of equation (24) is
By = —FchPal eR 2p, Ht-+ay, (29)
a, being a constant.
From (28) we see that the xz, component of velocity, ca,, consists
of two parts, a constant part c*p, 7-1, connected with the momentum
by the classical relativistic formula, and an oscillatory part
Mohad e—2HinFT-1,
whose frequency is high, being 2H/h, which is at least 2mc?/h. Only
the constant part would be observed in a practical measurement of
velocity, such a measurement giving the average velocity through a
time-interval much larger than h/2mc*. The oscillatory part secures
that the instantaneous value of ¢, shall have the eigenvalues +c. The
oscillatory part of x, is small, being, according to (29),
—fehiPad e~2Hit 2 = Mich(a,—cp, HY)H-,
which is of the order of magnitude fi/me, since (a,—cp, H-) is of the
order of magnitude unity.
70. Existence of the spin
In § 67 we saw that the correct wave equation for the electron in
the absence of an electromagnetic field, namely equation (7) or (10), is
equivalent to the wave equation (6) which is suggested from analogy
264 RELATIVISTIC THEORY OF THE ELECTRON § 70
with the classical theory. This equivalence no longer holds when
there is a field. The wave equation to be expected from analogy with
the classical theory in this case is
€
(eo ! Ad) (P cA) mectlys = 0, (30)
Cc
in. which the operator is just the classical relativistic Hamiltonian. If
we multiply (11) by some factor on the left to make it resemble
(30) as closely as possible, namely the factor
Po +f Acto(e, Dp +54) 1 pg me,
we get
2 e,\? e é
[Pet Ge) —[o.p+5A) —mtet—oi[ (ro +5 40](e.P-+54)—
—(2,p+2A}[ro+£40)]}o=0. (0
We now use the general formula that, if B and C are any two
three-dimensional vectors that commute with o,
(o, B)(o, C) = > {o? B,C +6, 0, B,Cy+o.0, By Ci},
123
the summation referring to cyclic permutations of the suffixes 1, 2, 3,
or (¢, B)(o, C) = (B, C)-+4 3 o5(B, O,— B,C)
123
= (B,C)+i(6,BxC). (32)
Taking B = C == p-+e/c.A, we find, since
(p+) x (p+