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* 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 nector ki 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 ki for −i∂ /∂xl. Solely on account of the invariance against the proper Lorentz group it must be of the typical form
(1)
This typical form shall mean that there may be as many different terms of the same type present, as there are quantities U * and U −. Furthermore, among the U * may occur the U + as well as the (U +)*, whereas other U may satisfy reality conditions U = U*. 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
(2)
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
(3)
where again a possible even number of k factors is omitted and no distinction between U and U* is made. Under the substitution (2) they remain unchanged, T → 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
(4)
and hence change the sign under the substitution (2), S → − S . Particularly is this the case for the current vector si . To the transformation ki → −ki , belongs for arbitrary wave packets the transformation xi → −xi , 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 sk , 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 “+∈ class” with j integral k half-integral, (4) the “ −∈ class” with j half-integral k integral.
The multiplication of the classes (1), ... , (4), follows from the rule ∈2 = 1 and the commutability of the multiplication. This law remains unchanged if ∈ is replaced by −∈.
We can summarize the multiplication law between the different classes in the following multiplication table:
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
We shall therefore cite the complex-conjugate quantities explicitly. (One could even choose the U+∈ suitably so that all quantities of the −∈ class are of the form (U+∈ )*).
Instead of (1) we obtain now as typical form
(5)
since a factor k or −i∂ /∂ x always changes the expression from one of the classes +∈ or −∈ into the other. As above, an even number of k factors have been omitted.
Now we consider instead of (2) the substitution
(6)
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+∈ , (U−∈ )* transform in the same way. Furthermore, it does not interfere with possible reality conditions of the type U+∈ = (U−∈ )*. or U−∈ = (U+∈ )*. 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
(7)
Furthermore, the tensors of odd rank (vectors, etc.) must be of the form
(8)
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:
(9)
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,gu but also that the question is left open whether the theory is also invariant with respect to space reflections x’ = − x, x’0 = x0). 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 Fierzgv 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 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.
§ 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.gw 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.gx 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.gy This method is especially convenient in the absence of interaction, where all fields U(r ) satisfy the wave equation of the second order
where
and κ 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 v
olume V of the eigenfunctions by
(10)
or in the limitV→∞
(11)
By to we understand the positive root
(12)
The D function is uniquely determined by the conditions:
(13)
For κ = 0 we have simply
(14)
This expression also determines the singularity of D(x, x0) on the light cone for κ ≠ 0. But in the latter case D is no longer different from zero in the inner part of the cone. One finds for this regiongz
with
(15)
The jump from + to − of the function F on the light cone corresponds to the δ 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 > x0 > −r).
The form of the factor d3k/k0, is determined by the fact that d3k/ k0 is invariant on the hyper-boloid (k) of the four-dimensional momentum space (κ, k0). 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,
(16)
For κ = 0 one finds
(17)
In general it follows
(18)
Here N0 stands for Neumann’s function and 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 that all physical quantities at finite distances exterior to the light cone (for | < |x’ − x”|) are commutable.ha 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 D1. 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.hb
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 D1 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
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
= odd number of derivatives of the function
(19a)
and similarly for integral spin
= even number of derivatives of the function
(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
(19)
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(x’ − x”,). 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
(20)
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
with the + sign. 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*(r) vanish.hc
Hence we come to the result: For integral spin the quantization according to the exclusion principle is not possible. For this result it is essential, that the use of the D1 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, 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 Paulihd 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.
EXCLUSION PRINCIPLE AND QUANTUM MECHANICS
BY
WOLFGANG PAULI
Nobel Lecture, December 13, 1946
The history of the discovery of the “exclusion principle”, for which I have received the honor of the Nobel Prize award in the year 1945, goes back to my students days in Munich. While, in school in Vienna, I had already obtained some knowledge of classical physics and the then new Einstein relativity theory, it was at the University of Munich that I was introduced by Sommerfeld to the structure of the atom - somewhat strange from the point of view of classical physics. I was not spared the shock which every physicist, accustomed to the classical way of thinking, experienced when he came to know of Bohr’s “basic postulate of quantum theory” for the first time. At that time there were two approaches to the difficult problems connected with the quantum of action. One was an effort to bring abstract order to the new ideas by looking for a key to translate classical mechanics and electrodynamics into quantum language which would form a logical generalization of these. This was the direction which was taken by Bohr’s “correspondence principle”. Sommerfeld, however, preferred, in view of the difficulties which blocked the use of the concepts of kinematical models, a direct interpretation, as independent of models as possible, of the laws of spectra in terms of integral numbers, following, as Kepler once did in his investigation of the planetary system, an inner feeling for harmony. Both methods, which did not appear to me irreconcilable, influenced me. The series of whole numbers 2, 8, 18, 32 . . . giving the lengths of the periods in the natural system of chemical elements, was zealously discussed in Munich, including the remark of the Swedish physicist, Rydberg, that these numbers are of the simple form 2 n2, if n takes on all integer values. Sommerfeld tried especially to connect the number 8 and the number of corners of a cube.
A new phase of my scientific life began when I met Niels Bohr personally for the first time. This was in 1922, when he gave a series of guest lectures at Göttingen, in which he reported on his theoretical investigations on the Periodic System of Elements. I shall recall only briefly that the essential progress made by Bohr’s considerations at that time was in explaining, by means of the spherically symmetric atomic model, the formation of the intermediate shells of the atom and the general properties of the rare earths. The question, as to why all electro
ns for an atom in its ground state were not bound in the innermost shell, had already been emphasized by Bohr as a fundamental problem in his earlier works. In his Göttingen lectures he treated particularly the closing of this innermost K-shell in the helium atom and its essential connection with the two non-combining spectra of helium, the ortho- and para-helium spectra. However, no convincing explanation for this phenomenon could be given on the basis of classical mechanics. It made a strong impression on me that Bohr at that time and in later discussions was looking for a general explanation which should hold for the closing of every electron shell and in which the number 2 was considered to be as essential as 8 in contrast to Sommerfeld’s approach.
The Dreams That Stuff is Made of Page 39