§ 5. THE ANGULAR MOMENTUM INTEGRALS FOR MOTION IN A CENTRAL FIELD
We shall consider in greater detail the motion of an electron in a central field of force. We put A = 0 and e’A0 = V (r), an arbitrary function of the radius r, so that the Hamiltonian in (14) becomesF ≡ p0 + V + ρ1(σ, p) + ρ3mc.
We shall determine the periodic solutions of the wave equation F ψ = 0, which means that p0 is to be counted as a parameter instead of an operator; it is, in fact, just 1/c times the energy level.
We shall first find the angular momentum integrals of the motion. The orbital angular momentum m is defined bym = x × p,
and satisfies the following “Vertauschungs” relations
(17)
together with similar relations obtained by permuting the suffixes. Also m commutes with r, and with pr, the momentum canonically conjugate to r.
We have
and so
(18)
Thus m is not a constant of the motion. We have further
with the help of (8), and soσF - Fσ = -2iρ1 σ × p.
Hence
Thus m + h σ (= M say) is a constant of the motion. We can interpret this result by saying that the electron has a spin angular momentum of h σ, which, added to the orbital angular momentum m, gives the total angular momentum M, which is a constant of the motion.
The Vertauschungs relations (17) all hold when M’s are written for the m’s. In particularM × M = ihM and M2M3 = M3M2.
M3 will be an action variable of the system. Since the characteristic values of m3 must be integral multiples of h in order that the wave function may be single-valued, the characteristic values of M3 must be half odd integral multiples of h. If we put
(19)
j will be another quantum number, and the characteristic values of M3 will extend from (j − ) h to (−j + )h.gn Thus j takes integral values.
One easily verifies from (18) that m2 does not commute with F, and is thus not a constant of the motion. This makes a difference between the present theory and the previous spinning electron theory, in which m2 is constant, and defines the azimuthal quantum number k by a relation similar to (19). We shall find that our j plays the same part as the k of the previous theory.
§ 6. THE ENERGY LEVELS FOR MOTION IN A CENTRAL FIELD
We shall now obtain the wave equation as a differential equation in r, with the variables that specify the orientation of the whole system removed. We can do this by the use only of elementary non-commutative algebra in the following way.
In formula (16) take B = C = m. This gives
(20)
Hence
Up to the present we have defined j only through j2, so that we could now, if we liked, take jh equal to (σ, m) + h. This would not be convenient since we want j to be a constant of the motion while (σ, m) + h is not, although its square is. We have, in fact, by another application of (16),(σ, m) (σ, p) = i (σ, m × p)
since (m, P) = 0, and similarly(σ, p) (σ, m) = i (σ, p × m),
so that
or{(σ, m) + h} (σ, p) + (σ, p) {(σ, m)} = 0.
Thus (σ , m) + h anticommutes with one of the terms in F, namely, ρ1 (σ, p), and commutes with the other three. Hence ρ3 {(σ, m) + h} commutes with all four, and is therefore a constant of the motion. But the square of ρ3 {(σ, m) + h} must also equal j2h2. We therefore take
(21)
We have, by a further application of (16)(σ, x) (σ, p) = (x, p) + i(σ, m).
Now a permissible definition of pr is(x, p) = rpr + ih,
and from (21)(σ, m) = ρ3jh - h.
Hence (σ, m) = ρ3jh − h.
(22)
Introduce the quantity ε defined by
(23)
Since r commutes with ρ1 and with (σ, x), it must commute with ε. We thus haver2ε2 = [ρ1 (σ, x)]2 = (σ, x)2 = x2 = r2
orε2 = 1.
Since there is symmetry between x and p so far as angular momentum is concerned, ρ1 (σ, x), like ρ1 (σ, p), must commute with M and j. Hence ε commutes with M and j. Further, ε must commute with pr, since we have(σ, x) (x, p) − (x, p) (σ, x) = ih (σ, x),
which givesrε (rpr + ih) - (rpr + ih) rε = ihrε,
which reduces toεpr − prε = 0.
From (22) and (23) we now haverερ1 (σ, p) = rpr + iρ3jh
or
Thus
(24)
Equation (23) shows that ε anticommutes with ρ3. We can therefore by a canonical transformation (involving perhaps the x’s and p’s as well as the σ’s and ρ’s) bring ε into the form of the ρ2 of § 2 without changing ρ3, and without changing any of the other variables occurring on the right-hand side of (24), since these other variables all commute with ε. iερ3 will now be of the form iρ2ρ3 = −ρ1, so that the wave equation takes the formFψ ≡ [p0 + V + ρ2 pr − ρ1jh/r + ρ3mc] ψ = 0.
If we write this equation out in full, calling the components of ψ referring to the first and third rows (or columns) of the matrices ψα and ψβ respectively, we get
The second and fourth components give just a repetition of these two equations. We shall now eliminate ψα. If we write hB for p0 + V + mc, the first equation becomes
which gives on differentiating
This reduces to
(25)
The values of the parameter p0 for which this equation has a solution finite at r = 0 and r = ∞ are 1/c times the energy levels of the system. To compare this equation with those of previous theories, we put ψβ = rχ, so that
(26)
If one neglects the last term, which is small on account of B being large, this equation becomes the same as the ordinary Schroedinger equation for the system, with relativity correction included. Since j has, from its definition, both positive and negative integral characteristic values, our equation will give twice as many energy levels when the last term is not neglected.
We shall now compare the last term of (26), which is of the same order of magnitude as the relativity correction, with the spin correction given by Darwin and Pauli. To do this we must eliminate the ∂χ/∂r term by a further transformation of the wave function. We put
which gives
(27)
The correction is now, to the first order of accuracy
where Bh = 2mc (provided p0 is positive). For the hydrogen atom we must put V = e2/cr. The first order correction now becomes
(28)
If we write −j for j + 1 in (27), we do not alter the terms representing the unperturbed system, so
(28’)
will give a second possible correction for the same unperturbed term.
In the theory of Pauli and Darwin, the corresponding correcting term is
when the Thomas factor is included. We must remember that in the Pauli-Darwin theory, the resultant orbital angular momentum k plays the part of our j. We must define k bym2 = k(k + 1)h2
instead of by the exact analogue of (19), in order that it may have integral characteristic values, like j. We have from (20)(σ , m)2 = k (k + 1)h 2 − h (σ , m)
or
hence(σ, m) = kh or − (k + 1)h.
The correction thus becomes
which agrees with (28) and (28’). The present theory will thus, in the first approximation, lead to the same energy levels as those obtained by Darwin, which are in agreement with experiment.
THE CONNECTION BETWEEN SPIN AND STATISTICS
BY
WOLFGANG PAULI
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 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 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 clas
ses which have a commutable multiplication like +1, −1, +∈, −∈ with ∈2 = 1.
§ 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 2π which we shall simply denote by ħ . This convention means that all quantities are brought to the dimension of the power of a length by multiplication with powers of ħ and c. The reciprocal length corresponding to the rest mass m is denoted by κ = mc /ħ.
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 Si the quantity So*is the complex conjugate of the charge density S0). in general U*iκ... signifies: the complex-conjugate of Uiκ... multiplied with (−1)n , where n is the number of occurrences of the digit 4 among the i, k, . . . (e.g. S4 = iS0, S = iS).
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 take 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).
Reprinted with permission from Physical Review, Volume 58, p. 716, (1940).
© 1940 by the American Physical Society
Dirac’s spinors uρ , with ρ = 1, . . . , 4 have always a Greek index running from 1 to 4, and means the complex-conjugate of uρ , in the ordinary sense.
Wave functions, insofar as they are ordinary vectors or tensors, are denoted in general with capital letters,Ui,Uiκ ...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 ϕi , fiκ = −fκi and giκ = gκi respectively.
The energy-momentum tensor Tiκ, is so defined, that the energy-density W and the momentum density Gκ are given in natural units by W = −T44 and Gκ = −iTκ 4 with k = 1, 2, 3.
§ 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.go The proper Lorentz group is that continuous linear group the transformations of which leave the form
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.)gp 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 (, ) 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 ( , 0) and (0, ) each of which consists of two components. If U(j, k) transforms according to the representation
then U * (k, j ) transforms according to the complex-conjugate representation Λ*. Thus for k = j, Λ* = Λ. 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 Λ and Λ* would have to be added. In view of §1 we represent generally with U * the quantity the transformation of which is equivalent to Λ* if the transformation of U is equivalent to Λ.
The most important operation is the reduction of the product of two quantitiesU1 (j1,k1) · U2 (j2, k2)
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
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, withl = 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,gq 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 ki , 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 ∑ r eigenfunctions always belong to a given arbitrary ki. 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 ki 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.gr 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 ki. 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, . . ..gs This is only the case for m = 0. For photons, s = 1, j = 0 is not possible for one single photon.gt 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.
§ 3. PROOF OF THE INDEFINITE CHARACTER OF THE CHARGE IN CASE OF INTEGRAL AND OF THE ENERGY IN CASE OF HALF-INT
EGRAL 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.
The Dreams That Stuff is Made of Page 38