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P.A.M.~Dirac, Proc. Roy. Soc. {\bf A 133,} 60 \hfill {\large \bf 1931}\\
\vspace{2cm}
\begin{center}
{\large \bf Quantised Singularities in the Electromagnetic Field}
\end{center}
\begin{center}
{\Large P.A.M.~ Dirac}\\
Received May 29, 1931\\
\end{center}
\vspace{2cm}
\section*{\it \S~1. Introduction}
The steady progress of physics requires for its theoretical formulation a
mathematics that gets continually more advanced. This is only natural and to be
expected. What, however, was not expected by the scientific workers of the last
century was the particular form that the line
of advancement of the mathematics would take, namely, it was expected that the
mathematics would get more and more complicated, but would rest on a permanent
basis of axioms and definitions, while actually the modem physical developments
have required a mathematics that continually shifts its foundations and gets
more abstract. Non-euclidean
geometry and non-commutative algebra, which were at one time considered to be purely
fictions of the mind and pastimes for logical thinkers, have now been
found to be very necessary for the description of general facts of the
physical world. It seems likely that this
process of increasing abstraction will continue in the future and that advance
in physics is to be associated with a continual modification and generalisation
of the axioms at the base of the mathematics rather than with a logical
development of any one mathematical scheme on a fixed foundation.
There are at present fundamental problems in theoretical physics awaiting
solution, e.g., the relativistic formulation of quantum mechanics and the nature
of atomic nuclei (to be followed by more difficult ones such as the problem
of life), the solution of which problems
will presumably require a more drastic revision of our fundamental concepts
than any that have gone before. Quite likely these changes will be so great
that it will be beyond the power
of human intelligence to get the necessary new ideas by direct attempts to
formulate the experimental data in mathematical terms. The theoretical worker
in the future will therefore have to proceed in a more indirect way. The most
powerful method of advance that can be
suggested at present is to employ all the resources of pure mathematics in
attempts to perfect and generalise the mathematical formalism that forms the
existing basis of theoretical physics, and after each success in this direction,
to try to interpret the new mathematical
features in terms of physical entities (by a process like Eddington's Principle of
Identification).
A recent paper by the author \footnote{Proc. Roy. Soc.,' A, vol. 126, p. 360 (1930).}
may possibly be regarded as a small step according to this
general scheme of advance. The mathematical formalism at that time involved a serious
difficulty through its prediction of negative kinetic energy values for an
electron. It was proposed to get over this difficulty, making use of Fault's
Exclusion Principle which does not allow more than one electron in any state,
by saying that in the physical world almost all the
negative-energy states are already occupied, so that our ordinary electrons of positive energy
cannot fall into them. The question then arises to the physical interpretation
of the negative--energy states, which on this view really exist. We should expect
the uniformly filled distribution of negative--energy states to be completely
unobservable to us, but an unoccupied one of these states, being something
exceptional, should make its presence felt as a kind of
hole. It was shown that one of these holes would appear to us as a particle with
a positive energy and a positive charge and it was suggested that this particle
should be identified with a proton. Subsequent investigations, however, have
shown that this particle necessarily has the same mass as an
electron \footnote{H. Weyl,' Gruppentheorie and Quantenmechanik,' 2nd ed. p.
234 (1931).} and also
that, if it collides with an electron, the two will have a chance of annihilating
one another much too great to be consistent with the known stability
of matter. \footnote{I. Tamm,' Z. Physik,' vol.
62, p. 545 (1930);' J. B. Oppenheimer, ' Phys. Rev.,' vol. 35, p. 939 (1930);
P. Dirac, ' Proc. Camb. Philos. Soc.,' vol. 26, p. 361 (1930).}
It thus appears that we must abandon the identification of the holes with protons
and must find some other interpretation for them. Following
Oppenheimer, \footnote{J. R. Oppenheimer,' Phys. Rev.,' vol. 35, p. 562 (1930).}
we can assume that in the
world as we know it, {\it all,} and not merely nearly all, of the negative--energy
states for electrons are occupied. A hole, if there were one, would be a new kind
of particle, unknown to experimental physics, having the same mass and opposite
charge to an electron. We may call such a particle an anti--electron. We should
not expect to find any of them in nature, on
account of their rapid rate of recombination with electrons, but if they could
be produced experimentally in high vacuum they would be quite stable and amenable
to observation. An encounter between two hard $\gamma$--rays (of energy at least
half a million volts) could lead to the creation simultaneously of an electron
and anti-electron, the probability of occurrence of this
process being of the same order of magnitude as that of the collision of the
two $\gamma$--rays on the
assumption that they are spheres of the same size as classical
electrons. This probability is negligible, however, with the intensities of
$\gamma$--rays at present available.
The protons on the above view are quite unconnected with electrons. Presumably the
protons will have their own negative--energy states, all of which normally are
occupied, an unoccupied one appearing as an anti--proton. Theory at present is
quite unable to suggest a reason why there should be any differences between
electrons and protons.
The object of the present paper is to put forward a new idea which is in many
respects comparable with this one about negative energies. It will be concerned
essentially, not with electrons and protons, but with the reason for the existence
of a smallest electric charge. This smallest charge is known to exist
experimentally and to have the value $e$ given approximately
by\footnote{$h$ means Planck's divided by $2 \pi$.}
\begin{equation}
hc/e^2 = 137.
\end{equation}
The theory of this paper, while it looks at first as though it will give a
theoretical value for $e,$ is found when worked out to give a connection between
the smallest electric charge and the smallest magnetic pole. It shows, in fact,
a symmetry between electricity and magnetism quite foreign to current views.
It does not, however, force a complete symmetry, analogous to the fact that the
symmetry between electrons and protons is not forced when we adopt
Oppenheimer's interpretation. Without this symmetry, the ratio on the left--hand
aide of (1) remains, from the theoretical standpoint, completely undetermined
and if we insert the experimental value 137 in our theory, it introduces
quantitative differences between electricity and magnetism so large that one can
understand why their qualitative similarities
have not been discovered experimentally up to the present.
\section*{\it
\S~ 2. Non--integrable Phases for Wave Functions.}
We consider a particle whose motion is represented by a wave function $\psi$
which is a function of $x, y, z$ and $t.$ The precise form of the wave equation
and whether it is relativistic or not, are not important for the present theory.
We express $\psi$ in the form
\begin{equation}
\psi = Ae^{i \gamma},
\end{equation}
where $A$ and $\gamma$ are real functions of $x, y, z$ and $t,$ denoting the
amplitude and phase of the wave function. For a given state of motion of the
particle, $\psi$ will be determined except for an
arbitrary constant numerical coefficient, which must be of modulus unity if we
impose the condition that shall be normalised.
The indeterminacy in $\psi$ then consists in the possible addition of an
arbitrary constant to the phase $\gamma.$ Thus the value of $\gamma$ at a
particular point has no physical meaning and only the
difference between the values of y at two different points is of any importance.
This immediately suggests a generalisation of the formalism. We may assume that
$\gamma$ has no definite value at a particular point, but only a definite
difference in values for any two points. We may go further and assume that this
difference is not definite unless the two points are neighbouring. For two
distant points there will then be a definite phase difference
only relative to some curve joining them and different curves will in general
give different phase differences. The total change in phase when one goes round
a closed curve need not vanish.
Let us examine the conditions necessary for this non--integrability of phase not
to give rise to ambiguity in the applications of the theory. If we multiply $\psi$
by its conjugate complex $\phi$
we get the density function, which has a direct physical meaning. This density
is independent of the phase of the wave function, so that no trouble will be
caused in this connection by any indeterminacy of phase. There are other more
general kinds of applications, however, which
must also be considered. If we take two different wave functions $\psi_m$ and
$\psi_n$ we may have
to make use of the product $\phi_m \psi_n$. The integral
$$
\int \phi_m \psi_n dx dy dz
$$
is a number, the square of whose modulus has a physical meaning, namely, the
probability of agreement of the two states. In order that the integral may have
a definite modulus the integrand, although it need not have a definite phase at
each point, must have a definite phase difference between any two points,
whether neighbouring or not. Thus the change in phase in
$\phi_m \psi_n$ round a closed curve must vanish. This requires that the change
in phase in $\psi_n$ round
a closed curve shall be equal and opposite to that $\phi_m$ and hence the same
as that in $\psi_m.$
We thus get the general result: — {\it The change in phase of a wave function
round any closed curve must be the same for all the wave functions.}
It can easily be seen that this condition, when extended so as to give the same
uncertainty of phase for transformation functions and matrices representing
observables (referring to representations in which $x, y$ and $z$ are diagonal)
as for wave functions, is sufficient to insure
that the non--integrability of phase gives rise to no ambiguity in all applications
of the theory. Whenever a $\psi_n$ appears, if it is not multiplied into a $\phi_m$,
it will at any rate be multiplied into something of a similar nature to a $\phi_m$,
which will result in the uncertainty of phase cancelling out, except for a constant
which does not matter. For example, if $\psi_n$ is to be transformed to another
representation in which, say, the observables $\xi,$
are diagonal, it must be multiplied by the transformation function $(\xi,xyzt)$
and integrated with respect to $x, y$ and $z.$ This transformation function will
have the same uncertainty of phase as a $\phi$, so that the transformed wave
function will have its phase determinate, except
for a constant independent of $\xi$. Again, if we multiply $\psi_n$ by a matrix
$(x'y'z't| \alpha | x'' y'' z'' t),$
representing an observable $\alpha$, the uncertainty in the phase as concerns
the column [specified by $x'', y'', z'', t$] will cancel the uncertainty in
$\psi_n$ and the uncertainty as concerns the row will
survive and give the necessary uncertainty in the new wave function $\alpha
\psi_n$. The superposition principle for wave functions will be discussed a
little later and when this point
is settled it will complete the proof that all the general operations of quantum
mechanics can be carried through exactly as though there were no uncertainty
in the phase at all.
The above result that the change in phase round a closed curve must be the same
for all wave functions means that this change in phase must be something
determined by the dynamical system itself (and perhaps also partly by the
representation) and must be independent of which state of the system is considered.
As our dynamical system is merely a simple particle, it appears that the
non--integrability of phase must be connected with the
field of force in which the particle moves.
For the mathematical treatment of the question we express $\psi$, more generally
than (2), as a product
\begin{equation}
\psi = \psi_1 e ^{i \beta},
\end{equation}
where $\psi_1$ is any ordinary wave function (i.e., one with a definite phase
at each point) whose modulus is everywhere equal to the modulus of $\psi$.
The uncertainty of phase is thus
put in the factor $e^{i \beta}$. This requires that $\beta$ shall not be a
function of $x, y, z, t$ having a definite value at each point, but $\beta$
must have definite derivatives
$$
\kappa_x = \frac{\partial \beta}{\partial x}, \qquad
\kappa_y = \frac{\partial \beta}{\partial y}, \qquad
\kappa_z = \frac{\partial \beta}{\partial z}, \qquad
\kappa_0 = \frac{\partial \beta}{\partial t},
$$
at each point, which do not in general satisfy the conditions of integrability
$\partial \kappa_x/\partial y = \partial \kappa_y/\partial x,$
etc. The change in phase round a closed curve will now be, by Stokes' theorem,
\begin{equation}
\int ({\bf \kappa, ds}) = \int ({\rm curl}~ {\bf \kappa, dS}),
\end{equation}
where $ds$ (a 4--vector) is an element of arc of the closed curve and $dS$
(a 6--vector) is an element of a two--dimensional surface whose boundary is the
closed curve. The factor $\psi_1$ does not enter at all into this change in phase.
It now becomes clear that the non--integrability of phase is quite consistent
with the principle of superposition, or, stated more explicitly, that if we take
two wave functions $\psi_m$ and $\psi_n$ both having the same change in phase
round any closed curve, any linear combination of them $c_m \psi_m + c_n
\psi_n$ must also have this same change in phase round every closed
curve. This is because $\psi_m$ and $\psi_n$ will both be expressible in the
form (3) with the same factor $e^{i \beta}$ (i.e., the same $\kappa$'s) but
different $\psi_1$'s, so that the linear combination will be
expressible in this form with the same $e^{i \beta}$ again, and this
$e^{i\beta}$ determines the change in phase
round any closed curve. We may use the same factor $e^{i \beta}$ in (3) for
dealing with all the wave functions of the system, but we are not obliged to do
so, since only curl $\kappa$ is fixed and we may use $\kappa$'s differing from
one another by the gradient of a scalar for treating the different
wave functions.
From (3) we obtain
\begin{equation}
- ih~ \frac{\partial}{\partial x}~ \psi = e^{i \beta} \left( - ih~
\frac{\partial}{\partial x} + h \kappa_x \right) ~ \psi_1,
\end{equation}
\noindent •
with similar relations for the $y, z$ and $t$ derivatives. It follows that if
$\psi$ satisfies any wave equation, involving the momentum and energy
operators ${\bf p}$
and $W, \psi_1$ will satisfy the
corresponding wave equation in which ${\bf p}$ and $W$ have been replaced
by ${\bf p} + h {\bf \kappa}$ and $W - h {\bf \kappa_0}$ respectively.
Let us assume that $\psi$ satisfies the usual wave equation for a free particle
in the absence of any field. Then $\psi_1$ will satisfy the usual wave equation
for a particle with charge $- e$ moving
in an electromagnetic field whose potentials are
\begin{equation}
{\bf A} = hc/e \cdot {\bf \kappa}, \qquad {\bf A_0} = -h/e \cdot {\bf \kappa_0}.
\end{equation}
Thus, since $\psi_1$ is just an ordinary wave function with a definite phase,
our theory reverts to the usual one for the motion of an electron in an
electromagnetic field. This gives a physical meaning to our non--integrability
of phase. We see that we must have the wave function $\psi$
always satisfying the same wave equation, whether there is a field or not, and
the whole effect of the field when there is one is in making the phase non--integrable.
The components of the 6-vector curl $\kappa$ appearing in (4) are, apart from
numerical coefficients, equal to the components of the electric and magnetic
fields ${\bf E}$ and ${\bf H.}$ They are,
written in three-dimensional vector--notation,
\begin{equation}
{\rm curl} ~{\bf \kappa} = \frac{e}{hc} ~{\bf H}, \qquad
{\rm grad}~ {\bf \kappa_0} - \frac{\partial \kappa}{\partial t} = \frac{e}{h}~{\bf
E.}
\end{equation}
The connection between non-integrability of phase and the electromagnetic field
given in this section is not new, being essentially just Weyl's Principle of
Gauge Invariance in its
modern form. \footnote{H. Weyl,' Z. Physik,' vol. 56, p. 330 (1929).} It is also
contained in the work of Iwanenko and Fock, \footnote{D. Iwanenko and V. Fock,'
C. R.,' vol. 188, p. 1470 (1929); V. Fock,' Z. Physik.' vol. 57, p. 261 (1929). The more general kind of
non-integrability considered by these authors does not seem to
have any physical application.} who consider a more
general kind of non--integrability based on a general theory of parallel
displacement of half--vectors. The present treatment is given in order to
emphasise that non--integrable phases are perfectly compatible with all the general
principles of quantum mechanics and do not in any
way restrict their physical interpretation.
\section*{\it
\S~ 3. Nodal Singularities.}
We have seen in the preceding section how the non--integrable derivatives
$\kappa$ of the phase of the wave function receive a natural interpretation in
terms of the potentials of the electromagnetic field, as the result of which our
theory becomes mathematically equivalent
to the usual one for the motion of an electron in an electromagnetic field and
gives us nothing new. There is, however, one further fact which must now be taken
into account, namely, that a phase is always undetermined to the extent of an
arbitrary integral multiple of $2 \pi$. This requires a reconsideration of the
connection between the $\kappa$'s and the potentials and
leads to a new physical phenomenon.
The condition for an unambiguous physical interpretation of the theory was that the
change in phase round a closed curve should be the same for all wave functions.
This change was then interpreted, by equations (4) and (7), as equal to
(apart from numerical factors) the total flax: through the closed curve of the
6--vector ${\bf E, H}$ describing the electromagnetic field.
Evidently these conditions must now be relaxed. The change in phase round a closed
curve may be different for different wave functions by arbitrary multiples of $2
\pi$ and is thus not
sufficiently definite to be interpreted immediately in terms of the electromagnetic
field.
To examine this question, let us consider first a very small closed curve.
Now the wave equation requires the wave function to be continuous
(except in very special circumstances which can be disregarded here) and hence the
change in phase round a small closed curve must be small.
Thus this change cannot now be different by multiples of $2 \pi$ for different
wave functions. It must have one definite value and may therefore be interpreted
without ambiguity in terms of the flux of the 6--vector $E, H$ through the small
closed curve, which flux must also be small.
There is an exceptional case, however, occurring when the wave function vanishes,
since then its phase does not have a meaning. As the wave function is complex,
its vanishing will require two conditions, so that in general the points at which
it vanishes will lie along a line. \footnote{We are here considering, for simplicity
in explanation, that the wave function is in three
dimensions. The passage to four dimensions makes no essential change in the theory.
The nodal lines then become two--dimensional nodal surfaces, which can be encircled by curves in the same way as lines
are in three dimensions.}
We call such a line a nodal line. If we now take a wave function having a nodal line
passing through our small closed curve, considerations of continuity will no
longer enable us to infer that the change in phase round the small closed curve
must be small. All we shall be able to say is that the change in phase will be
close to $2 \pi n$ where $n$ is some integer, positive or
negative. This integer will be a characteristic of the nodal line.
Its sign will be
associated with a direction encircling the nodal line, which in turn may be
associated with a direction along the nodal line.
The difference between the change in phase round the small closed curve and the
nearest $2 \pi n$ must now be the same as the change in phase round the closed
curve for a wave function with no nodal line through it. It is therefore this
difference that must be interpreted in terms of the flux of the 6--vector ${\bf
E, H}$
through the closed curve. For a closed curve in
three--dimensional space, only magnetic flux will come into play and hence we
obtain for the change in phase round the small closed curve
$$
2 \pi n + e/hc \cdot \int~ ({\bf H, dS}).
$$
We can now treat a large closed curve by dividing it up into a network of small
closed curves lying in a surface whose boundary is the large closed curve. The
total change in phase round the large closed curve will equal the sum of all the
changes round the small closed curves and will therefore be
\begin{equation}
2 \pi \sum n + e/hc \cdot \int~ ({\bf H, dS}),
\end{equation}
the integration being taken over the surface and the summation over all nodal
lines that pass through it, the proper sign being given to each term
in the sum. This expression consists of two parts, a part $e/hc \cdot
\int~ ({\bf H, dS})$
which must be the same for all wave functions and a part $2 \pi \sum n$ which may
be different for different wave functions.
Expression (8) applied to any surface is equal to the change in phase round the
boundary of the surface. Hence expression (8) applied to a closed surface must
vanish. It follows that $\sum n$, summed for all nodal lines crossing a closed
surface, must be the same for all wave
functions and must equal $- e/2 \pi hc$ times the total magnetic flux crossing
the surface.
If $\sum n$ does not vanish, some nodal lines must have end points inside the
closed surface, since a nodal line without such end point must cross the surface
twice (at least) and will contribute equal and opposite amounts to $\sum
n$ at the two points of crossing. The value of $\sum n$
for the closed surface will thus equal the sum of the values of $n$ for all nodal
lines having end points inside the surface. This sum must be the same for all
wave functions. Since this result applies to any closed surface, it follows {\it
that the end points of nodal lines must be the same
for all wave functions. These end points are then points of singularity in the
electromagnetic field.} The total flux of magnetic field crossing a small closed
surface surrounding one of these points is
$$
4 \pi \mu = 2 \pi nhc/e,
$$
where $n$ is the characteristic of the nodal line that ends there, or the sum of
the characteristics of all nodal lines ending there when there is more than one.
Thus at the end point there will be a magnetic pole of strength
$$
\mu = \frac{1}{2}nhc/e.
$$
Our theory thus allows isolated magnetic poles, but the strength of such poles
must be quantised, the quantum $\mu_0$ being connected with the electronic charge
$e$ by
\begin{equation}
hc/e \mu_0 = 2.
\end{equation}
This equation is to be compared with (1). The theory also requires
a quantisation
of electric charge, since any charged particle moving in the field of a pole of
strength $\mu_0$ must have for
its charge some integral multiple (positive or negative) of $e,$ in order that
wave functions describing the motion may exist.
\section*{\it
\S~ 4. Electron in Field of One--Quantum Pole.}
The wave functions discussed in the preceding section, having nodal lines ending on
magnetic poles, are quite proper and amenable to analytic treatment by methods
parallel to the usual ones of quantum mechanics. It will perhaps help the reader
to realise this if a simple example is discussed more explicitly.
Let us consider the motion of an electron in the magnetic field of a
one--quantum pole when there is no electric field present. We take polar
co--ordinates $\tau, \theta, \phi$, with the magnetic pole as origin.
Every wave function must now have a nodal line radiating out
from the origin.
We express our wave function $\psi$ in the form (3), where $\beta$ is some
non--integrable phase having derivatives $\kappa$ that are connected with the
known electromagnetic field by equations
(6). It will not, however, be possible to obtain $\kappa$'s satisfying these
equations all round the magnetic pole. There must be some singular line radiating
out from the pole along which these equations are not satisfied, but this line
may be chosen arbitrarily. We may choose it to
be the same as the nodal line for the wave function under consideration, which
would result in $\psi_1$ being continuous. This choice, however, would mean
different $\kappa$'s for different wave
functions (the difference between any two being, of course, the four--dimensional
gradient of a scalar, except on the singular lines). This would perhaps be
inconvenient and is not really necessary. We may express all our wave functions
in the form (3) with the same $e^{i \beta}$, and
then those wave functions whose nodal lines do not coincide with the singular
line for the $\kappa$'s
will correspond to $\psi_1$'s having a certain kind of discontinuity on this
singular line, namely, a discontinuity just cancelling with the discontinuity
in $e^{i \beta}$ here to give a continuous product.
The magnetic field ${\bf H,}$ lies along the radial direction and is of magnitude
$\mu_0/ \tau^2$, which by (9) equals $^1/_2 hc/ e \tau^2$. Hence, from equations
(7), curl ${\bf \kappa}$ is radial and of magnitude $1/2 \tau^2$. It may
now easily be verified that a solution of the whole of equations (7) is
\begin{equation}
\kappa_0 = 0, \qquad \kappa_{\tau} = \kappa_{\theta} = 0, \qquad
\kappa_{\phi} = 1/2 \tau \cdot \tan \frac{1}{2}~ \theta ,
\end{equation}
where $\kappa_{\tau}, \kappa_{\theta}, \kappa_{\phi}$, are the components of
$\kappa$ referred to the polar co--ordinates. This solution is
valid at all points except along the line $\theta = \pi$, where $\kappa_{\phi}$,
become infinite in such a way that $\int ({\bf \kappa, ds})$ round a small curve
encircling this line
is $2 \pi$. We may refer all our wave functions to this set of $\kappa$'s.
Let us consider a stationary state of the electron with energy $W.$ Written non-
relativistically, the wave equation is
$$
- h^2 / 2 m \cdot \bigtriangledown^2 \psi = W \psi.
$$
If we apply the rule expressed by equation (5), we get as the wave equation
for $\psi_1$
\begin{equation}
- h^2 / 2m \cdot \left\{ \bigtriangledown^2 + i ({\bf \kappa, \bigtriangledown})
+ i ({\bf \bigtriangledown, \kappa}) - \kappa^2 \right\} ~\psi_1 = W \psi_1.
\end{equation}
The values (10) for the $\kappa$'s give
$$
({\bf \kappa, \bigtriangledown}) = ({\bf \bigtriangledown, \kappa}) = \kappa_{\phi}~
\frac{1}{\tau \sin \theta }~ \frac{\partial}{\partial \phi} = \frac{1}{4
\tau^2} ~ \sec^2 ~\frac{1}{2}~ \theta ~ \frac{\partial}{\partial \phi}
$$
$$
{\bf \kappa^2} = \kappa_{\phi}^2 = \frac{1}{4 \tau^2}~ \tan^2~ \frac{1}{2}~ \theta,
$$
so that equation (11) becomes
$$
- \frac{h^2}{2m}~ \left\{ \bigtriangledown^2 + \frac{i}{2 \tau^2}~
\sec^2~ \frac{1}{2}~ \theta ~ \frac{\partial}{\partial \phi} - \frac{1}{4
\tau^2}~ \tan^2~ \frac{1}{2} ~ \theta \right\}~ \psi_1 = W \psi_1.
$$
We now suppose $\psi_1$ to be of the form of a function $f$ of $\tau$ only
multiplied by a function 8 of 6
and $\phi$ only, i.e.,
$$
\psi_1 = f (\tau) S (\theta \phi).
$$
This requires
\begin{equation}
\left\{ \frac{d^2}{d \tau^2} + \frac{2}{\tau}~ \frac{d}{d \tau} - \frac{\lambda}
{\tau^2} \right\}~ f = - \frac{2 m W}{h^2}~ f,
\end{equation}
\begin{equation}
\left\{ \frac{1}{\sin \theta} ~ \frac{\partial}{\partial \theta}
\sin \theta \frac{\partial}{\partial \theta}~
+ \frac{1}{\sin^2
\theta}~ \frac{\partial^2}{\partial \phi^2} + \frac{1}{2} i ~ \sec^2 ~
\frac{1}{2}~ \theta ~ \frac{\partial}{\partial \phi} \right.
$$
$$
\left.
- \frac{1}{4}~ \tan^2~
\frac{1}{2}~ \theta \right\}~ S = - \lambda S,
\end{equation}
where $\lambda$ is a number.
From equation (12) it is evident that there can be no stable states for which the
electron is bound to the magnetic pole, because the operator on the left--hand
side contains no constant with the dimensions of a length. This result is what
one would expect from analogy with the classical theory. Equation (13) determines
the dependence of the wave function on angle. It
may be considered as a generalisation of the ordinary equation for spherical
harmonies.
The lowest eigenvalue of (13) is $\lambda = 1/2$, corresponding to which there
are two independent wave functions
$$
S_a = \cos \frac{1}{2}~ \theta, \qquad S_b = \sin \frac{1}{2}~ \theta e^{i
\phi},
$$
as may easily be verified by direct substitution. The nodal line for $S_a$ is
$\theta = \pi$, that for $S_b$, is
$\theta = 0$. It should be observed that $S_a$ is continuous everywhere, while
$S_b$, is discontinuous
for $\theta = \pi$, its phase changing by $2 \pi$ when one goes round a small
curve encircling the line $\theta = \pi$. This is just what is necessary in order
that both $S_a$ and $S_b$, when multiplied by the $e^{i \beta}$
factor, may give continuous wave functions $\psi.$ The two $\psi$'s that we get
in this way are both on the same footing and the difference in behaviour of
$S_a$ and $S_b$, is due to our having chosen $\kappa$'s
with a singularity at $\theta = \pi$.
The general eigenvalue of (13) is $\lambda = n^2 + 2 n + \frac{1}{2}$.
The general solution of this wave
equation has been worked out by I. Tamm. \footnote{Appearing
probably in `Z. Physik.'}
\section*{\it
\S~ 5. Conclusion.}
Elementary classical theory allows us to formulate equations of motion for an
electron in the field produced by an arbitrary distribution of electric charges
and magnetic poles. If we wish to put the equations of motion in the
Hamiltonian form, however, we have to introduce the electromagnetic potentials,
and this is possible only when there are no isolated magnetic
poles. Quantum mechanics, as it is usually established, is derived from the
Hamiltonian form
of the classical theory and therefore is applicable only when there are no
isolated magnetic poles.
The object of the present paper is to show that quantum mechanics does not really
preclude the existence of isolated magnetic poles. On the contrary, the present
formalism of quantum mechanics, when developed naturally without the
imposition of arbitrary restrictions, leads inevitably to wave equations whose
only physical interpretation is the motion of an electron in the field of a
single pole. This new development requires {\it no change
whatever} in the formalism when expressed in terms of abstract symbols denoting
states and observables, but is merely a generalisation of the possibilities of
representation of these abstract symbols by wave functions and matrices.
Under these circumstances one would be surprised if Nature had made no use of it.
The theory leads to a connection, namely, equation (9), between the quantum of
magnetic pole and the electronic charge. It is rather disappointing to find
this reciprocity between electricity and magnetism, instead of a purely
electronic quantum condition, such as (1). However, there appears to be no
possibility of modifying the theory, as it contains no
arbitrary features, so presumably the explanation of (1) will require some
entirely new idea.
The theoretical reciprocity between electricity and magnetism is perfect.
Instead of discussing the motion of an electron in the field of a fixed
magnetic pole, as we did in \S~ 4,
we could equally well consider the motion of a pole in the field of fixed charge.
This would require the introduction of the electromagnetic potentials $B$ satisfying
$$
{\bf E} = {\rm curl}~ {\bf B}, \qquad {\bf H} = \frac{1}{c}~ \frac{\partial {\bf
B}}{\partial t}
+ {\rm grad}~ {\bf B_0},
$$
to be used instead of the $A$'s in equations (6). The theory would now run quite
parallel and would lead to the same condition (9) connecting the smallest pole
with the smallest charge.
There remains to be discussed the question of why
isolated magnetic poles are not observed.
The experimental result (1) shows that there must be some
cause of dissimilarity between electricity and magnetism (possible connected
with the cause of dissimilarity between electrons and protons) as the result
of which we have, not $\mu_0 = e$, but $\mu_0 = 137/2 \cdot e.$ This
means that the attractive force between two one--quantum poles of opposite
sign is $(137/2)^2 = 4692 \frac{1}{4}$ times that between electron
and proton. This very large force may perhaps account for why poles of
opposite sign have never yet been separated.
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