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\begin{document}
W. Pauli, Phys. Rev., {\bf Vol. 58,} 716 \hfill {\large \bf 1940}\\
\vspace{2cm}
\begin{center}
{\large The Connection Between Spin and Statistics}\footnote{This paper is part of a report which was prepared by the
author for the Solvay Congress 1939 and in which slight improvements have
since been made. In view of the unfavorable times, the Congress did
not Cake place, and the publication of the reports has been postponed
for an indefinite length of time. The relation between the present
discussion of the connection between spin and statistics, and the
somewhat less general one of Belinfante, based on the concert of charge
invariance, has been cleared up by W. Pauli and J. Belinfante, Physica
7, 177 (1940).}\\
\end{center}
\vspace{0.5cm}
\begin{center}
W. Pauli\\
Princeton, New Jersey\\
(Received August 19, 1940)
\end{center}
\vspace{0.5cm}
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\noindent
Reprinted in ``Quantum Electrodynamics'', edited by Julian
Schwinger\\
\centerline{--- ---~~~$\diamond~\diamondsuit~\diamond$~~~--- ---}
\vspace{0.5cm}
\begin{abstract}
In the following paper we conclude for the relativistically invariant wave equation
for free particles: From postulate (I), according to which the energy must be
positive, the necessity of {\it Fermi-Dirac} statistics for particles with arbitrary
half-integral spin; from postulate (II), according to which observables on different
space-time points with a space-like distance are commutable, the necessity of
{\it Einstein-Base} statistics for particles with arbitrary integral spin. It has
been found useful to divide the quantities which are irreducible against Lorentz
transformations into four symmetry classes which have a commutable multiplication
like $+1, - 1, + \epsilon, - \epsilon$ with $\epsilon^2 = 1.$
\end{abstract}
\section*{\S~1. UNITS AND NOTATIONS}
Since the requirements of the relativity theory and
the quantum theory are fundamental for every theory,
it is natural to use as units the vacuum velocity of
light $c$, and Planck's constant divided by $2 \pi$ which we
shall simply denote by $\hbar$. This convention means that
all quantities are brought to the dimension of the
power of a length by multiplication with powers of $\hbar$
and $c$. The reciprocal length corresponding to the rest
mass $m$ is denoted by $\kappa = mc/ \hbar$.
As time coordinate we use accordingly the length
of the light path. In specific cases, however, we do
not wish to give up the use of the imaginary time
coordinate. Accordingly, a tensor index denoted by
small Latin letters $i$, refers to the imaginary time
coordinate and runs from 1 to 4. A special convention
for denoting the complex conjugate seems desirable.
Whereas for quantities with the index 0 an asterisk
signifies the complex-conjugate in the ordinary sense
(e.g., for the current vector $S_i$ the quantity $S_0^{\ast}$ is the
complex conjugate of the charge density $S_0$). in
general $U^{\ast}_{i \kappa \ldots}$ signifies:
the complex-conjugate of $U_{i \kappa \ldots}$ multiplied with $(-1)^n$,
where $n$ is the number of occurrences of
the digit 4 among the $i, k, \ldots$ (e.g. $S_4 = i S_0$,
$S^{\ast}_4 = i S_0^{\ast}$).
Dirac's spinors $u_{\rho}$, with $\rho = 1, \ldots, 4$ have always a
Greek index running from 1 to 4, and $u_{\rho}^{\ast}$ means the
complex-conjugate of $u_{\rho}$, in the ordinary sense.
Wave functions, insofar as they are ordinary
vectors or tensors, are denoted in general with capital
letters, $U_i, U_{i \kappa} \ldots$ The symmetry character of these
tensors must in general be added explicitly. As
classical fields the electromagnetic and the
gravitational fields, as well as fields with rest mass
zero, take a special place, and are therefore denoted
with the usual letters $\varphi_i,~ f_{i \kappa} = - f_{\kappa i}$ and
$g_{i \kappa} = g_{\kappa i}$ respectively.
The energy-momentum tensor $T_{i \kappa}$, is so defined,
that the energy-density $W$ and the momentum density
$G_{\kappa}$ are given in natural units by $W = - T_{44}$ and $G_{\kappa}
= - i T_{\kappa 4}$ with $k = 1, 2, 3.$
\section*{\S~2. IRREDUCIBLE TENSORS. DEFINITION OF SPINS}
We shall use only a few general properties of those
quantities which transform according to irreducible
representations of the Lorentz group.\footnote{See B. L. v. d. Waerden,
{\it Die gruppentheoretische Methode in der Quantentheorie} (Berlin, 1932).}
The proper Lorentz group is that continuous linear group the
transformations of which leave the form
$$
\sum \limits^4_{k = 1} x^2 _{k} = {\bf x}^2 - x_0^2
$$
invariant and in addition to that satisfy the condition
that they have the determinant $+ 1$
and do not reverse the time. A tensor or spinor which
transforms irreducibly under this group can be
characterized by two integral positive numbers $(p, q)$.
(The corresponding ``angular momentum quantum
numbers'' $(j, k)$ are then given by $p = 2 j + 1,$ $q = 2k + 1$,
with integral or half-integral $j$ and $k$.)\footnote{In the spinor calculus
this is a spinor with $2j$ undotted and $2k$ dotted indices.} The quantity
$U(j, k)$ characterized by $(j, k)$ has $p \cdot q = (2j + 1)(2k + 1)$
independent components. Hence to $(0, 0)$ corresponds
the scalar, to $(\frac{\displaystyle 1}{\displaystyle 2}, \frac{\displaystyle 1}{\displaystyle
2})$ the vector, to $(1, 0)$ the self-dual
skew-symmetrical tensor, to $(1, 1)$ the symmetrical
tensor with vanishing spur, etc. Dirac's spinor it,
reduces to two irreducible quantities $(\frac{\displaystyle 1}{\displaystyle
2}, 0)$ and $(0, \frac{\displaystyle 1}{\displaystyle 2})$
each of which consists of two components. If $U(j, k)$
transforms according to the representation
$$
U'_{r} = \sum \limits_{s = 1}^{(2j + 1)(2k + 1)} \Lambda _{r s} U_s,
$$
then $U^{\ast} (k, j)$ transforms according to the complex-conjugate
representation $\Lambda^{\ast}$. Thus for $k = j,$ $\Lambda^{\ast} = \Lambda$.
This is true only if the components of $U(j, k)$ and
$U(k, j)$ are suitably ordered. For an arbitrary choice
of the components, a similarity transformation of $\Lambda$
and $\Lambda^{\ast}$ would have to be added. In view of \S~1 we
represent generally with $U^{\ast}$ the quantity the
transformation of which is equivalent to $\Lambda^{\ast}$ if the
transformation of $U$ is equivalent to $\Lambda$.
The most important operation is the reduction of
the product of two quantities
$$
U_1 (j_1, k_1) \cdot U_2 (j_2, k_2)
$$
which, according to the well-known rule of the
composition of angular momenta, decompose into
several $U(j, k)$ where, independently of each other $j, k$
run through the values
$$
j = j_1 + j_2,~ j_1 + j_2 - 1, \ldots , |j_1 - j_2|
$$
$$
k = k_1 + k_2,~ k_1 + k_2 - 1, \ldots , |k_1 - k_2|.
$$
By limiting the transformations to the subgroup of
space rotations alone, the distinction between the two
numbers $j$ and $k$ disappears and $U(j, k)$ behaves
under this group just like the product of two
irreducible quantities $U(j)U(k)$ which in turn reduces
into several irreducible $U(l)$ each having $2l + 1$ components, with
$$
l = j + k,~ j + k - 1, \ldots , |j - k|.
$$
Under the space rotations the $U(l)$ with integral $l$
transform according to single-valued representation,
whereas those with half-integral $l$ transform
according to double-valued representations. Thus the
unreduced quantities $T(j, k)$ with integral (half-integral) $j + k$ are
single-valued (double-valued).
If we now want to determine the spin value of the
particles which belong to a given field it seems at
first that these are given by $l = j + k$. Such a definition
would, however, not correspond to the physical facts,
for there then exists no relation of the spin value with
the number of independent plane waves, which are
possible in the absence of interaction) for given
values of the components $k$ in the phase factor
$\mbox{exp}~ i({\bf k x})$. In order to define the spin in an appropriate
fashion,\footnote{see M. Fierz, Helv. Phys. acta {\bf 12}, 3 (1939); also
L. de Broglie, Comptes rendus {\bf 208}, 1697 (1939); {\bf 209}, 265 (1939).}
we want to consider first the case in which
the rest mass $m$ of all the particles is different from
zero. In this case we make a transformation to the rest
system of the particle, where all the space
components of $k_i$, are zero, and the wave function
depends only on the time. In this system we reduce
the field components, which according to the field
equations do not necessarily vanish, into parts
irreducible against space rotations. To each such part,
with $r = 2s + 1$ componentsi belong $r$ different
eigenfunctions which under space rotations transform
among themselves and which belong to a particle
with spin $s$. If the field equations describe particles
with only one spin value there then exists in the rest
system only one such irreducible group of
components. From the Lorentz invariance, it follows,
for an arbitrary system of reference, that $r$ or $\sum r$
eigenfunctions always belong to a given arbitrary $k_i$.
The number of quantities $U(j, k)$ which enter the
theory is, however, in a general coordinate system
more complicated, since these quantities together
with the vector $k_i$ have to satisfy several conditions.
In the case of zero rest mass there is a special
degeneracy because, as has been shown by Fierz, this
case permits a gauge transformation of the
second kind.\footnote{By ``gauge-transformation of the first kind'' we understand a
transformation $U \rightarrow Ue^{i \alpha}$ $U^{\ast} \rightarrow U^{\ast}
e^{- i \alpha}$ with an arbitrary space and
time function $\alpha$. By ``gauge-transformation of the second kind'' we
understand a transformation of the type
$$
\varphi_k \rightarrow \varphi_k - \frac{1}{\epsilon} i \frac{\partial \alpha}{\partial
x_k}
$$
as for those of the electromagnetic potentials.}
If the field now describes only one
kind of particle with the rest mass zero and a certain
spin value, then there are for a given value of $k_i$.
only two states, which cannot be transformed into
each other by a gauge transformation. The definition
of spin may, in this case, not be determined so far as
the physical point of view is concerned because the
total angular momentum of the field cannot be
divided up into orbital and spin angular momentum
by measurements. But it is possible to use the
following property for a definition of the spin. If we
consider, in the $q$ number theory, states where only
one particle is present, then not all the eigenvalues
$j(j + 1)$ of the square of the angular momentum are
possible. But $j$ begins with a certain minimum value
$s$ and takes then the values $s, s + 1, \ldots.$ \footnote{The general proof for
this has been given by M. Fierz, Helv. Phys. Acta 13, 45 (1940).}
This is only the case for $m = 0$. For photons, $s = 1,$ $j = 0$ is not
possible for one single photon. \footnote{See for instance W. Pauli in the article
``Wellen-mechanik'' in the Handbuch der Physik, Vol. {\bf 24/2}, p. 260.}
For gravitational quanta $s = l$ and the values $j = 0$ and $j = 1$ do not
occur.
In an arbitrary system of reference and for
arbitrary rest masses, the quantities $U$ all of which
transform according to double-valued (single-valued) representations with
half-integral (integral) $j + k$ describe only particles with half-integral
(integral) spin. A special investigation is required
only when it is necessary to decide whether the
theory describes particles with one single spin value
or with several spin values.
\section*{\S~3. PROOF OF THE INDEFINITE CHARACTER OF
THE CHARGE IN CASE OF INTEGRAL AND OF THE ENERGY IN
CASE OF HALF-INTEGRAL SPIN}
We consider first a theory which contains only $U$
with integral $j + k,$ i.e., which describes particles with
integral spins only. It is not assumed that only
particles with one single spin value will be
described, but all particles shall have integral spin.
We divide the quantities $U$ into two classes:
(1) the ``$+ 1$ class'' with $j$ integral, $k$ integral;
(2) the ``$-1$ class'' with $j$ half-integral, $k$ half-integral.
The notation is justified because, according to the
indicated rules about the reduction of a product into
the irreducible constituents under the Lorentz group,
the product of two quantities of the $+1$ class or two
quantities of the $- 1$ class contains only quantities of
the $+1$ class, whereas the product of a quantity of the
$+1$ class with a quantity of the $-1$ class contains
only quantities of the $- 1$ class. It is important that
the complex conjugate $U^{\ast}$ for which $j$ and $k$ are
interchanged belong to the same class as $U$. As can
be seen easily from the multiplication rule, tensors
with even (odd) number of indices reduce only to
quantities of the $+1$ class ($-1$ class). The
propagation vector $k_i$ we consider as belonging to
the $-1$ class, since it behaves after multiplication
with other quantities like a quantity of the $- 1$ class.
We consider now a homogeneous and linear
equation in the quantities $U$ which, however, does
not necessarily have to be of the first order.
Assuming a plane wave, we may put $k_i$ for $- i \partial / \partial x_l$.
Solely on account of the invariance against the
{\it proper} Lorentz group it must be of the typical form
\begin{equation}
\sum k U^+ = \sum U^-, \quad \sum k U^- = \sum U^+.
\end{equation}
This typical form shall mean that there may be as
many different terms of the same type present, as
there are quantities $U^{\ast}$ and $U^-$. Furthermore,
among the $U^{\ast}$ may occur the $U^+$ as well as the
$(U^+)^{\ast}$, whereas other $U$ may satisfy reality conditions $U = U^{\ast}$.
Finally we have omitted an even number of $k$ factors. These may be present in
arbitrary number in the term of the sum on the left- or right-hand side of these
equations. It is now evident that these equations remain invariant under
the substitution
\begin{equation}
\begin{array}{l}
k_i \rightarrow - k_i; \quad U^+ \rightarrow U^+, \quad [(U^+) \rightarrow
(U^+)^{\ast}];\\
U^- \rightarrow - U^-, \quad [(U^-)^{\ast} \rightarrow - (U^-)^{\ast} \rightarrow
- (U^-)^{\ast}].
\end{array}
\end{equation}
Let us consider now tensors $T$ of even rank (scalars,
skew-symmetrical or symmetrical tensors of the 2nd
rank, etc.), which are composed quadratically or
bilinearly of the $U's$. They are then composed solely
of quantities with even $j$
and even $k$ and thus are of the typical form
\begin{equation}
T \sim \sum U^+ U^+ + \sum U^- U^- + \sum U^+ k U^-,
\end{equation}
where again a possible even number of $k$ factors is
omitted and no distinction between $U$ and $U^{\ast}$ is
made. Under the substitution (2) they remain
unchanged, $T \rightarrow T.$
The situation is different for tensors of odd rank $S$
(vectors, etc.) which consist of quantities with half-integral $j$ and half-integral
$k$. These are of the typical form
\begin{equation}
S \sim \sum U^+ kU^+ + \sum U^- kU^- + \sum U^-
\end{equation}
and hence change the sign under the substitution (2),
$S \rightarrow - S$. Particularly is this the case for the current
vector $s_i$. To the transformation $k_i \rightarrow - k_i$, belongs
for arbitrary wave packets the transformation $x_i \rightarrow - x_i$, and it
is remarkable that from the invariance of
Eq. (I) against the proper Lorentz group alone there
follows an invariance property for the change of sign
of all the coordinates. In particular, the indefinite
character of the current density and the total charge
for even spin follows, since to every solution of the
field equations belongs another solution for which
the components of $s_k$, change their sign. The
definition of a definite particle density for even spin
which transforms like the 4-component of a vector is
therefore impossible.
We now proceed to a discussion of the somewhat
less simple case of half-integral spins. Here we
divide the quantities $U$, which have half-integral $j + k$,
in the following fashion: (3) the ``$+ \epsilon$ class'' with $j$
integral $k$ half-integral, (4) the `` $- \epsilon$ class'' with $j$
half-integral $k$ integral.
The multiplication of the classes $(1), \ldots , (4),$
follows from the rule $\epsilon^2 = 1$ and the commutability of
the multiplication. This law remains unchanged if $\epsilon$
is replaced by $- \epsilon$.
We can summarize the multiplication law between
the different classes in the following multiplication
table:
\begin{center}
\begin{tabular}{r|r r r r}
& 1&$-1$&$\epsilon$& $- \epsilon$\\
\hline
~~1~~&~~1~~&~~$-1$~~&~~ $\epsilon$~~& ~~$- \epsilon$\\
$-1$&$- 1$& $+1$& $- \epsilon$& $+ \epsilon$\\
$\epsilon$& $- \epsilon$& $- \epsilon$& $+1$& $- 1$\\
$- \epsilon$& $- \epsilon$& $\epsilon$& $- 1$& $+1$\\
\end{tabular}
\end{center}
We notice that these classes have the multiplication law of Klein's ``four-group.''
It is important that here the complex-conjugate
quantities for which $j$ and $k$ are interchanged do not
belong to the same class, so that
$$
\begin{array}{l l}
U^{+ \epsilon},~ (U^{- \epsilon})^{\ast} \quad \mbox{belong to the} \quad
&+ \epsilon \quad \mbox{class}\\
U^{- \epsilon},~ (U^{+ \epsilon})^{\ast} &- \epsilon \quad \mbox{class}.
\end{array}
$$
We shall therefore cite the complex-conjugate
quantities explicitly. (One could even choose the
$U^{+ \epsilon}$ suitably so that {\it all} quantities of the $- \epsilon$
class are of the form $(U^{+ \epsilon})^{\ast}).$
Instead of (1) we obtain now as typical form
\begin{equation}
\begin{array}{l}
\sum k U^{+ \epsilon} + \sum k (U^{- \epsilon})^{\ast} = \sum U^{- \epsilon}
+ \sum (U^{+ \epsilon})^{\ast}\\
\sum kU^{- \epsilon} + \sum k (U^{+ \epsilon})^{\ast} = \sum U^{+ \epsilon}
+ \sum (U^{- \epsilon})^{\ast},
\end{array}
\end{equation}
since a factor $k$ or $- i \partial / \partial x$ always changes the
expression from one of the classes $+ \epsilon$ or $- \epsilon$ into the
other. As above, an even number of $k$ factors have
been omitted.
Now we consider instead of (2) the substitution
\begin{equation}
\begin{array}{l}
k_i \rightarrow - k_i; \quad U^{+ \epsilon} \rightarrow i U^{+ \epsilon};
\quad (U^{- \epsilon})^{\ast} \rightarrow i (U^{- \epsilon})^{\ast};\\
(U^{+ \epsilon} \rightarrow - i (U^{+ \epsilon})^{\ast}; \quad U^{- \epsilon}
\rightarrow - i U^{- \epsilon}.
\end{array}
\end{equation}
This is in accord with the algebraic requirement of
the passing over to the complex conjugate, as well as
with the requirement that quantities of the same class
as $U^{+ \epsilon}$, $(U^{- \epsilon})^{\ast}$ transform in the same way.
Furthermore, it does not interfere with possible
reality conditions of the type $U^{+ \epsilon} = (U^{- \epsilon})^{\ast}$. or
$U^{- \epsilon} = (U^{+ \epsilon})^{\ast}$.
Equations (5) remain unchanged under the
substitution (6).
We consider again tensors of even rank (scalars,
tensors of 2nd rank, etc.), which are composed
bilinearly or quadratically of the $U$ and their
complex-conjugate. For reasons similar to the above
they must be of the form
\begin{equation}
\begin{array}{l}
T \sim \sum U^{+ \epsilon} U^{+ \epsilon} + \sum U^{- \epsilon} U^{- \epsilon}
+ \sum U^{+ \epsilon} k U^{- \epsilon} + \sum U^{+ \epsilon} (U^{- \epsilon})^{\ast}\\
+ \sum U^{- \epsilon} (U^{+ \epsilon})^{\ast} + \sum (U^{- \epsilon})^{\ast}
k U^{- \epsilon}
+ \sum (U^{+ \epsilon})^{\ast} kU^{+ \epsilon} + \sum (U^{- \epsilon})^{\ast}
k (U^{+ \epsilon})^{\ast} \\
\sum (U^{- \epsilon})^{\ast} (U^{- \epsilon})^{\ast}
+ \sum (U^{+ \epsilon})^{\ast} (U^{+ \epsilon})^{\ast}.
\end{array}
\end{equation}
Furthermore, the tensors of odd rank (vectors, etc.) must be of the form
\begin{equation}
\begin{array}{l}
S \sim \sum U^{+ \epsilon} k U^{+ \epsilon} + \sum U^{- \epsilon} k U^{-
\epsilon} + \sum U^{+ \epsilon} U^{- \epsilon} + \sum U^{+ \epsilon} k
(U^{- \epsilon})^{\ast}\\ + \sum U^{- \epsilon} k (U^{+ \epsilon})^{\ast}
+ \sum U^{- \epsilon} (U^{- \epsilon})^{\ast}
+ \sum U^{+ \epsilon} (U^{+ \epsilon})^{\ast} + \sum (U^{- \epsilon})^{\ast}
k (U^{- \epsilon})^{\ast} + \\
\sum (U^{+ \epsilon})^{\ast} k (U^{+ \epsilon})^{\ast}
+ \sum (U^{- \epsilon})^{\ast} (U^{+ \epsilon})^{\ast}.
\end{array}
\end{equation}
{\it The result of the substitution (6) is now the
opposite of the result of the substitution (2): the
tensors of even rank change their sign, the
tensors of odd rank remain unchanged:}
\begin{equation}
T \rightarrow - T; \quad S \rightarrow + S.
\end{equation}
In case of half-integral spin, therefore, a positive
definite energy density, as well as a positive definite
total energy, is impossible. The latter follows from
the fact, that, under the above substitution, the
energy density in every space-time point changes its
sign as a result of which the total energy changes
also its sign.
It may be emphasized that it was not only
unnecessary to assume that the wave equation is of
the first order, \footnote{But we exclude operation like $(k^2 + \kappa^2)^{1/2}$,
which operate at finite distances in the coordinate space.}
but also that the question is left
open whether the theory is also invariant with
respect to space reflections ${\bf x}' = - {\bf x}, x_0' = x_0)$.
This scheme covers therefore also Dirac's two
component wave equations (with rest mass zero).
These considerations do not prove that for
integral spins there always exists a definite energy
density and for half-integral spins a definite charge
density. In fact, it has been shown by
Fierz \footnote{M. Fierz, Helv. Phys. Acta 12, 3 (1939).}
that this is not the case for spin $> 1$ for the densities. There
exists, however (in the $c$ number theory), a definite
total charge for half-integral spins and a definite
total energy for the integral spins. The spin value $\frac{\displaystyle
1}{\displaystyle 2}$ is discriminated through the possibility of a definite
charge density, and the spin values 0 and 1 are discriminated through the
possibility of defining a definite energy density. Nevertheless, the present
theory permits arbitrary values of the spin quantum
numbers of elementary particles as well as arbitrary
values of the rest mass, the electric charge, and the
magnetic moments of the particles.
\section*{\S~4. QUANTIZATION OF THE FIELDS IN THE ABSENCE OF
INTERACTIONS. CONNECTION BETWEEN SPIN AND STATISTICS}
The impossibility of defining in a physically
satisfactory way the particle density in the case of
integral spin and the energy density in the case of
half-integral spins in the $c$--number theory
is an indication that a satisfactory interpretation of
the theory within the limits of the one-body problem
is not possible. \footnote{The author therefore considers as not conclusive the
original argument of Dirac. according to which the field equation must be of
the first order.}
In fact, all relativistically invariant
theories lead to particles, which in external fields
can be emitted and absorbed in pairs of opposite
charge for electrical particles and singly for neutral
particles. The fields must, therefore, undergo a
second quantization. For this we do not wish to
apply here the canonical formalism, in which time is
unnecessarily sharply distinguished from space, and
which is only suitable if there are no supplementary
conditions between the canonical variables.\footnote{On account of the existence of
such conditions the canonical formalism is not applicable for spin $> 1$ and
therefore the discussion about the connection between spin and statistics by
J. S. de Wet, Phys. Rev. {\bf 57}, 646 (1940), which is based on that formalism is not
general enough.}
Instead, we shall apply here a generalization of this method
which was applied for the first time by Jordan and
Pauli to the electromagnetic field.\footnote{The consistent development of this
method leads to the ``many-time formalism'' of Dirac, which has been given
by P. A. M. Dirac, Quantum Mechanics (Oxford, second edition, 1935).}
This method is especially convenient in the absence of interaction,
where all fields $U^{(r)}$ satisfy the wave equation of the
second order
$$
\square U^{(r)} - \kappa^2 U^{(r)} = 0,
$$
where
$$
\square \equiv \sum \limits^4_{k = 1} \frac{\partial^2}{\partial x \kappa^2}
= \vartriangle - \frac{\partial^2}{\partial x_0^2}
$$
and $\kappa$ is the rest mass of the particles in units $hbar /c$.
An important tool for the second quantization is the
invariant $D$ function, which satisfies the wave
equation (9) and is given in a periodicity volume $V$
of the eigenfunctions by
\begin{equation}
D({\bf x}, x_0) = \frac{1}{V} \sum \mbox{exp} [i({\bf kx})] \frac{\sin
k_0x_0}{k_0}
\end{equation}
or in the limit $V \rightarrow \infty$
\begin{equation}
D({\bf x}, x_0) = \frac{1}{(2 \pi)^3} \int d^3 k \mbox{exp} [i({\bf kx})]
\frac{\sin k_0 x_0}{k_0}.
\end{equation}
By to we understand the positive root
\begin{equation}
k_0 = + (k^2 + \kappa^2)^{1/2}
\end{equation}
The $D$ function is uniquely determined by the
conditions:
$$
\square D - \kappa^2 D = 0; \quad D({\bf x}, 0) = 0;
$$
\begin{equation}
\left( \frac{\partial D}{\partial x_0}\right)_{x_0 = 0} = \delta({\bf x}).
\end{equation}
For $\kappa = 0$ we have simply
\begin{equation}
D({\bf x}, x_0) = \{ \delta (r - x_0) - \delta (r - x_0) \} /4 \pi
r.
\end{equation}
This expression also determines the singularity of
$D({\bf x}, x_0)$ on the light cone for $\kappa \ne 0$. But in the latter
case $D$ is no longer different from zero in the inner
part of the cone. One finds for this region \footnote{See P. A. M. Dirac,
Proc. Camb. Phil. Soc. {\bf 30}, 150 (1934).}
$$
D({\bf x}, x_0) = - \frac{1}{4 \pi r} \frac{\partial}{\partial r} F(r,
x_0)
$$
with
\begin{equation}
F(r, x_0) = \left\{
\begin{array}{ll}
J_0 [\kappa(x_0^2 - r^2)^{1/2} ]\quad&\mbox{for} \quad x_0 > r\\
0& \mbox{for} \quad r > x_0 > - r\\
-J_0 [\kappa (x_0^2 - r^2)^{1/2}]\quad&\mbox{for} \quad - r > x_0.
\end{array} \right\}
\end{equation}
The jump from $+$ to $-$ of the function $F$ on the
light cone corresponds to the $\delta$ singularity of $D$ on
this cone. For the following it will be of decisive
importance that $D$ vanish in the exterior of the cone
(i.e., for $r > x_0 > - r$).
The form of the factor $d^3 k / k_0$, is determined by the
fact that $d^3 k/k_0$ is invariant on the hyper-boloid ($k$)
of the four-dimensional momentum space (${\bf \kappa}, k_0$). It
is for this reason that, apart from $D$, there exists just
one more function which is invariant and which
satisfies the wave equation (9), namely,
\begin{equation}
D_1 ({\bf x}, x_0) = \frac{1}{(2 \pi)^3} \int d^3 k ~\mbox{exp} [i({\bf
kx})] \frac{\cos k_0 x_0}{k_0}.
\end{equation}
For $\kappa = 0$ one finds
\begin{equation}
D_1 ({\bf x}, x_0) = \frac{1}{2 \pi^2} \frac{1}{r^2 - x_0^2}.
\end{equation}
In general it follows
$$
D_1 ({\bf x}, x_0) = \frac{1}{4 \pi} \frac{1}{r} \frac{\partial}{\partial
r} F_1 (r, x_0)
$$
\begin{equation}
F_1 (r, x_0) = \left\{
\begin{array}{ll}
N_0[\kappa (x_0^2 - r^2)^{1/2}] \quad&\mbox{for} \quad x_0 > r\\
- i H_0^{(1)}[i \kappa(r^2 - x_0^2)^{1/2}] \quad& \mbox{for} \quad r >
x_0 > - r\\
N_0[\kappa(x_0^2 - r^2)^{1/2}] \quad& \mbox{for} \quad - r > x_0.
\end{array} \right\}
\end{equation}
Here $N_0$ stands for Neumann's function and $H_0^{(1)}$
for the first Hankel cylinder function. The strongest
singularity of $D$, on the surface of the light cone is in
general determined by (17).
We shall, however, expressively postulate in the
following {\it that all physical quantities at finite
distances exterior to the light cone (for $x_0' - x_0''| < |{\bf x' -
x''}|)$ are commutable}. \footnote{For the canonical quantization formalism this
postulate is satisfied
implicitly. But this postulate is much more general than the canonical
formalism.} It follows from this that
the bracket expressions of all quantities which satisfy
the force-free wave equation (9) can be expressed by
the function $D$ and (a finite number) of derivatives of
it without using the function $D_1$. This is also true for
brackets with the $+$ sign, since otherwise it would
follow that gauge invariant quantities, which are
constructed bilinearly from the $U^{(r)}$, as for example
the charge density, are noncommutable in two points
with a space-like distance. \footnote{See W. Pauli, Ann. de 1'Inst. H. Poincare {\bf
6} ,137 (1936), esp. \S~3.}
The justification for our postulate lies in the fact
that measurements at two space points with a space-like distance can never
disturb each other, since no
signals can be transmitted with velocities greater than
that of light. Theories which would make use of the
$D_1$ function in their quantization would be very much
different from the known theories in their
consequences.
At once we are able to draw further conclusions
about the number of derivatives of $D$ function which
can occur in the bracket expressions, if we take into
account the invariance of the theories under the
transformations of the restricted Lorentz group and if
we use the results of the preceding section on the
class division of the tensors. We assume the
quantities $U^{(r)}$ to be ordered in such a way that each
field component is composed only of quantities of
the same class.
We consider especially the bracket expression of a
field component $U^{(r)}$ with its own complex conjugate
$$
[U^{(r)} ({\bf x'}, x_0'), U^{\ast 9r)} ({\bf x''}, x_0'')].
$$
We distinguish now the two cases of half-integral
and integral spin. In the former case this expression
transforms according to (8) under Lorentz
transformations as a tensor of odd rank. In the
second case, however, it transforms as a tensor of
even rank. Hence we have for half-integral spin
$$
[U^{(r)} ({\bf x'}, x_0'), U^{\ast (r)} ({\bf x''}, x_0'')]
$$
~~~~~~~~~~~~~~~~~~~$=$ odd number of derivatives of the function
$$
D({\bf x' - x''}, x_0' - x_0'') \eqno(19a)
$$
and similarly for integral spin
$$
[U^{(r)} ({\bf x'}, x_0'), U^{\ast (r)} ({\bf x''}, x_0'')]
$$
~~~~~~~~~~~~~~~~~~~$=$ even number of derivatives of the function
$$
D({\bf x' - x''}, x_0' - x_0''). \eqno(19b)
$$
This must be understood in such a way that on the
right-hand side there may occur a complicated sum
of expressions of the type indicated. We consider
now the following expression, which is symmetrical
in the two points
\begin{equation}
X \equiv [U^{(r)} ({\bf x'}, x_0'), U^{\ast (r)} ({\bf x''}, x_0'')]
+ [U^{(r)} ({\bf x''}, x_0''), U^{\ast (r)} ({\bf x'}, x_0')].
\end{equation}
Since the $D$ function is even in the space coordinates
odd in the time coordinate, which can be seen at
once from Eqs. (11) or (15), it follows from the
symmetry of $X$ that $X =$ even number of space-like
times odd numbers of time-like derivatives of $D({\bf x' - x''}, x_0' -
x_0'')$. This is fully consistent with the
postulate (19a) for half-integral spin, but in
contradiction with (19b) for integral spin unless $X$
vanishes. We have therefore the result for integral
spin
\begin{equation}
[U^{(r)} ({\bf x'}, x_0'), U^{\ast (r)} ({\bf x''}, x_0'')]
+[U^{(r)} ({\bf x''}, x_0''), U^{\ast (r)} ({\bf x'}, x_0')] = 0.
\end{equation}
So far we have not distinguished between the two
cases of Bose statistics and the exclusion principle.
In the former case, one has the ordinary bracket with
the --- sign, in the latter case,
according to Jordan and Wigner, the bracket
$$
[A, B]_+ = AB + BA
$$
with the $+$ sign. {\it By inserting the brackets with the +
sign into (20) we have an algebraic contradiction,}
since the left-hand side is essentially positive for
$x' = x''$ and cannot vanish unless both $U^{(r)}$ and $U^{\ast (r)}$
vanish. \footnote{This contradiction may be seen also by resolving $U^{(r)}$ into
eigenvibrations according to
$$
U^{\ast (r)} ({\bf x}, x_0) = V^{-1/2} \sum_k \{ U_+^{\ast} (k)~ \mbox{exp}
[i \{ - ({\bf kx}) + k_0 x_0 \} ]
+ U_- (k)~ \mbox{exp} [i \{({\bf kx}) - k_0 x_0\} ] \}
$$
$$
U^{(r)} ({\bf x}, x_0) = V^{- 1/2} \sum_k \{ U_+ (k) ~\mbox{exp} [i \{({\bf
kx}) - k_0 x_0 \}]
+ U_-^{\ast} (k) \mbox{exp} [i \{ - ({\bf kx}) + k_0 x_0 \} ] \}.
$$
The equation (21) leads then, among others, to the relation
$$
[U_+^{\ast} (k), U_+(k)] + [U_- (k), U_-^{\ast} (k)] = 0,
$$
a relation, which is not possible for brackets with the $+$ sign unless
$U_{\pm} (k)$ and $U_{\pm}^{\ast}(k)$ vanish.}
Hence we come to the result: {\it For integral spin the
quantization according to the exclusion principle is
not possible. For this result it is essential, that the
use of the $D_1$ function in place of the $D$ function be,
for general reasons, discarded.}
On the other hand, it is formally possible to
quantize the theory for half-integral spins according
to Einstein-Bose-statistics, {\it but according to the
general result of the preceding section the energy of
the system would not be positive.} Since for physical
reasons it is necessary to postulate this, we must
apply the exclusion principle in connection with
Dirac's hole theory.
For the positive proof that a theory with a positive
total energy is possible by quantization according to
Bose-statistics (exclusion principle) for integral
(half-integral) spins, we must refer to the already
mentioned paper by Fierz. In another paper by Fierz
and Pauli \footnote{M. Fierz and W. Pauli, Proc. Roy. Soc. {\bf A173,} 211 (1939).}
the case of an external electromagnetic
field and also the connection between the special
case of spin 2 and the gravitational theory of
Einstein has been discussed.
In conclusion we wish to state, that according to
our opinion the connection between spin and
statistics is one of the most important applications of
the special relativity theory.
\end{document}
%\encode August 7, 2002 BY NIS;