Connection Between Spin and Statistics.pdf

W. Pauli, Phys. Rev., Vol. 58, 716

1940

The Connection Between Spin and Statistics 1

W. Pauli

Princeton, New Jersey

(Received August 19, 1940)

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Reprinted in “Quantum Electrodynamics”, edited by Julian Schwinger

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Abstract

In the following paper we conclude for the relativistically invari-

ant wave equation for free particles: From postulate (I), according

to which the energy must be positive, the necessity of Fermi-Dirac

statistics for particles with arbitrary half-integral spin; from postulate

(II), according to which observables on dierent space-time points with

a space-like distance are commutable, the necessity of 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, +,− with 2 = 1.

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).

§ 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 which we shall simply

denote by ~. This convention means that all quantities are brought to the

dimension of the power of a length by multiplication with powers of ~ and

c. The reciprocal length corresponding to the rest mass m is denoted by

= mc/~.

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 is the

complex conjugate of the charge density S 0 ). in general U i... signiﬁes: the

complex-conjugate of U i... multiplied with (−1) n , where n is the number of

occurrences of the digit 4 among the i,k,... (e.g. S 4 = iS 0 , S 4 = iS 0 ).

Dirac’s spinors u , with = 1,..., 4 have always a Greek index running

from 1 to 4, and u means the complex-conjugate of u , in the ordinary

sense.

Wave functions, insofar as they are ordinary vectors or tensors, are de-

noted in general with capital letters, U i ,U i ... 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

' i , f i = −f i and g i = g i respectively.

The energy-momentum tensor T i , is so deﬁned, that the energy-density

W and the momentum density G are given in natural units by W = −T 44

and G = −iT 4 with k = 1, 2, 3.

§ 2. IRREDUCIBLE TENSORS. DEFINITION

OF SPINS

We shall use only a few general properties of those quantities which trans-

form according to irreducible representations of the Lorentz group. 2

The

See B. L. v. d. Waerden, Die gruppentheoretische Methode in der Quantentheorie

(Berlin, 1932).

proper Lorentz group is that continuous linear group the transformations of

which leave the form

4 X

x k = x 2 −x 0

k=1

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 = 2j + 1, q = 2k + 1, with integral

or half-integral j and k.) 3 The quantity U(j,k) characterized by (j,k) has

p·q = (2j + 1)(2k+ 1) independent components. Hence to (0, 0) corresponds

the scalar, to ( 2 , 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 ( 2 , 0) and (0, 2 ) each of

which consists of two components. If U(j,k) transforms according to the

representation

(2j+1)(2k+1)

U 0 r =

rs U s ,

s=1

then U (k,j) transforms according to the complex-conjugate representation

. Thus for k = j, = . 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 and would have to be added. In view of

§ 1 we represent generally with U the quantity the transformation of which

is equivalent to if the transformation of U is equivalent to .

The most important operation is the reduction of the product of two

quantities

U 1 (j 1 ,k 1 ) ·U 2 (j 2 ,k 2 )

which, according to the well-known rule of the composition of angular mo-

menta, 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,...,|j 1 −j 2 |

k = k 1 + k 2 , k 1 + k 2 − 1,...,|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)

In the spinor calculus this is a spinor with 2j undotted and 2k dotted indices.

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,...,|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 exp i(kx). In order to deﬁne

the spin in an appropriate fashion, 4 we want to consider ﬁrst the case in

which the rest mass m of all the particles is dierent 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 dierent

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,

see M. Fierz, Helv. Phys. acta 12, 3 (1939); also L. de Broglie, Comptes rendus 208,

1697 (1939); 209, 265 (1939).

By “gauge-transformation of the ﬁrst kind” we understand a transformation U ! Ue i

U ! U e −i with an arbitrary space and time function . By “gauge-transformation of

for an arbitrary system of reference, that r or P 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. 5 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,.... 6 This is only the

case for m = 0. For photons, s = 1, j = 0 is not possible for one single

photon.

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 par-

ticles 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.

§ 3. PROOF OF THE INDEFINITE CHARAC-

TER 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

the second kind” we understand a transformation of the type

' k ! ' k − 1

i @

as for those of the electromagnetic potentials.

The general proof for this has been given by M. Fierz, Helv. Phys. Acta 13, 45 (1940).

See for instance W. Pauli in the article “Wellen-mechanik” in the Handbuch der

Physik, Vol. 24/2, p. 260.

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