Symmetry Properties in the Theory of Elementary Particles and Nuclear Processes

I. S. Shapiro

1. Introduction

In recent years the number of so-called elementary particles known to physicists has increased considerably. Many of these particles are unstable and decay into other particles (the neutron, \(\mu\)-, \(\pi\)-, \(\tau\)-mesons, \(V\)-particles, etc.). The number and types of particles appearing as decay products are determined, first of all, by the well-known conservation laws of charge, energy, momentum, and angular momentum. At the same time, however, modern theory indicates certain additional selection rules which in a number of cases make it possible to understand why a process permitted by the above-mentioned basic conservation laws does not in fact occur. The same is true for various kinds of nuclear transformations.

The existence of various selection rules is due to the invariance of the Lagrangian and of the equations of motion (wave equations) with respect to various groups of transformations and, first and foremost, with respect to the group of transformations of reference frames. It is very important that specifying a group of transformations of reference frames also determines the transformation laws of physical quantities when passing from one reference frame to another. This happens because to each transformation \(s\) of the reference frame there corresponds a certain transformation \(S\) of the given physical quantity and, moreover, to the product \(s_2 \cdot s_1\) of two transformations \(s_1\) and \(s_2\) there corresponds the product \(S_2 \cdot S_1\) of the transformations \(S_1\) and \(S_2\). If the transformation \(S\) is linear, and the correspondence \(s \to S\) is single-valued (mutual single-valuedness is not necessary), then one says that the group of transformations \(S\) is a representation of the group of transformations \(s\) of reference frames. The single-valuedness of the correspondence \(s \to S\) is evidently necessary in the case when the physical quantity under consideration—

value is among those directly measured in experiment. But if the given physical quantity is auxiliary (for example, the wave function of a particle, specified up to an arbitrary phase factor), then, in principle, under certain additional conditions, many-valuedness is also admissible. A very important case of the use in physics of a many-valued correspondence of the type indicated above is the two-valued, so-called spinor representation, according to which the wave functions of particles with half-integer spin transform.

The present article is devoted to a consideration of the symmetry properties of wave equations and of some selection rules based on them.

Very important symmetry properties follow from the invariance of the Lagrangian and, consequently, of the equations of motion with respect to reflections of the spatial and time coordinate axes. We shall begin with a consideration precisely of these transformations, having as our immediate aim to indicate the possible types of particles with a given spin.

2. TYPES OF PARTICLES. INTRINSIC PARITY

Let us first consider transformations of rotations and reflections with respect to the origin of the three spatial coordinate axes. The spin of a particle is determined by the transformation properties of its wave function with respect to rotations of the coordinate system. More precisely, in specifying the spin of a particle, we fix an irreducible) representation of the group of three-dimensional rotations according to which its wave function transforms*). We shall restrict ourselves here to the study of particles with spins 0, 1, and \(1/2\), since particles with higher spins have not yet been observed.

If we know the behavior of the wave function under rotations of the coordinate system, this still does not completely determine the transformation law under reflection of the axes. The reason is that transformations of reflection of the coordinate axes cannot be reduced to rotations. Therefore, in prescribing the law of transformation of the wave function of a particle with a definite spin under reflection of the coordinate axes, there is a certain freedom. We say “certain,”

*) Let the quantity \(\Psi\) transformed according to the given representation (in our case, the wave function) have components \(\Psi_1,\ldots,\Psi_n\). Then the representation is called irreducible if one cannot find \(m<n\) linear combinations of the components \(\Psi_1,\ldots,\Psi_n\) which transform among themselves under all transformations of the given group.

**) In the present article only ordinary, finite-dimensional representations of the Lorentz group are considered. For infinite-dimensional representations see the work of Gelfand and Naimark \(^{1}\) and of Gelfand and Yaglom \(^{1a}\). For the theory of representations of the group of three-dimensional rotations we refer the reader to the article of Gelfand and Shapiro \(^{2}\).

so that reflection and rotation transformations are nevertheless connected with one another. Namely, any rotation can be obtained as the product of an even number of transformations of mirror reflection of axes with respect to different planes passing through the origin. Accordingly, the transformation undergone by the wave function under rotations must also be representable in the form of the product of an even number of transformations associated with reflections of axes. This circumstance also restricts the freedom in choosing the law of transformation of the wave function under reflections, if its transformation properties with respect to the rotation group are given.

Let us denote the wave function of a particle by \(\Psi(x)\). In the general case \(\Psi(x)\) is a multicomponent quantity and is transformed under transformations of the coordinate system

\[ x' = sx, \tag{1} \]

where \(s\) denotes a matrix, and \(x\) is the four-dimensional radius-vector of a point*), according to the law

\[ \Psi'(x) = \Psi'(s^{-1}x') = S\Psi(x). \tag{2} \]

In equation (2) \(s^{-1}\), as usual, denotes the transformation inverse to \(s\), and \(S\) is some matrix, since we shall consider, with few exceptions, only linear transformations**).

If the spin of the particle is equal to 0, then its wave function consists of one component, and for all rotations \(S = 1\). The matrix \(S\) for reflections of spatial axes with respect to any plane passing through the origin must also consist of a single number \(\zeta\), which may also differ from 1. Consequently, under reflections of axes the wave function of a particle with zero spin is transformed according to the law

\[ \Psi'(x) = \zeta \Psi(x). \tag{3} \]

It can easily be found that the quantity \(\zeta\) must be the same for reflections with respect to any plane passing through the origin. Indeed, let us denote the set of parameters deter-

*) We take all components of the four-dimensional radius-vector \(x_1, x_2, x_3\) and \(x_0 = ct\) to be real. The square of the length of the vector is written in the form

\[ x^2 = x_1^2 + x_2^2 + x_3^2 - x_0^2 . \]

) Let us emphasize that, according to (2), the transformation of the wave function \(\Psi'(x)\) under consideration does not consist in replacing the argument \(x\) by \(x'\). The transformation (2) does not answer the question of how the function \(\Psi'(x') = \Psi(sx)\) is expressed in terms of \(\Psi(x)\). Therefore the form of the function \(\Psi'(x)\) may be completely arbitrary. In this paragraph we are interested in the behavior of \(\Psi(x)\) under transformations of the reference system**, but not under displacement from one point of space to another.

dividing the position of the reflecting plane by the letter \(a\). Then, since the group of all transformations \(S\) is a representation of the group of rotations and reflections of the coordinate axes, we must have

\[ \zeta(a'')\,\zeta(a')=1, \tag{4} \]

since the product of two reflections is a rotation.
From (4), putting \(a'=a''\), we obtain:

\[ \zeta(a')^2=1, \tag{5} \]

and, comparing (4) and (5), we find:

\[ \zeta(a')=\zeta(a'')=\zeta=\pm 1. \tag{6} \]

The number \(\zeta\) determines the intrinsic spatial parity of the particle. Thus one may indicate two types of particles with spin 0, but with different intrinsic spatial parity: scalar particles, corresponding to \(\zeta=+1\), and pseudoscalar particles, characterized by \(\zeta=-1\). The wave functions of these two types of particles are transformed differently under reflection of the spatial coordinate axes, but behave in absolutely the same way under rotations. An example of pseudoscalar particles may be, as follows from experimental data, the \(\pi^\pm\)- and \(\pi^0\)-mesons; as for scalar particles, it is possible that these are the \(V^0_2\)-particles, decaying according to the scheme\(^3\)

\[ V^0_2 \to \pi^+ + \pi^- . \]

The wave function of a particle with spin 1 has four components and transforms under rotations in the same way as the radius vector \(x\). Therefore, for rotations \(S=s\). Suppose that for reflections the matrix \(S=A'\), while the matrix \(s=A\). Then, by the same considerations as in the case of a particle with spin 0, we must have, for the product of any two reflections determined by the parameters \(a'\), \(a''\):

\[ A'(a'')\cdot A'(a')=A(a'')\cdot A(a') \tag{7} \]

and

\[ A'^2=1, \tag{8} \]

since \(A^2=1\). Multiplying (7) on the right by \(A'(a')\), we find:

\[ A'(a'')=A(a'')\cdot A(a')\cdot A'(a'). \tag{9} \]

It follows from (9) that the matrix \(A(a)\cdot A'(a)=B\) is independent of \(a\). We may write:

\[ A'=AB. \tag{10} \]

Taking (8) into account, we obtain:

\[ BA=AB^{-1}. \tag{11} \]

Now let us use the fact that the form of the matrix \(A\) is known. If, as the parameters \(a\), we choose the components of a unit vector perpendicular to the plane with respect to which reflection is performed, then

\[ x'_\mu=x_\mu-2a_\mu(x_\alpha a_\alpha),\qquad a_\alpha a_\alpha=1^{*}), \tag{12} \]

so that

\[ (s)_{ij}=(A)_{ij}=\delta_{ij}-2a_i a_j. \tag{13} \]

With the aid of (13) and (11) it is easy to obtain:

\[ B^2=1, \tag{14} \]

moreover the matrix \(B\) is diagonal:

\[ B=[-\zeta,\ -\zeta,\ -\zeta,\ 1]. \tag{15} \]

On the basis of (14) we have:

\[ \zeta^2=1,\qquad \zeta=\pm1. \tag{16} \]

Thus, also in the case of spin 1 we have two types of particles with different intrinsic spatial parity—vector \((\zeta=-1)\) and pseudovector, or axial-vector \((\zeta=+1)\). The wave functions of vector particles transform exactly as the radius vector. The wave functions of pseudovector particles transform in the same way as the radius vector only under rotations. The matrices corresponding to reflections of the spatial axes differ from the matrices giving the transformation of the radius vector in that the elements standing at the intersections of the first three rows and columns have the opposite signs. Examples of vector particles are quanta of the electromagnetic field. It is also possible that the \(V_{2}^{0}\)-particles mentioned above turn out to be vector particles. Considering particles with spins 0 and 1, we have arrived at the conclusion that in each case there are two types of particles, differing in their intrinsic spatial parity. It can be shown that the same situation holds in general for all particles with integral spin.

Let us now turn to the transformation properties of the wave functions of particles with spin \(^{1}/_{2}\). The behavior of the wave functions of such particles under reflections of the spatial axes turns out to be essentially different, and the consideration of the question somewhat more complicated. This occurs because the correspondence \(s\to S\) in the present case is not single-valued. To each transformation \(s\) of the system

\[ \text{*) } x_\alpha a_\alpha=x_1a_1+x_2a_2+x_3a_3-x_0a_0. \]

there correspond two matrices \(+S\) and \(-S\), which, according to (2), carry out the transformation of the wave function. The reason for the double-valuedness of the correspondence lies in the character of the dependence of the matrix \(S\) on the parameters determining the given transformation of the coordinate axes (for example, the components of the unit vector \(a_\mu\) in the case of reflection with respect to the plane perpendicular to \(a_\mu\), or, if rotations are in question, the angle of rotation about the axis of rotation and the unit vector directed along this axis). At the same time—and this is especially important for physics—the correspondence \(s \to S\) can be made single-valued in the case of infinitely small rotations*). Thus we have a representation of the rotation group only “in the infinitely small.” Such a representation is called spinor, and quantities transforming according to this representation are called spinors. To rotations of the coordinate system about an arbitrary axis through the angles \(\varphi\) and \(\varphi+2\pi\) there corresponds in equation (1) one and the same matrix \(s\). In contrast to this, the matrices \(S\) transforming a spinor have different signs under such a transformation of the coordinate system (see \(^{2}\) or \(^{5}\)). Since this situation remains valid also for \(\varphi=0\), one may say that, upon rotation of the coordinate system about any axis through the angle \(2\pi\), the spinor wave function changes sign. The double-valuedness of the spinor representation of the rotation group consists precisely in the fact that to one and the same matrix \(s(\varphi)=s(\varphi+2\pi)\) there correspond two matrices \(S(\varphi)\) and \(-S(\varphi)=S(\varphi+2\pi)\). Let us denote by the symbol \(A(a_\mu)\) the matrix transforming a spinor under reflection of the axes with respect to the plane passing through the origin of coordinates and perpendicular to the unit vector \(a_\mu\). In view of the double-valuedness of the spinor representation, we can no longer, as was the case for particles with spins 0 and 1, put forward as a categorical requirement the condition

\[ A(a_\mu)^2 = 1 . \tag{17} \]

Indeed, to the identical transformation of the coordinate axes

*) The construction of the angular-momentum tensor and the derivation of the corresponding conservation law from the invariance of the Lagrangian with respect to rotations presuppose, as a necessary condition, that an infinitesimal change of the coordinate system cause an infinitesimal change in the wave functions entering the Lagrangian (see, for example, 4). The latter is possible only with a single-valued choice of the operator of an infinitesimal rotation continuously depending on the parameters. If, even in the case of infinitesimal rotations, it were impossible to choose one of the two values of the matrix \(S\), then this, in other words, would mean that there are two equally valid possible values of the transformation of the function \(\pm \Psi'(x)\). It is clear that if, for example, \(\Psi'(x)-\Psi(x)=\delta\Psi(x)\) is an infinitesimal quantity, then \((-\Psi'(x))-\Psi(x)=-\delta\Psi(x)-2\Psi(x)\) is not such.

(s = 1) in spinor space there correspond two matrices \(S(0)=+1\) (rotation through the angle \(0\)) and \(S(2\pi)=-1\) (rotation through the angle \(2\pi\)). But a reflection performed twice with respect to one and the same plane may be regarded with equal right both as a rotation through the angle \(0\) and as a rotation through the angle \(2\pi\). Therefore, instead of (17) one could have required:

\[ A'(a_\mu)^2=-1. \tag{18} \]

Representations for whose matrices equations (17) and (18) hold are, as is easily seen, not equivalent.* Therefore wave functions which are transformed under reflections by the matrices \(A\) and \(A'\) correspond to different particles. We shall see (§ 4) that the difference in the transformation laws (17) and (18) can indeed manifest itself in experimentally observable effects. From what has been said it is clear that, whereas for particles with spins 0 and 1 the different behavior of the wave function under reflections reduces to a difference in internal parities, the wave functions of spinor particles differ in the sense of transformation under reflections in an entirely different way. Here it is not the matrices \(A\) and \(A'\) themselves that have different signs, but their squares. The concept of internal spatial parity in the sense in which it was defined above cannot be introduced for spinor particles. This circumstance is a consequence of the two-valuedness of the spinor representation and can be understood if one writes explicitly the matrix \(A(a_\mu)\) (or \(A'(a_\mu)\)) as a function of the components of the unit vector \(a_\mu\). Consider the product of two reflections \(A(a''_\mu)\cdot A(a'_\mu)\). This transformation is a rotation about some axis through an angle \(\varphi\):

\[ A(a''_\mu)\cdot A(a'_\mu)=S(\varphi). \]

It is not difficult to see that a rotation about this same axis through the angle \(\varphi+2\pi\) will correspond to the matrix

\[ S(\varphi+2\pi)=A(-a''_\mu)\cdot A(a'_\mu)=A(a''_\mu)\cdot A(-a'_\mu). \]

But according to the preceding,

\[ S(\varphi)=-S(\varphi+2\pi), \]

therefore

\[ A(a''_\mu)\cdot A(a'_\mu)=-A(-a''_\mu)\cdot A(a'_\mu). \tag{19} \]

* This means that there does not exist a nonsingular matrix \(P\) \((\operatorname{Det} P\ne 0)\) satisfying, for all \(A\) and \(A'\), the relation

\[ A'=PAP^{-1}. \]

Equation (19) indicates that the elements of the matrix \(A(a_\mu)\) are odd functions of the components of the unit vector \(a_\mu\). It is easy to verify that \(A(a_\mu)\) is a linear function of \(a_\mu\). If this were not so, then from (17) and (18) we would obtain some relation between the components \(a_\mu\) different from the only possible one

\[ a_\alpha a_\alpha = 1. \tag{20} \]

In equation (20) \(a_0 = 0\), since we considered transformations of the spatial axes of coordinate systems associated with one and the same reference body. Since, however, we are interested in the entire Lorentz group*), it is necessary to consider, for example, transformations consisting in reflection of the axes with respect to a plane passing through the origin of coordinates and the transition to a moving reference system**). Mathematically such a transformation will look like reflection with respect to a three-dimensional hyperplane perpendicular to the space-like unit vector \(a_\mu\). Bearing this in mind, we can now write for the matrix \(A(a_\mu)\) the following expression:

\[ A(a_\mu)=a_1 h_1+a_2 h_2+a_3 h_3+a_0 h_0. \tag{21} \]

The matrix \(A'(a_\mu)\), satisfying (18), has an analogous form (the \(h_\mu\) are replaced by \(h'_\mu\)). Let us now compare equation (22) with equation (13), which gives the form of the matrix \(s\) of the coordinate transformation of a point under reflection with respect to a hyperplane perpendicular to a space-like vector. In equation (13) the components \(a_\mu\) enter quadratically, so that \(s(a_\mu)=s(-a_\mu)\). And so it must be, since the vectors \(a_\mu\) and \(-a_\mu\) determine one and the same hyperplane with respect to which the reflection is performed. As for the matrix (21), \(A(a_\mu)=-A(-a_\mu)\). Consequently, to one and the same matrix \(s(a_\mu)\) there correspond two matrices \(\pm A(a_\mu)\).

Thus, in contrast to particles with integer spin, the matrices \(\pm A(a_\mu)\) give transformations of the wave function of one and the same particle with half-integer spin under reflection of the spatial axes. Among particles with spin \(1/2\) there are therefore no “spinor” and “pseudospinor” particles in the sense in which we spoke, for example, of scalar and pseudoscalar particles. At the same time, however, if two different spinor particles are considered (for example, neutron and proton, electron and neutri-

*) The Lorentz group, as is known, includes all three-dimensional and four-dimensional rotations, as well as transformations of reflection with respect to the origin of coordinates and the spatial axes.

**) All other Lorentz transformations that include reflections of spatial axes are obtained as the product of an odd number of transformations of the type indicated in the text.

... ), one can speak of their relative internal parity. It is said that two spinor particles possess different relative parity if the signs of the matrices \(A\), transforming the wave functions of the particles under each given reflection of the spatial axes, are different. The distinction between the relative internal parity for spinor particles and the internal parity considered above for particles with spins 0 and 1 consists in the fact that in the case of spin \(1/2\) one can fix only the difference or coincidence of the signs of the matrices \(A\), but for each of the particles the matrix \(A\) may be taken with either the sign \(+\) or the sign \(-\)*).

We shall now find the matrices \(h_\mu\). This is easy to do if one uses relation (17). Substituting (21) into (17), we find:

\[ h_1^2=h_2^2=h_3^2=+1,\qquad h_0^2=-1, \tag{22} \]

\[ h_i h_j+h_j h_i=2\delta_{ij}\quad (i,\ j=1,\ 2,\ 3),\qquad h_jh_0+h_0h_j=0. \tag{23} \]

The conditions (22) and the anticommutation relations (23) are satisfied, as is well known, by the matrices \(\gamma_\mu\) entering the Dirac equation:

\[ \left(\gamma_\alpha \frac{\partial}{\partial x_\alpha}+m\right)\Psi(x)=0^{**}). \tag{24} \]

It can be shown (see \({}^{7}\)) that there exist only two possible choices of the matrices \(h_\mu\):

either

\[ h_\mu=\gamma_\mu, \tag{25} \]

or

\[ h_\mu=\gamma_\mu\gamma_5,\qquad \gamma_5=\gamma_1\gamma_2\gamma_3\gamma_0, \tag{26} \]

where

\[ \gamma_\mu\gamma_5+\gamma_5\gamma_\mu=0,\qquad \gamma_5^2=-1. \tag{27} \]

If in equation (24) \(m\ne 0\), then possibility (25) has to be discarded, since otherwise equation (24) will not be invariant with respect to reflection of the coordinate axes. Indeed, invariance of equation (24) means that if \(\Psi(x)\) satisfies (24), then \(\Psi'(s^{-1}x')=A\Psi(x)\) is a solution of the equation obtained from (24) by replacing \(\dfrac{\partial}{\partial x_\mu}\) by \(\dfrac{\partial}{\partial x'_\mu}\). Let us perform a reflection with respect to a plane passing through the origin of coordinates and perpendicular to one of the axes, for example the \(j\)-th.

*) The possibility of introducing the concept of relative internal parity for spinor particles was pointed out to the author by Landau (see paper \({}^{6}\)).

**) We use units in which \(\hbar=c=1\) (\(\hbar\) is Planck’s constant divided by \(2\pi\); \(c\) is the speed of light in vacuum).

Then, if \(h_j=\gamma_j\), we obtain:

\[ \left(-\gamma_j\frac{\partial}{\partial x_j} +\sum_{k\ne j}\gamma_k\frac{\partial}{\partial x_k}+m\right)\gamma_j\Psi' = -\gamma_j\left(\gamma_\alpha\frac{\partial}{\partial x_\alpha}-m\right)\Psi'=0 \]

or

\[ \left(\gamma_\alpha\frac{\partial}{\partial x_\alpha}-m\right)\Psi'=0, \]

which does not coincide with (24)*). If, however, \(h_\mu\) is chosen according to (26), then, carrying out the same operations, we find:

\[ \left(-\gamma_j\frac{\partial}{\partial x_j} +\sum_{k\ne j}\gamma_k\frac{\partial}{\partial x_k}+m\right)\gamma_j\gamma_5\Psi' = \]

\[ =\gamma_j\gamma_5-\left(\gamma_\alpha\frac{\partial}{\partial x_\alpha}+m\right)\Psi'=0, \]

whence it follows

\[ \left(\gamma_\alpha\frac{\partial}{\partial x_\alpha}+m\right)\Psi'=0 \]

in complete agreement with (24).

We have carried out all calculations for matrices \(A\) satisfying (17). But one could also have chosen matrices \(A'\) satisfying equation (18). In this case we would obtain:

\[ h'_\mu=i h_\mu=i\gamma_\mu\gamma_5 . \tag{28} \]

Thus, there exist two types of particles with spin \(1/2\), whose wave functions behave differently under reflection of spatial axes. The transformations of the wave functions of these two types of particles are completely specified by the matrices (26) and (28), and differ in that the square of the reflection with respect to a hyperplane perpendicular to a space-like vector is, in one case, equal to \(+1\) and signifies the identity transformation, while in the other it is \(-1\), which coincides with the matrix transforming a spinor under rotation of the coordinate system through an angle \(2\pi\) about an arbitrary axis. Often, in describing the group of rotations and reflections, use is made of the fact that every reflection is a product

\[ \text{*) If }h_\mu\text{ is chosen according to (25), then the equation invariant with respect to reflections will be:} \]

\[ \left(\gamma_\alpha\frac{\partial}{\partial x_\alpha}+m\gamma_5\right)\Psi=0. \tag{24a} \]

This equation leads to the incorrect relation \(P^2-E^2=m^2\) between the energy \(E\) and the momentum \(P\) of a free particle of mass \(m\).

rotation into the transformation of reflection of all three spatial axes with respect to the coordinate planes (inversion with respect to the origin). The inversion matrices will be:

\[ T=\pm h_1h_2h_3=\pm\gamma_0, \tag{29} \]

\[ T'=\pm h'_1h'_2h'_3=\pm i\gamma_0. \tag{30} \]

For the squares of these matrices we have:

\[ T^2=-1, \tag{31} \]

\[ T'^2=+1. \tag{32} \]

Thus, the two different types of spinors can be characterized by the fact that for one of them (equations (26) and (31)) the square of inversion of the spatial axes is equivalent to a rotation through the angle \(2\pi\), while for the other (equations (28), (32)) it is equivalent to the identity transformation.

In contrast to the situation that obtains for particles with a given integer spin and with different intrinsic parity, it is difficult, from the point of view of the modern theory, to allow the coexistence of spinor particles of both of the above types. Indeed, let us imagine that there are two particles at rest with spin \(1/2\), belonging to the different types (31) and (32). The angular momentum of such a system as a whole may be equal to 0 or to 1. Suppose that the former is the case. Then the wave function of the system, being one-component, at the same time can be neither a scalar nor a pseudoscalar: under reflection of the spatial axes the wave function acquires the factor \(i\), and under the performance of \(n\) reflections—the factor \(i^n\), equal for even \(n\) to \(\pm 1\). It is easy to see that the correspondence \(s\to S\) in this case will be two-valued; but, in contrast to spinors, the transformations of the wave function of the system under consideration do not constitute a representation of the rotation group even in the infinitesimal. For a quantity with such transformation properties one cannot define uniquely the operator of an infinitesimal rotation, and in connection with this difficulties arise in introducing the concept of angular momentum (see the footnote on p. 12). From the foregoing it follows that, for spinor particles, a very peculiar situation takes shape within the framework of the modern theory: all existing spinor particles must belong to one or the other of the two indicated types. The question of exactly which type real particles with spin \(1/2\) belong to can be answered only by experiment (see § 4).*)

*) In this connection let us point out the untenability of purely speculative conclusions contained in work \(^{8}\). We note that a number of considerations on the impossibility of the coexistence of two different types of spinor particles is also given in work \(^{9}\).

Until now we have considered reflections only of spatial coordinate axes. From the formal mathematical point of view there also exists the transformation of reflection of the time axis, or in the general case reflection with respect to a hyperplane perpendicular to a timelike vector. Passage to the new frame of reference obtained as a result of reflection of the time axis means that an event previously characterized by the time coordinate \(x_0\) is now assigned the coordinate \(x'_0=-x_0\). It is very important that time reflections cannot be obtained by the successive performance of Lorentz rotations and inversions of spatial axes. This circumstance can easily be made clear if one turns to Fig. 1 and takes into account that, under Lorentz rotations, rotations in the plane \((x_j, x_0)\) are performed only in the interval of angles determined by the position of the generators of the light cone. The reason for the impossibility of reducing time reflections to transformations of the Lorentz group lies in the physical singling out of the time axis, which finds its mathematical expression in the difference of signs with which the squares of the spatial and time coordinates enter the fundamental form. Consequently, by introducing time reflections into consideration, we supplement the Lorentz group with new transformations not contained in it. The Lorentz group extended with allowance for time reflections will be called the full Lorentz group. The full Lorentz group comprises, besides translations, all transformations of four-dimensional space that leave invariant the square of the distance between two world points. It is essential that all known Lagrangians for quantized fields and the wave equations obtained from them prove to be invariant with respect to the full group

Fig. 1. Lorentz rotations in the plane \((x_j, x_0)\):
a) possible rotation; b) impossible rotation.

Lorentz, if, in accordance with general mathematical requirements, one defines the transformation laws of the wave functions and of the other physical quantities entering into these equations. One may say that physical equations initially obtained without imposing the requirement of invariance with respect to time reflections in fact potentially possess such invariance. What, then, can a special consideration of time reflections give in such a case, if taking them into account does not lead to the appearance of other equations, different from those already known? Taking account of time reflections leads to the appearance of a new characteristic of a particle—its intrinsic time parity. The latter may, generally speaking, not coincide with the space parity, since, for fixed transformation properties of the wave function with respect to the Lorentz group, there exists a certain freedom in specifying the transformation law under time reflections. Thus, particles with a given spin and intrinsic space parity may also possess different time intrinsic parities. This leads to an increase in the number of theoretically conceivable types of particles7, 10. Let us note that, since time reflections are not derivable from transformations of the ordinary Lorentz group, one cannot, proceeding from the principle of relativity alone, arrive at the conclusion that time reflections must necessarily be taken into account in theoretical physics. The transition from the Lorentz group to the full Lorentz group is, in fact, a hypothesis suggested by the mathematical apparatus—a very natural hypothesis, it is true, but nevertheless a hypothesis whose consequences must be subjected to experimental verification*.

Time reflections are often connected with the question of the behavior of a system “in the past.” The reason for such an interpretation is the fact that the time coordinate \(x_0 = t\) of a world point, as a result of reflection of the time axis, changes its sign \((x'_0 = -x_0 = -t)\). One must not forget, however, that the transformation of the wave function \(\Psi(x)\) in passing from one reference frame to another by no means consists in replacing the argument \(x\) by \(x'\) (in the present case \(x_0\) by \(-x_0\)), but is determined by equation (2), according to which the argument of the transformed function remains unchanged. In quantum mechanics the legitimacy of the concept of “reversibility”

* Schwinger10a, for example, drew attention to the fact that the requirement of invariance of the Lagrangian with respect to time reflections entails the necessity of quantizing the spinor field according to the exclusion principle (see the end of this paragraph).

the wave function into the past generally requires special analysis (see, for example, \(^{11}\)); but even if this circumstance is disregarded, the solution of the problem reduces to the analytic continuation of the solution of the wave equation, for the given boundary conditions, into the region of values \(x_0<0\), or, speaking quite roughly, to the substitution in the wave function of \(x_0=-t\) instead of \(x_0=t\). Nothing of the sort, as we have seen, occurs in the case of a transformation of the coordinate system. Under time reflections only the coordinates assigned to world points are changed, or, in other words, one and the same events are considered from the point of view of different reference systems, differing in the direction of the time axis. In reversing the wave function into the past, however, other events, preceding the given ones, are to be considered.

Let us now consider the laws of transformation of wave functions under time reflections. It was indicated above that the behavior of wave functions under time reflections is not completely determined by the transformation properties with respect to the Lorentz group, since transformations of time reflection are not contained in this group. The resulting freedom in the choice of a transformation law under time reflections is limited, first, by the requirement of linearity and, second, by the fact that the product of two time reflections belongs to the Lorentz group, being a four-dimensional rotation. The coordinates of a world point under reflection relative to the hyperplane perpendicular to the timelike vector \(a_\rho\) transform according to the law

\[ x'_\mu=(\delta_{\mu\alpha}+2a_\mu a_\alpha)x_\alpha,\qquad a_\alpha a_\alpha=-1. \tag{33} \]

Taking (33) into account and carrying out exactly the same reasoning as in the consideration of reflections of spatial axes, we shall find that, independently of the behavior of the wave function under inversion of the spatial axes, particles with spins 0 and 1 may be characterized by an internal time parity \(\zeta_0=\pm 1\). In the case of spin 0, the time parity \(\zeta_0=+1\) corresponds to a time scalar, while \(\zeta_0=-1\) corresponds to a time pseudoscalar. Thus four types of spin-0 particles are possible: completely scalar \((\zeta=+1,\ \zeta_0=+1)\), spatial scalars and time pseudoscalars \((\zeta=+1,\ \zeta_0=-1)\), spatial pseudoscalars and time scalars \((\zeta=-1,\ \zeta_0=+1)\), and completely pseudoscalar \((\zeta=-1,\ \zeta_0=-1)\). Likewise, for spin 1 there exist four types of particles, characterized by different combinations of the values of \(\zeta\) and \(\zeta_0\).

The spinor transformation matrix under reflection with respect to a hyperplane perpendicular to a timelike vector is obtained from formula (21), if the vector \(a_\mu\) is regarded as timelike. In particular, the matrix corresponding to reflection of the time axis with respect to a hyperplane perpendicular to it will be

\[ T_0=A(\delta_{\mu 0})=\pm h_0=\pm \gamma_1\gamma_2\gamma_3,\quad T_0^2=-1 \tag{34} \]

or

\[ T'_0=A'(\delta_{\mu 0})=\pm ih_0=\pm i\gamma_1\gamma_2\gamma_3,\quad {T'_0}^{\,2}=+1^{*}). \tag{35} \]

It is very important that the choice of the matrices (34) or (35) for the transformation of a spinor under reflection of the time axis can be made independently of the transformation law under inversion of the spatial axes. Therefore four types of particles with spin \(1/2\) are possible, corresponding to the different possible combinations of the matrices (29), (30), (34), and (35): \((T,T_0)\), \((T,T'_0)\), \((T',T'_0)\), \((T',T_0)\). Among the spinor particles of the four types indicated there may be particles with different relative intrinsic parities; moreover, a mismatch of their relative intrinsic spatial and temporal parities is possible. A peculiar feature of the spinor field is, besides everything else, the fact that from unquantized spinor wave functions one cannot construct a Lagrangian invariant with respect to the full Lorentz group. Indeed, it is not difficult to verify that the Lagrangian of a free spinor field

\[ L=-\frac{1}{2}\left\{\overline{\Psi}\gamma_\mu\frac{\partial\Psi}{\partial x_\mu} -\frac{\partial\overline{\Psi}}{\partial x_\mu}\gamma_\mu\Psi\right\} -\chi\overline{\Psi}\Psi,\ldots, \tag{36} \]

where

\[ \overline{\Psi}=\Psi^{+}i\gamma_0, \tag{36a} \]

changes sign under the transformations \(T_0\) or \(T'_0\).

If, however, one introduces quantization of the spinor field according to the exclusion principle and at the same time defines in the corresponding way the transformation of the creation and annihilation operators of particles, then the indicated deficiency is removed. To show this, let us pass to the momentum representation, putting, as usual,

\[ \Psi(x)=\frac{1}{(2\pi)^{3/2}}\sum_{r=1,2}\int \{a_r(\mathbf{p})u_+^r(\mathbf{p})e^{ipx}+ \]

\[ +\,b_r^+(\mathbf{p})u_-^r(\mathbf{p})e^{-ipx}\}\,d^3p,\ldots, \tag{37} \]

\(^*\) The matrices (34) and (35) were first found by Cartan\(^{12}\). In the physics literature they were first considered by Racz\(^{13}\).

where \(u_+^r(\mathbf p), u_-^r(\mathbf p)\) are solutions of the Dirac equation for positive and negative frequencies, the index \(r\) indicates the spin state, and \(a_r(\mathbf p)\) and \(b_r^+(\mathbf p)\) are, respectively, the operators of particle annihilation and antiparticle creation. We rewrite equation (37) in the following form, more convenient for us, as an integral over the four-dimensional volume in \(p\)-space:

\[ \Psi(x)=\sum_{r=1,2}\int c_r(p_\mu)u^r(p_\mu)e^{ipx}\delta(p^2+m^2)d^4p . \tag{37a} \]

Here the following notation has been used:

\[ \delta(p^2+m^2)=\left(\mathbf p^2+m^2\right)^{-\frac12} \left\{\delta\left(p_0-\sqrt{\mathbf p^2+m^2}\right)+ \delta\left(p_0+\sqrt{\mathbf p^2+m^2}\right)\right\}, \tag{37b} \]

where

\[ (2\pi)^{-3/2}p_0^{1/2}c_r(\mathbf p,p_0)=a_r(\mathbf p),\qquad u^r(\mathbf p,p_0)=u_+^r(\mathbf p), \tag{37c} \]

\[ (2\pi)^{-3/2}p_0^{-1/2}c_2(-\mathbf p,-p_0)=b_r^+(\mathbf p),\qquad u^r(-\mathbf p,p_0)=u_-^r(\mathbf p), \tag{37d} \]

if \(p_0=\sqrt{\mathbf p^2+m^2}\).

In this notation the Lagrangian density \(L\) is written in the form

\[ L=\int d^4p\,d^4p'\left\{ a^+(p_\mu)a(p'_\mu)e^{i(p'-p)x}\cdot \left[\bar u(p_\mu)\widehat D u(p'_\mu)+\bar u(p_\mu)\widehat D' u(p'_\mu)\right]\right. \]

\[ \left. +a^+(p'_\mu)a(p_\mu)e^{-i(p'-p)x}\cdot \left[\bar u(p'_\mu)\widehat D' u(p_\mu)+\bar u(p'_\mu)\widehat D u(p'_\mu)\right]\right\}; \tag{38} \]

\[ \widehat D=\widehat p+m;\qquad \widehat p=ip_\mu\gamma_\mu;\qquad \widehat D'=i\widehat p'+m . \]

At the same time, to simplify the notation, we have omitted in (38) the summation over the variables \(r\) and \(r'\). After carrying out the transformation of time-axis inversion we obtain:

\[ \Psi(x)\to\Psi'(x)=\sum_{r=1,2}\int c'_r(p_\mu)\bigl(u^r(p_\mu)\bigr)'e^{ipx}\delta(p_\mu^2+m^2)d^4p, \tag{39} \]

where

\[ u^{r'}=T_0 u^r \tag{39а} \]

or

\[ u^{r'}=T'_0 u^r. \tag{39б} \]

Replacing \(\Psi\) in (36) by \((\Psi')\) and requiring

\[ L=L', \tag{39в} \]

it is easy to find that a sufficient condition for (39в) to hold is

\[ \begin{aligned} c_r^{+}(p_\mu)'&=c_r(p_\mu),\\ c_r(p_\mu)'&=c_r^{+}(p_\mu), \end{aligned} \tag{39г} \]

if the spinor field is quantized according to the exclusion principle.

It is also not difficult to verify that the energy density of the spinor field does not change sign under the transformations (39).

As for fields with integral spins, there the Lagrangian is invariant under time reflections not only in the quantized theory, but also in the classical theory.

The results obtained in this paragraph on the possible types of particles with a given spin are collected, for clarity, in Table I,

Table I

Possible types of particles with a given spin

Spin	Type	Intrinsic parity: spatial \((\xi)\)	Intrinsic parity: temporal \((\xi_0)\)	Square of inversion of spatial axes	Square of inversion of the time axis
\(0;\ 1\)	\(S,\ S_0;\ V,\ V_0\)	\(+1;\ -1\)	\(+1;\ -1\)	\(+1\)	\(+1\)
\(0;\ 1\)	\(S,\ P_0;\ V,\ A_0\)	\(+1;\ -1\)	\(-1;\ +1\)	\(+1\)	\(+1\)
\(0;\ 1\)	\(P,\ P_0;\ A,\ A_0\)	\(-1;\ +1\)	\(-1;\ +1\)	\(+1\)	\(+1\)
\(0;\ 1\)	\(P,\ S_0;\ A,\ V_0\)	\(-1;\ +1\)	\(+1;\ -1\)	\(+1\)	\(+1\)
\(1/2\)	\(T,\ T_0\)	—	—	\(-1\)	\(-1\)
\(1/2\)	\(T,\ T'_0\)	—	—	\(-1\)	\(+1\)
\(1/2\)	\(T',\ T'_0\)	—	—	\(+1\)	\(+1\)
\(1/2\)	\(T',\ T_0\)	—	—	\(+1\)	\(-1\)

in which the letters \(S,\ P,\ V,\ A\) denote, respectively, a spatial scalar, pseudoscalar, vector, and pseudovector (axi-

al vector). Letters with the subscript 0 characterize the behavior of the wave functions under time reflections.

Let us briefly consider the interaction of various kinds of particles with one another. As is known, the possible Hamiltonians of interactions of spinor fields with one another and with particles of integral spin have the form of a scalar product of tensors composed of quantized spinor wave functions and of functions describing fields with integer spin. The tensors (spin-tensors) that can be formed from any two spinors \(\Psi_1, \Psi_2\) of the same type are listed in Table II, where \(\Psi^+\) denotes the Hermitian-conjugate matrix to \(\Psi\), and the tilde \((\sim)\) the transposition operation. The symbol \(B_{\mu\nu}\) denotes a bivector (an antisymmetric tensor of second rank of the type \(x_\mu y_\nu - x_\nu y_\mu\), where \(x_\mu, y_\mu\) are four-dimensional vectors), and the letters \(P\) or \(P_0\) placed before it indicate that the sign of the matrix transforming the given spin-tensor under inversion of the spatial or the time axes is opposite to the sign of the matrix carrying out the transformation of the bivector.

Spin-tensors bilinear with respect to \(\Psi_1^+\) and \(\Psi_2\) have the same transformation properties regardless of the type to which the spinors belong, provided only that their internal relative parities coincide (this assumption applies to the whole of Table II). For tensors containing \(\Psi_1\) and \(\Psi_2\), however, the situation is different. A spin-tensor that is a scalar if, for example, \(\Psi_1\) and \(\Psi_2\) belong to the type \((T, T'_0)\), becomes a pseudoscalar upon transition to the type \((T', T_0)\), etc. (this circumstance was clarified by Zharkov\(^{16}\), and also in Ref. \(^{17}\)). Spin-tensors of the type \((\Psi_1^+ F \Psi_2)\) enter into Hamiltonians of interactions as a result of which a particle with integral spin is created or absorbed, a spinor particle in state 1 is created, and a spinor particle in state 2 disappears. Spin-tensors \((\Psi_1 F \Psi_2)\) make it possible to construct Hamiltonians describing processes of decay of a particle with integral spin into two particles with half-integral spin, for example, the decay of a \(\pi^\pm\)-meson into a \(\mu^\pm\)-meson and a neutrino

\[ \pi^\pm \to \mu^\pm + \nu . \]

We note that this same decay process can be interpreted as the creation of a \(\mu\)-meson and the absorption of a neutrino from a negative level, i.e. as the emission of an antineutrino. If the spinors are assumed to belong to the type \(T'T'_0\), then, for a nonzero neutrino mass, the use of the “neutrino” and “antineutrino” variants leads to different results for the effects predicted by the theories. The latter testifies—

exists concerning the fundamental distinguishability of the neutrino and antineutrino, as was pointed out by Markov \(^{18}\)*).

Table II

Irreducible spin-tensors from two spinors

Spin-tensor	Spinor type \(T, T_0\)	Spinor type \(T, T'_0\)	Spinor type \(T', T'_0\)	Spinor type \(T', T_0\)
\(\widetilde{\Psi}_1\delta\Psi_2\)	\(P, S_0\)	\(P, P_0\)	\(S, P_0\)	\(S, S_0\)
\(\widetilde{\Psi}_1\delta\gamma_\mu\gamma_\nu\gamma_\rho\Psi_2^{(*)}\)	\(A, V_0\)	\(A, A_0\)	\(V, A_0\)	\(V, V_0\)
\(\widetilde{\Psi}_1\delta\gamma_\mu\gamma_\nu\Psi_2^{(**)}\)	\(P_0B_{\mu\nu}\)	\(B_{\mu\nu}\)	\(PB_{\mu\nu}\)	\(PP_0B_{\mu\nu}\)
\(\widetilde{\Psi}_1\delta\gamma_\mu\Psi_2\)	\(V, A_0\)	\(V, V_0\)	\(A, V_0\)	\(A, A_0\)
\(\widetilde{\Psi}_1\delta\gamma_5\Psi_2\)	\(S, P_0\)	\(S, S_0\)	\(P, S_0\)	\(PP_0\)
\(\Psi_1^{+}\gamma_0\Psi_2\)	For all types: \(S, P_0\)	For all types: \(S, P_0\)	For all types: \(S, P_0\)	For all types: \(S, P_0\)
\(\Psi_1^{+}\gamma_0\gamma_\mu\Psi_2\)	\(V, A_0\)	\(V, A_0\)	\(V, A_0\)	\(V, A_0\)
\(\Psi_1^{+}\gamma_0\gamma_\mu\gamma_\nu\Psi_2^{(**)}\)	\(P_0B_{\mu\nu}\)	\(P_0B_{\mu\nu}\)	\(P_0B_{\mu\nu}\)	\(P_0B_{\mu\nu}\)
\(\Psi_1^{+}\gamma_0\gamma_5\gamma_\mu\Psi_2\)	\(A, V_0\)	\(A, V_0\)	\(A, V_0\)	\(A, V_0\)
\(\Psi_1^{+}\gamma_0\gamma_5\Psi_2\)	\(P, S_0\)	\(P, S_0\)	\(P, S_0\)	\(P, S_0\)

\[ (*)\ \mu\ne\nu\ne\rho. \]

\[ (**)\ \mu\ne\nu. \]

Matrices:

\[ \gamma_i\gamma_j+\gamma_j\gamma_i=2\delta_{ij}\quad (i,j=1,2,3);\qquad \gamma_\mu\gamma_5+\gamma_5\gamma_\mu=0; \]

\[ \delta=\gamma_1\gamma_3;\qquad \gamma_\mu\gamma_0+\gamma_0\gamma_\mu=-2\delta_{\mu0};\qquad \gamma_5=\gamma_1\gamma_2\gamma_3\gamma_0;\qquad \gamma_5^2=-1. \]

If we denote by \(\Phi_{\mu\ldots}\) the tensors obtained from the quantized wave functions of particles with integral spin, then the dens-

*) The works of Markov and collaborators \(^{19,20}\) devoted to this question served as the starting point for clarifying the transformation properties of various spin-tensors depending on the type of spinors entering into them (see \(^{16}\)).

the part of the interaction Hamiltonian of these particles with the spinor field will have the form*)

\[ H' = g\Phi_{\mu\ldots}\left(\Psi_1^{+}F_{\mu\ldots}\Psi_2\right)+\text{Herm. conj.} \tag{40} \]

or

\[ H' = g\Phi_{\mu\ldots}\left(\widetilde{\Psi}_1 F'_{\mu\ldots}\Psi_2\right)+\text{Herm. conj.}, \tag{41} \]

where \(g\) is an interaction constant, and the spin-tensors must be chosen so that the Hamiltonian \(H'\) is invariant with respect to the Lorentz group. The tensors \(\Phi_{\mu\ldots}\) for a particle with spin 0 are:

\[ \Phi=\psi, \tag{42} \]

\[ \Phi_\mu=\frac{\partial\psi}{\partial x_\mu}. \tag{43} \]

For a particle with spin 1 one may write:

\[ \Phi_\mu=\psi_\mu, \tag{44} \]

\[ \Phi_{\mu\nu}=\frac{\partial\psi_\mu}{\partial x_\nu}-\frac{\partial\psi_\nu}{\partial x_\mu}. \tag{45} \]

Analogously to (40) and (41), one can construct the interaction Hamiltonian of four particles with spin \(1/2\). This Hamiltonian will have the form

\[ H'=\sum_F g_F\left(\Psi_1^{+}F_{\mu\ldots}\Psi_2\right)\left(\varphi_1^{+}F'_{\mu\ldots}\varphi_2\right)+\text{Herm. conj.} \tag{46} \]

or

\[ H'=\sum_F g_F\left(\widetilde{\Psi}_1 F'_{\mu\ldots}\Psi_2\right)\left(\widetilde{\varphi}F'''_{\mu\ldots}\varphi_2\right)+\text{Herm. conj.} \tag{47} \]

3. SPATIAL PARITY IN A STATE WITH GIVEN ANGULAR MOMENTUM

The spatial parity of a wave function characterizes its behavior under the successive performance of two transformations: inversion \(T\) of the spatial coordinate axes and replacement of the arguments \(x_i\,(i=1,2,3)\) by \(-x_i\) (this transformation will hereafter be denoted by the symbol \(R\)). The transformation \(D=TR=RT\) therefore takes the function \(\Psi'(x)\) into \(\Psi'(x')\), when \(x'_i=-x_i,\ x'_0=x_0\):

\[ \Psi'(-x_i,x_0)=D\Psi'(x_i,x_0). \tag{48} \]

*) We do not consider Hamiltonians containing derivatives of spinors, since the experimental data do not require this.

If the function \(\Psi(x)\) is an eigenfunction of the operator \(D\), then

\[ D\Psi(x)=\xi\Psi(x), \tag{49} \]

where \(\xi\) is the corresponding eigenvalue.

The operator \(D\), as follows from § 2, is defined uniquely only for wave functions of particles with integral spin. In this case one always has \(D^2=1\), and the number \(\xi\), equal to \(\pm 1\), is called the spatial parity of the wave function.

The wave functions of a particle in a state with definite angular momentum (momentum of the amount of motion) are eigenfunctions of the operator \(D\) and, for integral spin, may be even \((\xi=+1)\) and odd \((\xi=-1)\). A one-to-one connection between the parity of the wave function and the total angular momentum \(j\) of the particle exists only for spin equal to 0.

In this case

\[ \xi=(-1)^j\zeta, \tag{50} \]

where \(\zeta\) is the intrinsic spatial parity of the particle (see § 2). If the spin of the particle is different from zero, then there are at least two states with one and the same angular momentum but different spatial parity. The existence of such states can be explained most simply if one considers a particle with nonzero rest mass and nonrelativistic velocity. In this case it is meaningful to divide the total angular momentum \(j\) into spin \((\sigma)\) and orbital \((l)\) momenta, and for given \(j\) and \(\sigma\) several values of \(l\) are possible:

\[ l=j+\sigma,\quad j+\sigma-1,\ldots,(j-\sigma). \tag{51} \]

The number \(l\) determines the weight of the spherical functions \(Y_l^m(\vartheta,\varphi)\) (\(\vartheta,\varphi\) are polar angles), entering into the expressions for the components of the wave function of a particle with spin, and all the components that remain in the nonrelativistic approximation contain spherical functions with the same \(l\) (see \(^{2,5,21}\)).

Since

\[ RY_l^m(\vartheta,\varphi)=(-1)^lY_l^m(\vartheta,\varphi), \tag{52} \]

then, for a given intrinsic parity \(\zeta\), the number \(\xi\) will be given by equation (50), if in the latter \(j\) is replaced by \(l\). In the relativistic theory the number \(l\), determined by formula (51), ceases to be a physically observable quantity*), but the number of sta-

*) Formally, this occurs because the relativistic wave function is reducible with respect to the group of three-dimensional rotations. Therefore not all components of the relativistic wave function are expressed through

with a given total angular momentum still follows from relation (51), and the states corresponding to numbers \(l\) of the same parity possess the same spatial parity of the wave functions (see Appendix I).

What has been said above is fully valid only for particles with nonzero rest mass. In the case of particles with spin and zero rest mass, the number of states with a given angular momentum may be smaller than that prescribed by equation (51). Thus, for example, for quanta of the electromagnetic field (spin \(\sigma=1\), \(\zeta=\zeta_0=+1\)) there exist only two (and not three, as might have been expected on the basis of (51)) states with fixed angular momentum \(j\) and its projection on one of the axes*). These two states of the electromagnetic field are called electric and magnetic multipole radiation; for electric multipole radiation

\[ \xi=(-1)^j, \tag{53} \]

whereas for magnetic multipole radiation

\[ \xi=(-1)^{j+1}. \tag{54} \]

The operator \(D\) commutes with the Hamiltonian of any isolated system; consequently the spatial parity \(\xi\) of its wave function, being an integral of motion, is conserved in time. This accounts for the existence of selection rules with respect to spatial parity, which operate both in the decay of elementary particles and in various nuclear reactions. Owing to these selection rules, only those processes occur in which the spatial parities of the wave functions of the initial and final states of the isolated system coincide.

When considering the decay process of an elementary particle with nonzero rest mass, we can, evidently, always regard the particle as being at rest. Then the spatial parity of its wave function will coincide with its intrinsic parity, and the effect of the selection rules under consideration will appear in the fact that the spatial parity \(\xi\) of the wave function of the system of particles—the decay products—must be equal to the intrinsic parity \(\zeta\) of the decayed particle. If the symbol \(\xi_i\) denotes the spatial—

spherical functions of equal weight \(l\). In view of the circumstance indicated, the relativistic wave function of a particle with spin (see \(^{23,21}\)) is not an eigenfunction of the orbital angular-momentum operator.

*) The reason which in this case causes the reduction in the number of possible states as compared with (51) lies in the additional requirement of invariance of the mathematical expectations of observable quantities under the gradient transformation of the calibration of the potentials (see \(^{22}\)).

...the corresponding parity of the wave function of the \(i\)-th particle, one may write:

\[ \xi=\prod_{i=1}^{i=N}\xi_i, \tag{55} \]

and

\[ \zeta=\prod_{i=1}^{i=N}\xi_i. \tag{56} \]

As an example of the application of the parity selection rules, let us consider the decay of particles with integer spin into two and three particles with spins \(0, 1\). The simplest case is the decay of a particle with zero spin into two bosons whose spins are also equal to zero.*) For such a process, from equation (56) it follows that

\[ \zeta=\zeta_1\zeta_2, \tag{57} \]

where \(\zeta_1,\ \zeta_2\) are the intrinsic spatial parities of the particles—the decay products. Indeed, according to (50) we have

\[ \xi_1=(-1)^{j_1}\zeta_1,\qquad \xi_2=(-1)^{j_2}\zeta_2, \tag{58} \]

where \(j_1,\ j_2\) are the angular momenta of the particles—the decay products. Since the angular momentum of the whole system as a whole is equal to zero, it must necessarily be

\[ j_1=j_2=j. \tag{58a} \]

Using now equations (56), (58), (58a), we obtain (57). From the selection rule (57) it is immediately clear that decays of scalar \((S)\) and pseudoscalar \((P)\) particles into two particles with zero spins according to the schemes

\[ S\to S,\ P;\qquad P\to S,\ S;\qquad P\to P,\ P \tag{59} \]

are absolutely forbidden. It is precisely for this reason that the \(V^0_2\)-particle, if its decay products are two \(\pi\)-mesons, cannot be pseudoscalar.

Let us now consider the decay of a boson with spin \(\sigma\) into two bosons with zero spins. In the coordinate system of the center of inertia the sum of the momenta of the particles—the decay products—is equal to zero. For this reason the wave function of the final state of the system will depend only on the difference of the spatial coordinates of both particles. At the same time, the wave function of the final state, being a bilinear combination of the wave functions of particles with zero spin, can be only a scalar or a pseudoscalar.

*) A boson is a particle obeying Bose–Einstein statistics and, consequently, possessing integer spin.

Taking into account what has been said, as well as the law of conservation of angular momentum for the wave function of the final state, one may write:

\[ \Psi(x_j,x_0)=Y_\sigma^m(\vartheta,\varphi)\,R(r,x_0), \]

where \(\vartheta\) and \(\varphi\) are the polar angles of the vector \(x_j\), and \(r\) is its absolute value (the distance between the particles). Equation (56) for the given case takes the form

\[ \zeta=(-1)^\sigma \zeta_1\zeta_2. \tag{60} \]

If \(\zeta_1=\zeta_2\), which is realized, for example, in the decay of a \(V_2^0\)-particle into two \(\pi\)-mesons \((\zeta_1=\zeta_2=-1)\), then

\[ \zeta=(-1)^\sigma . \tag{61} \]

It follows from (61), in particular, that a \(V_2^0\)-particle decaying into two \(\pi\)-mesons cannot be pseudovectorial \((\zeta=+1,\ \sigma=1)\).

One may also point out the applicability of the selection rule (60) in nuclear physics if, instead of \(\zeta,\zeta_1,\zeta_2\), one substitutes the spatial parities \(\xi,\xi_1,\xi_2\) of the wave functions describing the initial state of the system and the internal states of nuclei with equal zero spins that are produced as a result of the reaction. Let us consider, for example, the photodisintegration of an even-even nucleus into an \(\alpha\)-particle and a residual nucleus with equal zero angular momentum.* Using (53), (54), and (60), it is easy to show that, when the spatial parities of the wave functions characterizing the internal states of the nuclei participating in the reaction and of the \(\alpha\)-particle are equal, photodisintegration will be effected only by quanta of the electric \(2^j\)-poles contained in the plane photon wave incident on the target. From the general theory it follows that the main contribution to the reaction cross section will then belong to the dipole \((j=1)\) quanta. If, however, the spatial parities of the wave functions of the initial and residual nuclei are different, then photodisintegration can be caused only by quanta of magnetic \(2^j\)-poles, chiefly by magnetic dipole radiation, which will lead to a substantial decrease in the effective cross section of the process as compared with the case of identical spatial parities.

Above we considered decay into two particles with zero spins. Passing to cases in which among the decay products there are particles with spin 1, we first of all point out the absolute prohibition of the decay of a boson with spin 0 into a photon (spin 1) and a boson with spin 0. This selection rule, formulated in work \(^{24}\), is a consequence of the equality to zero of the rest mass of the electromagnetic field. The point is that, for a particle with zero

\[ \text{* An even-even nucleus is one consisting of an even number of neutrons and an even number of protons. Such nuclei possess zero spins in their ground states.} \]

mass at fixed momentum, only two independent states are possible, in which the projection of the angular momentum on the direction of motion is equal to \(\pm \sigma\), where \(\sigma\) is the spin of the particle (see \(^{22}\))*). In the general case of a nonzero rest mass, the projection of the total angular momentum on the direction of motion can take \(2\sigma+1\) values from \(+\sigma\) to \(-\sigma\); higher values of the projection are excluded by axial symmetry (the axis of symmetry is the direction of motion of the particle). In a decay into two particles, their emission takes place along one straight line. According to what has been set forth, the projection of the angular momentum of a boson with zero spin, formed as a result of the decay, on this straight line will be zero, whereas the projection of the angular momentum of a photon is equal either to \(+1\) or to \(-1\). Consequently, the projection of the total angular momentum of the whole system must be equal in absolute value to 1, which is impossible, since the initial particle had spin 0. The decay of a boson with spin 0 into a particle with zero spin and a photon is therefore forbidden, owing to nonconservation of angular momentum, although at first sight it seems that this conservation law can be satisfied if the particle with zero spin produced in the decay acquires a nonzero orbital angular momentum (for example, equal to 1).

If a boson with spin 1 has a nonzero rest mass, then the selection rules are

\[ \zeta = -\zeta_1 \zeta_2 . \tag{62} \]

By the selection rules for parity and angular momentum, the decays of scalar and pseudoscalar particles according to the schemes \(^{23-25}\)

\[ S \to P, V;\qquad S \to S, A;\qquad P \to P, A, \tag{62a} \]

are therefore forbidden, where all decay products may have a rest mass different from zero. These selection rules are most simply obtained on the basis of the general theory of representations of the group of three-dimensional rotations and reflections (see Appendix II).

In the decay of particles with arbitrary integer spin into two vector particles (see \(^{24}\)), generally speaking, no “unexpected” features arise that are due to the selection rules for spatial parity and angular momentum. However, in the special case where the decay products are two photons, there occur additional restrictions first considered by Landau \(^{26}\) (see also \(^{26, 27, 28}\)), connected, first, with the fact that the electromagnetic field is characterized by zero rest mass and, second, with the fact that two photons are identical Bose particles. Considering the decay of a boson into two photons, we shall

*) For a photon, these two independent states are states with different polarization (linear or circular).

one should proceed from the fact that the wave function of the photon must be a vector. Therefore the wave function of a system of two photons can be represented in the form of a tensor of the second rank, constructed bilinearly from the components of the electromagnetic field of the photons*). For what follows it is essential that the total momentum of the photons formed as a result of the decay of a particle at rest is equal to zero. Therefore the wave function will depend on the difference of the photon coordinates

\[ x_j^{(2)}-x_j^{(1)}=x_j=a_j r, \tag{63} \]

where \(a_j\) is a unit vector parallel to the radius vector \(x_j\), and \(r\) is the absolute value of the latter.

With respect to the group of three-dimensional rotations and reflections, every three-dimensional tensor decomposes into the following irreducible parts: a scalar \(S\), the antisymmetric tensor of the second rank \(A_{ik}\), equivalent to a pseudovector, and the symmetric tensor of the second rank \(S_{ik}\) with zero trace \(\left(\sum_i S_{ii}=0\right)\). Let us consider the states described by each of these irreducible wave functions. First of all, from the condition of permutation symmetry we obtain:

\[ \begin{aligned} S(-a_j)&=S(a_j);\\ A_{ik}(-a_j)&=A_{ki}(a_j)=-A_{ik}(a_j);\\ S_{ik}(-a_j)&=S_{ki}(a_j)=S_{ik}(a_j). \end{aligned} \tag{64} \]

Since the intrinsic parity of a tensor of the second rank, composed bilinearly from the components of vectors, is \(+1\), on the basis of (64) we have:

\[ DS=+S,\qquad DA_{ik}=-A_{ik},\qquad DS_{ik}=+S_{ik}, \tag{65} \]

because the substitution of \(-a_j\) for \(a_j\) is equivalent to the transformation \(R\) considered at the beginning of this paragraph. According to (65), \(S\) and \(S_{ik}\) describe even \((\xi=+1)\), while \(A_{ik}\) describes odd \((\xi=-1)\) states of a system of two photons. Let us first consider the odd states. The tensor \(A_{ik}\), of whose six components only three are independent, is essentially a pseudovector parallel to the unit vector \(a_j\). The latter follows from the transversality of the photon field (a consequence of the equality to zero of the rest mass) and can be most simply established if \(A_{ik}\) is represented in the form of the vector product of the vectors \(E_j^{(1)}\), \(E_j^{(2)}\), characterizing the elec-

*) In the subsequent exposition we shall mainly follow the work \(^{26}\).

magnetic fields of each of the photons:

\[ A_{ik}=E_i^{(1)}E_k^{(2)}-E_k^{(1)}E_i^{(2)}\;{}^{*}), \tag{66} \]

\[ a_jE_j^{(1)}=a_jE_j^{(2)}=0,\qquad A_{ik}a_i=a_{ik}a_k=0. \tag{67} \]

Thus,

\[ A_{lk}=A_i'=a_l\varphi(a_j)\varepsilon\quad (l\ne i,\ k;\quad i\ne k), \tag{68} \]

where \(\varphi(a_j)\) is a scalar function, and \(\varepsilon\) is a pseudoscalar independent of \(a_j\). Comparing (64) and (68), we find:

\[ \varphi(a_j)=\varphi(-a_j). \tag{69} \]

If we consider states with a given angular momentum \(I\), then the angular part of the function \(\varphi\) is expressed in terms of a spherical Laplace function of order \(I\), and according to (69) the moment \(I\) must be either even or zero. Consequently, the tensor \(A_{ik}\) describes states of systems of two photons with even angular momentum and an odd (\(\xi=-1\)) wave function. It is easy to see that in these states \((I=0,2,4,\ldots,\xi=-1)\) both photons are polarized in mutually perpendicular directions. Indeed, \(A'=E^{(1)}E^{(2)}\sin\vartheta\), where \(\vartheta\) is the angle between the vectors \(E_j^{(1)}, E_j^{(2)}\). Since, furthermore, for a photon there exist only two independent polarization states—parallel and perpendicular to any plane containing the wave vector—the angle \(\vartheta\) can be equal either to \(0\) or to \(\pi/2\). The first case, however, is impossible, since for \(\vartheta=0\), \(A_{ik}=0\). We shall immediately indicate some applications of the results obtained. If, for example, it were possible experimentally to establish the perpendicularity of the polarizations of two gamma quanta into which the \(\pi^0\)-meson decays, then, since its spin is zero, one could assert with certainty that the \(\pi^0\)-meson is a pseudoscalar particle.

A good illustration of the rules obtained may also be provided by two-photon annihilation of para-positronium (the spins of the electron and positron are antiparallel, the total angular momentum is \(0\)). The annihilation quanta in this case prove to be polarized in perpendicular directions\(^{29}\). The latter indicates that the wave function of positronium is a pseudoscalar. This same result can also be obtained by direct calculation (see Appendix III). We now turn to the consideration of even states, described by the scalar \(S\) and the symmetric tensor \(S_{ik}\).

\({}^{*})\) Everywhere below, in expressing the tensors \(A_{ik}, S_{ik}; S\) in terms of the photon field vectors \(E_j^{(1)}, E_j^{(2)}\), we omit normalization coefficients, which are immaterial for our discussion.

From equations (64), (65) it is clear that the scalar function describes even states with even angular momentum (including zero). The polarizations of the photons in these states are parallel, since the scalar \(S\) can be represented in the form of the scalar product of the vectors \(E_j^{(1)}\) and \(E_j^{(2)}\). As regards the symmetric tensor \(S_{ik}\) with zero trace, let us first note that it, just like the tensor \(A_{ik}\), satisfies the transversality condition

\[ S_{ik}a_i=S_{ik}a_k=0 \tag{70} \]

and can be represented in the form

\[ S_{ik}=E_i^{(1)}E_k^{(2)}+E_k^{(1)}E_i^{(2)}-\frac{2}{3}\delta_{ik}E_j^{(1)}E_j^{(2)} . \tag{71} \]

Using (67), (70), and (71), it is easy to establish that in the states described by the tensor \(S_{ik}\) the polarizations of the photons are perpendicular. Indeed,

\[ S_{ik}a_i=-\frac{2}{3}a_k E_j^{(1)}E_j^{(2)}=0, \tag{72} \]

whence

\[ E_j^{(1)}E_j^{(2)}=E^{(1)}E^{(2)}\cos\vartheta=0,\qquad \vartheta=\frac{\pi}{2}, \]

since among the components \(a_k\) at least one component is nonzero. If the tensor \(S_{ik}\) did not satisfy the transversality condition (70), it would be very simple to count the number of even \((\xi=+1)\) states described by it with given angular momentum \(I\). Indeed, as was indicated above, any symmetric tensor of second rank \(S'_{ik}\) with zero trace is irreducible with respect to the rotation group. Among its six mutually nonidentical components only five are independent (because of the condition that the trace be zero). Consequently, the spin angular momentum of a system with wave function \(S'_{ik}\) is equal to 2, since \(2\sigma+1=2\cdot2+1=5\). Using this, one can find all states with given \(I\) and \(\xi=+1\) from formula (51), if one takes into account that, because of \(\xi=+1\), the number \(l\) must be even. Proceeding in this way, we immediately find:

\[ N'_0=N'_1=1;\qquad N'_{I=2n}=3;\qquad N'_{I=2n+1}=2, \tag{73} \]

where \(N'_I\) is the number of even states described by the tensor \(S'_{ik}\). In such a count the additional restriction (70), which can lead, and in fact does lead, to a reduction in the number of states, has not been taken into account. In order to take (70) into account, let us pre—

we put \(S'_{ik}\) in the form

\[ S'_{ik}=S_{ik}+P_{ik}, \]

where \(S_{ik}\) satisfies, and \(P_{ik}\) does not satisfy, the transversality condition (70). Thus, the state of a certain imaginary system with wave function \(S'_{ik}\) has been represented by us as a superposition of states \(S_{ik}\) and \(P_{ik}\). Since we know the total number of states \(N'_I\) with given momentum \(I\) and parity \(\xi=\pm 1\), the number of states \(N_I\) for \(S_{ik}\) can be found by subtracting from \(N'_I\) the number of states \(N''_I\) described by the tensor \(P_{ik}\). This latter has the form

\[ P_{ik}=B_i a_k+B_k a_i, \]

where \(B_j\) is some vector. The number \(N''_I\) therefore coincides with the number of states described by the vector function \(B_j(a_j)\), characterized by odd numbers \(l\), since the tensor \(P_{ik}\) as a whole,

Table III

States of a system of two photons with zero total momentum

Total angular momentum	Even states \((\xi=+1)\): parallel polarization	Even states \((\xi=+1)\): perpendicular polarization	Odd states \((\xi=-1)\): parallel polarization	Odd states \((\xi=-1)\): perpendicular polarization
\(0\)	1	0	0	1
\(1\)	0	0	0	0
\(2n\)	1	1	0	1
\(2n+1\)	0	1	0	0

by assumption, is even under the replacement \(a_j\to -a_j\). With the help of equation (51), putting \(\sigma=1\) and \(l\) odd, we find:

\[ N''_0=N''_1=1,\qquad N'_{I=2n}=2,\qquad N_{I=2n+1}=1. \tag{74} \]

Subtracting (74) from (73), we obtain:

\[ N_0=N_1=0,\qquad N_{I=2n}=1,\qquad N_{I=2n+1}=1. \tag{75} \]

The list of the results obtained above is presented in Table III*).

) Note added in proof. Table III coincides with that given in work \(^{28}\), if certain corrections are made to the latter; see Yu. M. Shirokov, ZhETF 26*, 128 (1954).

As is seen from this table, for two photons with equal energies and momenta of opposite sign there exist, first, no states with angular momentum equal to 1, and, second, no odd states ($\xi=-1$) with odd angular momentum. The absence of states with angular momentum 1 means that the decay of a vector or pseudovector particle into two photons is absolutely forbidden. Hence, in particular, it follows that the spin of the $\pi^0$ meson decaying into two photons cannot be equal to 1 (as is known, the totality of experimental data definitely indicates spin 0). For the same reason, annihilation of ortho-positronium with emission of two photons is forbidden[^30].

Speaking of the relative polarization of the photons, we assumed that both photons are linearly polarized. It is, of course, possible to carry out the discussion also under the assumption of circular polarization of the photons. In this case, in the states $S$, $A_{ik}$ the circular polarizations have the same signs, while in the states $S_{ik}$ they have opposite signs.

We have considered the decay of a boson with arbitrary spin into two particles with integral spins. Decay into three particles with integral spins has so far been investigated theoretically insufficiently. Among the existing results on this question we shall indicate the following selection rule[^23],[^25]: the decay of a boson with zero spin into three bosons, also possessing zero spins, is absolutely forbidden if one or three of the four particles participating in the process are pseudoscalar. In other words, decays of the type $S \to S, S, P$, $S \to P, P, P$, $P \to S, S, S$ and the like are forbidden. This selection rule can be written in the form of the relation

\[ \xi=\xi_1\cdot \xi_2\cdot \xi_3 \tag{76} \]

and is proved in complete analogy with the selection rule (57) (see also Appendix II). From the selection rule (76) it follows, in particular, that a $\tau$ meson decaying into three $\pi$ mesons cannot be a scalar particle.

With regard to the decay of a boson with arbitrary spin into three particles with spins 1, the decay into three photons has been studied theoretically in the greatest detail. The application of the selection rules for angular momentum and spatial parity leads here to no substantial restrictions, unless, for example, one counts the fact that for neutral scalar and pseudoscalar particles decay into three photons with equal energies is absolutely forbidden[^31]. Considerably stronger prohibitions are found if, in addition to the selection rules described in this paragraph, one uses also charge-conjugation symmetry (see § 4). Then, in particular,

It turns out that the decay of some neutral particles into three photons is in general impossible.

Up to now we have been speaking of the spatial parity of particles with integer spin. The wave function of a particle with half-integer spin and given total angular momentum satisfies equation (49), just as is the case for bosons. However, one cannot speak of a definite parity of the wave function, since the operator \(D=TR\) is now determined only up to sign, because of the matrix \(T\) entering it as a factor (see § 2). At the same time it is obvious that if we consider the wave functions of two spinor particles with fixed intrinsic relative parities, then we can always establish equality or inequality of the eigenvalues \(\xi_1\) and \(\xi_2\) of the operator \(D\), although each of these quantities is by itself determined only up to sign. Let us also note that \(\xi_1\) and \(\xi_2\) are not necessarily equal to \(\pm 1\), since the operator \(D^2=T^2R^2=T^2\), depending on the type of particles, may be equal to either \(+1\) or \(-1\) (see § 2). The ratio of the quantities \(\xi_1\) and \(\xi_2\) will be called the relative spatial parity of the wave functions of particles with half-integer spin. The selection rules for spatial parity expressed by equation (55) can now be generalized to the case of systems with arbitrary spin, provided that for spinor particles \(\xi_i\) is understood to mean the ratio \(\xi_i/\xi\), and in the left-hand side of (55) the quantity \(\xi\) is replaced by \(+1\).

4. CHARGE CONJUGATION; SELECTION RULES FOR CHARGE PARITY

Consider the wave equation

\[ \left\{\sum_j\left(\frac{\partial}{\partial x_j}-ieA_j\right)^2 -\left(\frac{\partial}{\partial x_0}-ieA_0\right)^2-m^2\right\}\Psi=0 \tag{77} \]

for a particle with zero spin and charge \(e\), situated in an electromagnetic field with vector potential \(A_j,-A_0\) \((j=1,2,3)\). The equation for the function \(\Psi^*\), complex conjugate to \(\Psi\), according to (77) will be:

\[ \left\{\sum_j\left(\frac{\partial}{\partial x_j}+ieA_j\right)^2 -\left(\frac{\partial}{\partial x_0}+ieA_0\right)^2-m^2\right\}\Psi^*=0, \tag{78} \]

since \(A_j,A_0\) are assumed to be real. Equations (77) and (78) differ only in the sign of \(e\). One may therefore interpret \(\Psi^*\) as the wave function of a particle with the opposite sign of charge. The validity of such an interpretation is confirmed by considering the expression for the four-dimensional current-density vector.

and charge

\[ \left. \begin{aligned} j_k&=e\left(\frac{\partial\Psi^*}{\partial x_k}\Psi-\frac{\partial\Psi}{\partial x_k}\Psi^*\right),\\ j_0&=-e\left(\frac{\partial\Psi^*}{\partial x_0}\Psi-\frac{\partial\Psi}{\partial x_0}\Psi^*\right),\quad k=1,2,3, \end{aligned} \right\} \tag{79} \]

which changes sign when \(\Psi\) is replaced by \(\Psi^*\). Denoting \(\Psi_+=\Psi\), \(\Psi_-=\Psi^*\), we can therefore write

\[ \Psi_-=\Psi_+^*. \tag{80} \]

Relation (80) is Lorentz-invariant, since under transformations of the reference system \(\Psi\) and \(\Psi^*\) transform in the same way (they do not change at all if \(\Psi\) is a scalar, and change sign under inversion of the axes if \(\Psi\) is a pseudoscalar). We shall call the functions \(\Psi_+\) and \(\Psi_-\) conjugate with respect to the sign of the charge, and the transformation \(\Psi_+\to\Psi_-\) charge conjugation.

In the case of particles with spin \(1/2\), obeying the Dirac equation, the relation between charge-conjugate functions is somewhat more complicated*). The Dirac equation in the presence of an external electromagnetic field has, as is well known, the form

\[ \left\{\gamma_\alpha\left(\frac{\partial}{\partial x_\alpha}-ieA_\alpha\right)+m\right\}\Psi=0 \tag{81} \]

or

\[ \left\{\gamma_\alpha^*\left(\frac{\partial}{\partial x_\alpha}+ieA_\alpha\right)+m\right\}\Psi^*=0. \tag{82} \]

We see that \(\Psi^*\) does not satisfy equation (81) if in the latter only the sign before \(e\) is changed, since in (82) the matrices \(\gamma_\alpha^*\) occur instead of \(\gamma_\alpha\). It is possible, however, to obtain a function \(\Psi_-\), conjugate with respect to the sign of the charge to the function \(\Psi_+=\Psi\), if one multiplies equation (82) on the left by some nonsingular matrix \(C\) and requires

\[ C\gamma_\alpha^*=\gamma_\alpha C. \tag{83} \]

Choosing \(\gamma_\alpha\) so that, for example, the matrix \(\gamma_2\) is real and the other three matrices are purely imaginary**), we immediately find, using the commutation relations given in Table II:

\[ C=\gamma_2. \tag{84} \]

*) Concerning charge conjugation for other relativistically invariant equations, see the work of Gelfand and Yaglom\(^{32}\).

**) If the \(\gamma_\alpha\) are defined according to § 2, this can always be done, and \(\gamma_1,\gamma_2,\gamma_3\) will be Hermitian, while \(\gamma_0\) will be anti-Hermitian (the matrices \(\gamma_1,\gamma_2,\gamma_3\) are defined in the same way as in\(^{4}\); the matrix \(\gamma_0\) differs from \(\gamma_4\) in\(^{4}\) by the factor \(i\)).

It follows from this that

\[ C=C^*,\quad C=\widetilde{C},\quad C^2=1. \tag{85} \]

Thus, a nonsingular matrix \(C\) satisfying (83) does indeed exist. We may therefore write:

\[ \left. \begin{aligned} \Psi_-&=C\Psi_+^*,\\ \Psi_+&=C\Psi_-^* . \end{aligned} \right\} \tag{86} \]

It is easy to see that the Lorentz invariance of (86) holds only for spinors of type \(T, T_0\) (see § 2), for which the squares of the matrices corresponding to inversions of the spatial and time axes are equal to \(-1\).

The vector of current and charge density for a spinor field has the form

\[ j_k=e\Psi^+\gamma_0\gamma_k\Psi,\quad j_0=e\Psi^+\Psi,\quad k=1,2,3^*), \tag{87} \]

or

\[ j_k=e\widetilde{\Psi}_-C\gamma_0\gamma_k\Psi_+,\quad j_0=e\widetilde{\Psi}_-C\Psi_+. \tag{88} \]

If we now make in (88) the substitution \(\Psi_+\leftrightarrows\Psi_-\), then \(j_\alpha\) does not change sign, as was the case for particles with zero spin. On the other hand, under such a transformation the sign changes for the components of the energy-momentum tensor of the field

\[ \left. \begin{aligned} T_{44}&=\frac{1}{2i}\left\{ C\widetilde{\Psi}_-\frac{\partial\Psi_+}{\partial x_0} - \frac{\partial C\widetilde{\Psi}_-}{\partial x_0}\Psi_+ \right\},\\[6pt] T_{4k}&=\frac{1}{2i}\left\{ C\widetilde{\Psi}_-\frac{\partial\Psi_+}{\partial x_k} - \frac{\partial C\widetilde{\Psi}_-}{\partial x_k}\Psi_+ \right\}, \end{aligned} \right\} \tag{89} \]

which is inadmissible. As is well known (see, for example, \(^{4}\)), this defect is eliminated by quantizing the spinor field according to the exclusion principle (i.e. taking account of the Pauli exclusion principle), whereby the required behavior of \(j_\alpha\) and \(T_{\mu\nu}\) under the interchange \(\Psi_+\leftrightarrows\Psi_-\) is achieved by the fact that the operators \(\Psi_+\) and \(\Psi_-\), acting on the wave function of the occupation numbers, anticommute. Summarizing what has been set forth, one may therefore say that in the quantized theory of the spinor field the operators \(\Psi_+\) and \(\Psi_-\) correspond to a particle and an “antiparticle,” whereas the unquantized wave functions \(\Psi_+\)

\(*\) Let us recall that in our notation \(\Psi^+=\widetilde{\Psi}^*\).

and \(\Psi_{-}\) describe states with positive and negative energy and opposite signs of the momenta.

From the requirement of Lorentz invariance (86) it follows that the functions \(\Psi_{+}\) and \(\Psi_{-}\) transform identically under a change of reference frame, i.e., possess the same intrinsic parity in the sense in which this concept was defined in § 2. At the same time, it is interesting to note that the wave function of a system consisting of a particle and an antiparticle (for example, an electron and a positron), having zero relative orbital angular momentum, can be only either a vector (ortho-state) or a pseudoscalar (para-state)\(^*\). It is precisely this latter circumstance, as was indicated in § 3, that explains the perpendicularity of the polarizations in two-photon annihilation of parapositronium\(^**\).

Charge conjugation can be used to obtain certain additional selection rules if one requires invariance of the Hamiltonian with respect to this transformation, i.e., with respect to the replacement particle \(\rightleftarrows\) antiparticle. Let us consider the wave function \(\Psi_q\) of the occupation numbers for a system of particles, antiparticles, and the electromagnetic field. The index \(q=(N_f^{+},\,N_f^{-},\,N_s^{\gamma})\) indicates the state of the system as a whole, characterized by the presence of \(N_f^{+}\) particles, \(N_f^{-}\) antiparticles, and \(N_s^{\gamma}\) quanta of the electromagnetic field in the individual states \(f, s\); moreover, generally speaking, \(N_f^{+}\ne N_f^{-}\). Introduce an operator \(U\), defined by the relations

\[ U\Psi_q=\Psi_{q'},\quad q=(N_f^{+},\,N_f^{-},\,N_s^{\gamma}),\quad q'=(N_f^{-},\,N_f^{+},\,N_s^{\gamma}), \tag{90} \]

\[ UF_{\pm}U^{-1}=F_{\mp}, \tag{91} \]

where the state \(q'\) differs from the state \(q\) in that the number of antiparticles is now equal to \(N_f^{+}\), and the number of particles to \(N_f^{-}\). By the symbol \(F_{\pm}\) in equation (91) is denoted an arbitrary operator acting on the variables of the occupation numbers of particles (sign \(+\)) or antiparticles (sign \(-\)). Thus, the operator \(U\) effects the replacement

\(^*\) This assertion is true in general for any two particles with spin \(1/2\) and zero orbital angular momenta, if their wave functions are characterized by the same relative intrinsic parity and transform under inversion of axes according to the representation \(T=-1\) (for the proof see Appendix III).

\(^**\) In the paper \(^{33}\) the electron and positron are treated as particles with different intrinsic parity. We point out in this connection that the concept of intrinsic parity introduced in \(^{33}\) and in the present article are different. In the paper \(^{33}\), by the intrinsic relative parity of two spinor particles is meant the coincidence or difference of the signs of the factor acquired under inversion of axes by the single nonzero component of the wave function of the particle considered in the coordinate system in which the particle is at rest.

particle $\rightleftarrows$ antiparticle, and, obviously,

\[ U^2=1. \tag{92} \]

We shall write the Hamiltonian operator for the system under consideration in the form

\[ H=H_p+H_r+H'=H_0+H', \tag{93} \]

where $H_p$ is the Hamiltonian of the system of free particles and antiparticles, $H_r$ is the Hamiltonian of the electromagnetic field in vacuum, and $H'$ is the operator of the interaction energy of all charged particles with the electromagnetic field. It is easy to see that $U$ commutes with $H_p$, since replacing a free particle by the corresponding antiparticle, in view of the exact equality of their masses, does not change the energy of the system. The same also holds with respect to the commutation of $U$ with $H_r$, since the operator $U$ does not change the number of photons at all. Thus,

\[ [UH_0]_- = UH_0-H_0U=0. \tag{94} \]

The question of the commutation of $U$ and $H'$ must be considered in more detail. As is known, the operator $H'$ has the form

\[ H'=\int j_\alpha A_\alpha\,dv,\qquad dv=dx_1dx_2dx_3, \tag{95} \]

\[ \alpha=0,\ 1,\ 2,\ 3. \]

Here $j_\alpha$ and $A_\alpha$ are the operators of the current density and the potentials of the electromagnetic field, acting on the corresponding occupation numbers. The operator $U$ anticommutes with $j_\alpha$, since under the replacement particle $\rightleftarrows$ antiparticle the vector of the electric current density changes sign. In connection with this, if we require invariance of the Hamiltonian as a whole with respect to the charge-conjugation transformation, it is necessary to assume

\[ UA_\alpha=-A_\alpha U. \tag{96} \]

The operator $U$ can be sought in the form of the product of two commuting operators $U_p$ and $U_r$, acting respectively on the occupation numbers of particles and photons. We now need to determine whether it is possible to find such an operator $U_r$ as to satisfy (93) and (95) simultaneously. Since the potential operators $A_\alpha$ contain linearly the photon creation and absorption operators, we shall satisfy (95) if we define the action of $U_r$ on the wave function of the system according to the equation

\[ U_r\Psi_{N'}=(-1)^{N'}\Psi_{N'}, \tag{97} \]

where $N'$ is the total number of photons.

On the other hand, the operator $U_r$, defined according to (97), certainly commutes with $H_0$, since the latter does not change

the number of photons \(N^r\). Thus, the operator \(U=U_pU_r\), satisfying simultaneously (94) and (96), and hence also the relation

\[ [UH]_- = 0, \tag{98} \]

does indeed exist. As was already indicated above, \(U\) anticommutes with the current operator \(I_\alpha=\int j_\alpha\,dV\), whose fourth component \(I_0\) is the operator of the total charge of the system. For this reason \(U\) cannot have common eigenfunctions with \(I_0\), except in the case when the eigenvalue of \(I_0\) is equal to 0, i.e., when the total charge of the system is equal to 0. Since the actually realized states of physical systems are always states with a definite total charge, the wave functions \(\Psi_q\) will be eigenfunctions of the operator \(U\) only for neutral systems. Under this last condition

\[ U\Psi_q=\nu\Psi_q,\qquad \nu=\nu_p\nu_r,\qquad \nu_r=(-1)^{N^r} \tag{99} \]

and

\[ U_p\Psi_q=\nu_p\Psi_q, \tag{100} \]

where

\[ \nu_p=\pm 1 \tag{101} \]

in view of equation (92).

In the apparatus developed above (see\(^ {34}\)), so far as electrodynamics is concerned, there is nothing hypothetical. In fact, starting from the prediction of the theory of the existence of particles and antiparticles (i.e., particles differing only in the sign of the charge), we have found one more, in addition to those considered in § 3, integral of motion for a neutral system—the charge parity of the wave function \(\nu\). The selection rules with respect to charge parity that arise in this case are exact to the same degree as the selection rules with respect to spatial parity.

Let us now examine, from the point of view of the selection rules obtained with respect to charge parity, the annihilation of an electron and a positron in para- and ortho-states. The wave function of such a system in the para-state can be represented in the following form\(^*\):

\[ \Psi_{\text{para}}=F\Psi_0,\qquad F=\frac{1}{2}\sum_{m_\sigma^+,\,m_\sigma^-} \left\{\tilde u_+^{*}Q u_-^{*}-\tilde u_-^{*}Q u_+^{*}\right\}, \]

\[ m_\sigma^+ + m_\sigma^- = 0, \tag{102} \]

Here \(u_+\), \(u_-\) are the quantized amplitudes of the Dirac wave

\(^*\) For simplicity, we assume the electron and positron to be free. This assumption in no way affects the result (see\(^ {34}\)).

functions of the electron and positron with spin projections on the quantization axis \(m_+^+\), \(m_-^-\), \(Q\) is the matrix ensuring the necessary transformation properties of \(\Psi_{\text{para}}\) with respect to transformations of the reference frame, and \(\Psi_0\) is the wave function of the vacuum. Equation (102) indeed describes the state we need, since \(u_+^*\) and \(u_-^*\) contain the creation operators of an electron and a positron, which, acting on the function \(\Psi_0\), transform it into the function \(\Psi_q\), characterized by the occupation numbers \(N^+=N^-=1\), \(N^\gamma=0\). Transforming \(\Psi_{\text{para}}\) by means of the operator \(U\), we obtain:

\[ U\Psi_{\text{para}}=UFU^{-1}U\Psi_0=UFU^{-1}\Psi_0, \tag{103} \]

since the wave function of the vacuum may, without any restriction of generality, be taken to be charge-even. Using (91) and (92), we find:

\[ UFU^{-1}=\frac{1}{2}\sum_{m_\sigma^+,m_\sigma^-}\left\{u_-^*Qu_+^*-\tilde{u}_+^*\tilde{Q}u_-^*\right\}. \tag{104} \]

It was indicated above that the wave function of para-positronium is a pseudoscalar (see also Appendix III). Therefore the matrix \(Q=\delta\) (see Table II). Since

\[ \tilde{\delta}=-\delta, \tag{105} \]

then, substituting (105) into (104), we find:

\[ UFU^{-1}=F \tag{106} \]

or

\[ U\Psi_{\text{para}}=\Psi_{\text{para}}. \tag{107} \]

Equation (107) shows that the wave function of para-positronium is charge-even \((\nu=+1)\). Hence follows the impossibility of three-photon annihilation of para-positronium. Indeed, the charge parity of a system of three photons according to (96) will be \(\nu=\nu_\gamma=(-1)^3=-1\), whereas for para-positronium \(\nu=+1\). In an analogous way other states of positronium may also be investigated (the classification of these states is given in Table IV). One may indicate a simple method for finding the charge parity of positronium states, based on the results obtained above. From the anticommutation of the operators \(u_+\) and \(u_-\) it follows that the wave function of positronium in the configuration representation must be antisymmetric with respect to interchange of the spatial, spin, and charge coordinates of the electron and positron. If one restricts oneself to the nonrelativistic approximation and neglects the exchange annihilation interaction (see \(^{33}\)),

Table IV

Spatial and charge parities of positronium states

Spin state	Orbital angular momentum	Spatial parity $\xi$	Charge parity $\nu$	Prohibition of annihilation by charge parity: two-photon annihilation	Prohibition of annihilation by charge parity: three-photon annihilation
singlet	even	$-1$	$+1$		forbidden
singlet	odd	$+1$	$-1$	forbidden
triplet	even	$-1$	$-1$	forbidden
triplet	odd	$+1$	$+1$	*)	forbidden

*) In states with total angular momentum 1, two-photon annihilation is forbidden according to Table III. The same also holds for all states with odd total angular momentum and an odd wave function ($\xi=-1$). In these latter cases the prohibitions according to Tables III and IV coincide.

The results indicated in Table IV were also obtained by direct calculations in work $^{61}$.

then the positronium wave function can be represented in the form of a product of three functions:

\[ \Psi=\Psi_{\text{spin}}\Psi_{\text{orb}}\Psi_{\text{ch}}. \tag{108} \]

In the state ${}^{1}S_{0}$, for example, $\Psi_{\text{spin}}$ is antisymmetric, while $\Psi_{\text{orb}}$ is symmetric. Therefore, for the antisymmetry of $\Psi$ it is necessary that $\Psi_{\text{ch}}$ be symmetric. But in the case of a system of two particles, the symmetry of $\Psi_{\text{ch}}$ coincides with the charge parity $\Psi$. Thus it is clear that the wave function of para-positronium must be charge-even, whereas the wave function of ortho-positronium (state ${}^{3}S_{1}$), owing to the symmetry of $\Psi_{\text{spin}}$ and $\Psi_{\text{orb}}$, must be charge-odd. This non-rigorous method of determining the charge parity of positronium turns out, however, to be quite accurate, since the corrections to the wave function (108) caused by the presence of the retarded interaction do not change the charge parity, as is easy to show with the formalism developed above.

Taking into account the selection rules with respect to charge parity leads to a number of essential results also for the decay of neutral elemen-

PROPERTIES OF SYMMETRY IN THE THEORY OF ELEMENTARY PARTICLES

tary particles. Thus, for neutral scalar particles a three-photon decay turns out to be absolutely forbidden. This follows from the fact that the wave function of a neutral scalar particle must be charge-even (otherwise the operator \(U\) does not commute with the Hamiltonian of the system, which contains terms of interaction of the scalar field under consideration with charged scalar, spinor, or vector fields). Since the neutral pseudoscalar \(\pi^0\)-meson decays into two photons, its wave function, according to what was set forth above, must be charge-even. Hence it follows that for charge-odd neutral particles decays of the type are forbidden

\[ X^0 \to \pi^0 + \pi^0 . \]

It can also be shown that for a neutral charge-odd particle decay into an electron-positron pair is forbidden

\[ X^0 \to e^- + e^+ . \]

For charge-even neutral bosons, decays according to the scheme are forbidden

\[ Y^0 \to \text{any number of } \pi^0 + \text{an odd number of } \gamma\text{-quanta}. \]

Concerning other possible applications of the operation of charge conjugation, let us first of all point to the problem of beta decay. The general form of the Hamiltonian for the interaction of nucleons with the field of light particles (electron-neutrino) was given in § 2 (equations (46), (47)). If one requires invariance of the Hamiltonian with respect to the operation of charge conjugation for nucleons and light particles, then all constants \(g^F\) turn out to be real\(^{36,37}\). Still more essential is the requirement of invariance of the Hamiltonian, up to the energy of the Coulomb interaction, with respect to the charge-conjugation transformation only for the light particles. The meaning of this requirement is that the difference between the electron and positron spectra, for a given upper limit and degree of forbiddenness of the transition, is assumed to arise exclusively from the difference in the interaction of electrons and positrons with the Coulomb field of the nucleus—the product of beta decay. It is not difficult to verify that the indicated requirement of symmetry leads to the following result: in the Hamiltonian (46), only the constants \(g_S\), \(g_A\), \(g_P\), corresponding to scalar, pseudovector, and pseudoscalar interactions, or the constants \(g_V\), \(g_T\) (vector and tensor interactions), may simultaneously be different from zero\(^{36,37}\).

For particles with spin \(1/2\), a relativistically invariant relation between the charge-conjugate functions \(\Psi_+\) and \(\Psi_-\) exists, as was already stated above, only for spinors of type \(T_1T_0\) (see Table I). In this case, the existence of neu-

tral wave spinor fields, for which the particle and antiparticle are identical. In fact, for the spinor \(\Psi=\Psi_+ + \Psi_-\) we have:

\[ \Psi_{\text{antipart}}=C\Psi^*=C\Psi_+^*+C\Psi_-^* =C\cdot C\Psi_-+C\cdot C\Psi_+= \]

\[ =\Psi_+ + \Psi_-=\Psi. \tag{109} \]

The vector of the current and charge density for such a field automatically becomes zero (see 4), as a result of which the spinor (109) describes neutral particles with a magnetic moment equal to zero. Of the neutral particles known to us with spin \(1/2\), only the neutrino can prove to be a particle of this type (for the neutron such a possibility is excluded because of the presence of a magnetic moment). This circumstance can be used in the theory of beta decay, and for the processes of ordinary beta decay or \(K\)-capture the identification of the neutrino and antineutrino gives nothing new. However, for the so-called double beta decay, in which the nuclear charge changes at once by two units, the use of the neutrino field (109) proves to be very essential \(^{38,39}\). The point is that, from the point of view of perturbation theory, double beta decay is a second-order process consisting of two virtual transitions, in each of which the nuclear charge changes by one unit. If the ordinary variant of the theory is applied, then in each virtual transition an electron or a neutrino is emitted, so that as a result of the whole process four particles will be emitted—two electrons and two neutrinos. If, however, for the neutrino one takes wave functions of the type (109), then, in addition, the emission of a neutrino in one virtual transition and its absorption in the other is possible, so that, in the end, only two electrons will be emitted. The probability of double beta decay with the emission of only two electrons exceeds by more than \(10^6\) times the probability of double beta decay proceeding according to the ordinary scheme. Since the half-life for a transition energy of the order of \(2\div 3\) MeV amounts, in the latter case, to \(10^{23} - 10^{24}\) years, under the present state of experimental technique only double beta decay accompanied by the emission of two particles can be observed, and it is possible only when the neutrino and antineutrino, described by the wave function (109), are identical. But for the existence of a spinor field of the type (109) it is necessary that it belong to the type \(T,T_0\). Thus, double beta decay belongs to the number of observable effects in which the difference in the properties of spinor fields that transform differently under inversion of the axes of the coordinate system is manifested. Unfortunately, the experimental data presently available concerning the observation of double beta decay (see \(^{40,41}\)) are not entirely reliable.

... and require further experimental verification. Let us emphasize that for the theory it would be exceptionally valuable to establish reliably at least the very fact of the existence of double beta decay and to obtain an approximate (even only to within an order of magnitude) estimate of the half-life.

In addition to the examples given above of the use of the charge-conjugation transformation, the latter also finds application in deriving inexact selection rules based on perturbation theory. These selection rules are considered in § 6.

5. SYMMETRY IN ISOTOPIC-SPIN SPACE

The concept of isotopic spin first arose in nuclear physics in the consideration of beta-decay processes, as a result of which a neutron is transformed into a proton or a proton into a neutron. This circumstance suggested the idea that the neutron and proton are, as it were, different states of one and the same particle—the nucleon. Of course, such a statement can mean something more than a new name for old facts only if it is reflected in the quantitative apparatus of the theory.

As is well known, systems of identical particles are characterized in quantum mechanics by a definite permutation symmetry of the wave functions. In particular, the wave function of a system of identical particles with half-integral spin is antisymmetric with respect to permutation of the spatial and spin coordinates of the particles. It is therefore necessary to impose the requirement of antisymmetry of the wave function of any system of protons and neutrons both with respect to permutation of particles of the same name and with respect to the permutation proton \(\rightleftarrows\) neutron. For this purpose we introduce a variable \(m_\tau\), taking the values \(\pm 1/2\) according as the nucleon is in “proton” \(\left(m_\tau=+\frac{1}{2}\right)\) or “neutron” \(\left(m_\tau=-\frac{1}{2}\right)\) states. We may now say that, in the nonrelativistic approximation, the wave function of the nucleon will depend on the spatial coordinates \(x_j\), the spin variable \(m_\sigma=\pm \frac{1}{2}\) (the projection of the spin on the quantization axis), and the new variable \(m_\tau=\pm \frac{1}{2}\), which, by analogy with \(m_\sigma\), we shall call the projection of the nucleon’s isotopic spin on the quantization axis in isotopic space. The deeper content of what is so far a purely external analogy between \(m_\tau\) and \(m_\sigma\) becomes clear upon a detailed consideration of the properties of the permutation symmetry of the wave function of a system of nucleons, if:

a) one neglects the difference between the masses of the neutron and the proton;

b) neglect the energy of the Coulomb interaction of the protons in comparison with the binding energy of the nucleus;

c) assume that the specific nuclear forces acting between nucleons do not depend on the isotopic state of the nucleons.

Requirements a) and b) can be satisfied for light nuclei. As for assumption c), experimental data on nuclear masses and schemes of nuclear levels (see \(^{42}\)) indicate the validity of this postulate, at least for the interaction of nucleons with not very large energies (of the order of several tens of MeV). When the conditions enumerated above are fulfilled, the wave function of a system of nucleons may be represented in the form

\[ \Psi=\varphi\bigl(x_j^{(1)}\ldots x_j^{(A)};\,m_\sigma^{(1)}\ldots m_\sigma^{(A)}\bigr)\, \chi\bigl(m_\tau^{(1)}\ldots m_\tau^{(A)}\bigr), \tag{110} \]

where \(A\) is the number of nucleons. Since \(\Psi\) as a whole is antisymmetric with respect to the interchange of the variables \(x_j\), \(m_\sigma\), \(m_\tau\) for any two nucleons, the properties of permutation symmetry of the functions \(\varphi\) and \(\chi\) must be related to each other in a definite way (for example, if \(\varphi\) is antisymmetric in the variables of the \(k\)-th and \(l\)-th nucleons, then \(\chi\) must be symmetric in the variables \(m_\tau^{(k)}\), \(m_\tau^{(l)}\)). Since the variable \(m_\tau\) assumes only two values,

\[ m_\tau=\pm \frac{1}{2}, \]

it is clear that all variables with respect to which \(\chi\) is symmetric can be grouped into two rows \(\bigl(m_\tau^{(k)},\,m_\tau^{(l)},\ldots\bigr)\), \(\bigl(m_\tau^{(k')},\,m_\tau^{(l')},\ldots\bigr)\), with \(k\ne k'\), \(l\ne l'\), etc. It is well known (see, for example, \(^{5,43}\)) that the various types of permutation symmetry are described by the so-called Young diagrams, shown for our case in Fig. 2. In the cells of the rows and columns of a Young diagram are placed the variables with respect to which the function is respectively symmetric and antisymmetric.

Fig. 2. Young diagrams for the isotopic part \((\chi)\) of the wave functions of the ground states of some nuclei: \(D_1^2\), \(T=0\); \(He_2^4\), \(T=0\); \(Li_3^6\), \(T=1/2\); \(O_8^{14}\), \(T=1\).

Thus, the Young diagram for \(\chi\) will have no more than two rows, and that for \(\varphi\) no more than two columns. From what has been said it is evident that the type of permutation symmetry of \(\chi\) (and hence also of \(\varphi\)), for a fixed total number of nucleons, may be characterized by a single number—the difference between the numbers of cells in the first and second rows of the Young diagram. One half of this difference is called the isotopic spin of the system of nucleons. If the system consists of \(A\) nucleons, then the isotopic-

the spin for odd \(A\) will be equal to a half-integer, and for even \(A\)—to an integer. The isotopic spin of a single nucleon should evidently be taken as equal to \(1/2\). After all that has been said, it is already not difficult to understand the essence of the analogy between ordinary and isotopic spin. An entirely similar picture with respect to the permutation symmetry of the spin and coordinate parts of the wave function obtains for a system of identical particles with spin \(1/2\), for example for electrons in an atom (see \(^{5,43}\)). In this case too, half the difference of the numbers of boxes in the rows of the Young diagram for the spin function gives the value of the total spin of the system. In other words, the study of the permutation symmetry of the spin function, say, of a system of electrons, can be replaced by an investigation of the transformation properties of the wave function under rotations of the coordinate axes. The latter circumstance is also used in the theory of isotopic spin. Namely, the ordinary wave functions of the proton and neutron are regarded as components of a single nucleon wave function, with respect to which it is assumed that it transforms according to the spinor representation of the rotation group of some auxiliary three-dimensional space having no direct physical meaning. The permutation proton \(\leftrightarrows\) neutron corresponds to a rotation of the “quantization axis” in this space by \(180^\circ\), since under such a transformation the projection of the isotopic spin of the nucleon \(m_\tau\) changes sign \(\left(m_\tau = +\frac{1}{2} \to m_\tau = -\frac{1}{2}\right)\).

The operators of the components of the isotopic spin \(\tau_1, \tau_2, \tau_3\) are the ordinary Pauli matrices

\[ \tau_1=\frac{1}{2} \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},\quad \tau_2=\frac{1}{2} \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix},\quad \tau_3=\frac{1}{2} \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \tag{111} \]

\[ \tau_1^2+\tau_2^2+\tau_3^2=\frac{3}{4} =\frac{1}{2}\left(\frac{1}{2}+1\right)=\tau(\tau+1),\quad \tau=\frac{1}{2}, \tag{112} \]

with the sole difference that the matrix elements of these matrices are themselves two-row matrices if the nucleons are treated nonrelativistically (but with spin taken into account), and four-row matrices if relativistic functions are considered.

The operators of isotopic spin for a system of \(A\) nucleons will have the form

\[ T_j=\sum_{i=1}^{i=A}\tau_j^{(i)}, \tag{113} \]

\[ T_1^2+T_2^2+T_3^2=T(T+1), \tag{114} \]

where \(T\) is an integer or half-integer depending on whether \(A\) is even or odd. To all that has been said one may also add that

the addition of the isotopic spins \(T^{(1)}\) and \(T^{(2)}\) and of their projections \(M_T^{(1)}\) and \(M_T^{(2)}\) of two systems of nucleons is performed according to the same rules as the addition of ordinary spins:

\[ \left. \begin{gathered} T=T^{(1)}+T^{(2)},\, T^{(1)}+T^{(2)}-1,\ldots,\left|T^{(1)}-T^{(2)}\right|,\\ M_T=M_T^{(1)}+M_T^{(2)},\quad M_T=T,\,T-1,\ldots,-T. \end{gathered} \right\} \tag{115} \]

If, for a system of nucleons, the conditions a), b), c) listed earlier are satisfied, then the operator

\[ \sum_{j=1}^{j=3} T_j^2 \]

commutes with the Hamiltonian of the system, as a result of which the isotopic spin must be an integral of motion. For this reason there arise selection rules with respect to isotopic spin, analogous to the selection rules with respect to angular momentum. As an example, let us consider the reaction \(O^{16}(d,\alpha)N^{14}\). In this reaction, at a fixed deuteron energy, several energy groups of \(\alpha\)-particles are observed, corresponding to the formation of the residual nucleus \(N^{14}\) in various excited states. At the same time, no cases have been found of formation of the nucleus \(N^{14}\) in the first excited level (excitation energy \(2.3\) Mev), although by the selection rules for angular momentum and spatial parity such a course of the process is allowed. If, however, one assumes that the isotopic spin of the nucleus \(N^{14}\) in the indicated excited state is equal to \(1^*\), then the process under consideration will be forbidden by the selection rules for isotopic spin, since the isotopic spins of \(O^{16}\), of the deuteron, and of the \(\alpha\)-particle are equal to zero.\(^{**}\)

A number of interesting consequences of the concept of isotopic spin also occur in the region of gamma radiation of nuclei. The Hamiltonian of the interaction of a nucleus with electromagnetic radiation may be written in the form

\[ H'=\frac{e}{2m}\sum_{j=1}^{j=3}\sum_{i=1}^{i=A} \left\{ P_j^{(i)}A_j\left(x^{(i)}\right)\left(1+2\tau_3^{(i)}\right) + \left[ \mu_n\left(1-2\tau_3^{(i)}\right) \right. \right. \]

\[ \left. \left. +\mu_p\left(1+2\tau_3^{(i)}\right) \right]\sigma_j^{(i)}H_j \right\}, \tag{116} \]

* Such an assumption is consistent with the shell structure of the nucleus and with other experimental data (see 44).

** The equality to zero of the isotopic spins of \(O^{16}\) and of the \(\alpha\)-particle follows from the fact that in these nuclei the proton and neutron shells are filled. As for the deuteron, its ground state is \({}^3S_1\) (with a small admixture of \({}^3D_1\)). Hence it follows that the function \(\varphi\) in (110) is symmetric, while the isotopic function \(\chi\) is antisymmetric, i.e. \(T=0\).

SYMMETRY PROPERTIES IN THE THEORY OF ELEMENTARY PARTICLES

where \(P_j, A_j, \sigma_j, H_j\) are the operators of the components of momentum, vector potential, spin, and magnetic-field strength, \(\mu_p, \mu_n\) are the magnetic moments of the proton and neutron, \(e\) is the charge of the proton, and \(m\) its mass. On the basis of (116) one may write:

\[ H' = H'_0 + \sum_{i=1}^{i=A} f_i \tau_3^{(i)}, \tag{117} \]

where the operators \(H'_0\) and \(f_i\) do not depend on \(\tau_3^{(i)}\). The probability of emission by a nucleus of a gamma quantum is proportional, according to the general rules, to the square of the matrix element of \(H'\). Using (116), (117), it is not difficult to determine the cases in which the matrix elements of \(H'\) vanish. Indeed, the operator \(H'_0\) is a scalar in isotopic-spin space. Therefore the matrix element \(H'_0\) is equal to zero if the isotopic spin changes in the transition. The second term on the right-hand side of (117), under rotations in isotopic space, behaves as a component of a vector; therefore the selection rules for it will be exactly the same as for matrix elements of the operators \(\sigma_j\), the components of ordinary spin. On the basis of what has been stated, we obtain the following selection rules with respect to isotopic spin for gamma radiation of nuclei:

\[ \Delta T = 0, \pm 1,\quad \Delta M_T = 0. \tag{118} \]

Let us note further that for \(M_T = 0\) (i.e. for equal numbers of protons and neutrons) it follows from (116) that electric dipole transitions of nuclei are forbidden\(^{45,46*}\). In an analogous way one can readily obtain the selection rules with respect to isotopic spin for beta processes:

\[ \left. \begin{aligned} \Delta T &= 0,\ \pm 1\ (\text{except }0 \to 0),\\ \Delta M_T &= \pm 1. \end{aligned} \right\} \tag{119} \]

In addition to the selection rules with respect to isotopic spin, one can also indicate, for nuclei with \(M_T = 0\), selection rules with respect to isotopic parity,\(^{47}\) analogous to the selection rules with respect to charge parity considered in § 4. In doing so, of course, it is necessary to bear in mind that, in contrast to the selection rules with respect to charge parity, the selection rules with respect to isotopic parity are not exact and are valid only in the approximations indicated earlier: a), b), c). The same also holds for the selection rules with respect to isotopic spin.

*) In Ref. 46 the question of gamma radiation and photodisintegration of nuclei in connection with isotopic spin is considered in detail.

It is rather difficult to estimate in the general case the degree of inaccuracy of the selection rules under consideration. However, a calculation carried out for light nuclei with \(T=0^{48}\) (see also \(^{49}\)) shows that the admixture of states with \(T=1\) arising from the Coulomb interaction and the inequality of the proton and neutron masses amounts to only about \(0.25\%\).

Thus, from a test of the validity of the concept of isotopic spin by studying the disintegration of light nuclei one may expect to obtain valuable information on the degree of charge independence of nuclear forces.

In connection with what has been set forth above, the question arises of the existence of isotopic symmetry for other particles known to us, for example \(\pi^\pm\)- and \(\pi^0\)-mesons. If by isotopic symmetry one understands definite properties of the permutation symmetry of the wave function, as was the case for nucleons, then one must admit that the introduction of the concept of isotopic spin for \(\pi\)-mesons is associated with difficulties. The point is that the \(\pi\)-meson has three charge states, as a result of which the Young diagram for the function \(\chi\) may consist of three rows (the variable \(m_\zeta\) takes the three values \(0,\pm 1\)). But a Young diagram with a number of rows greater than 2 cannot be characterized by a single number*); therefore, to introduce isotopic spin on the basis only of the hypothesis concerning the properties of the permutation symmetry of the wave function of a system of \(\pi\)-mesons appears difficult. Nevertheless, people speak of the isotopic spin of \(\pi\)-mesons, taking it to be equal to 1 and combining the wave functions of \(\pi^\pm\)- and \(\pi^0\)-mesons into a single three-component function—a vector in isotopic space. It is quite obvious that the concept of isotopic spin for \(\pi\)-mesons lacks the physical definiteness which exists for the isotopic spin of nucleons. Still less intelligible from the indicated point of view is the notion of the isotopic spin of a system of different particles, for example a \(\pi\)-meson and a nucleon. The meaning of the concept of the isotopic spin of the \(\pi\)-meson, or, more precisely, the definition of this concept, reduces to the postulation of definite symmetry properties of the Hamiltonian of the interaction of the meson field with nucleons or with other particles.

Considering purely formally the wave functions of the nucleon and of the \(\pi\)-meson as a spinor and a vector of isotopic space and requiring invariance of the Hamiltonian with respect to rotations in this space, we may write:

\[ H' = g\Phi_\mu^{k}(U^+\tau_{-k}F_\mu U), \tag{120} \]

*) If one speaks of Young diagrams for ordinary spin functions, then for spin \(>1/2\) different Young diagrams may correspond to one and the same total spin of the system.

where

\[ \begin{gathered} k=\pm 1,0;\quad \tau_{\pm 1}=\tau_1\pm i\tau_2,\quad \tau_0=2\tau_3,\\ U=\begin{pmatrix}\Psi_p\\ \Psi_n\end{pmatrix},\\ \Phi_\mu^{+1}=\Phi_{+\mu}^{*}+i\Phi_{-\mu}^{*};\quad \Phi_\mu^{-1}=\Phi_{+\mu}^{*}-i\Phi_{-\mu}^{*};\\ \Phi_\mu^0=\Phi_{0\mu}+\Phi_{0\mu}^{*}, \end{gathered} \tag{121} \]

where \(\Phi_{+\mu}, \Phi_{-\mu}, \Phi_{0\mu}\) are any of the tensors (42), (43) for positive, negative, and neutral mesons, while the matrices \(F_\mu\) have the same meaning as in equations (40), and are assumed to be Hermitian. The Hamiltonian (120) is a scalar in isotopic space, since it is the scalar product of two isotopic vectors \(\Phi_\mu^k\) and \((U^{+}\tau_{-k}F_\mu U)\). Such a Hamiltonian will be invariant under rotations of the 3-axis in isotopic space through \(90^\circ\) (the replacement \(\pi^{\pm}\rightleftarrows \pi^0\)) and \(180^\circ\) (the replacement \(\pi^{\pm}\rightleftarrows \pi^{\mp}\) and neutron \(\rightleftarrows\) proton). A characteristic feature of the Hamiltonian (120) is that it implies a difference in the signs of the interaction constant of the neutral meson with protons and neutrons. Indeed,

\[ g\Phi_\mu^0(U^{+}\tau_0F_\mu U) = g\Phi_\mu^0\bigl((\Psi_p^{+}F_\mu\Psi_p)-(\Psi_n^{+}F_\mu\Psi_n)\bigr) = -g_p\Phi_\mu^0(\Psi_p^{+}F_\mu\Psi_p)+g_n\Phi_\mu^0(\Psi_n^{+}F_\mu\Psi_n), \tag{122} \]

so that

\[ g_p=-g_n. \tag{123} \]

The validity of (123) can be checked experimentally, for example, by studying the photoproduction of \(\pi^0\)-mesons on deuterons. In this case two processes are possible:

\[ \gamma+d\to d+\pi^0, \tag{124a} \]

\[ \gamma+d\to n+p+\pi^0. \tag{124b} \]

A calculation (see \({}^{30}\)) shows that, if (123) is valid, both of these processes have effective cross sections of the same order of magnitude; but if one rejects (123) and sets \(g_p=g_n\), then the cross section of process (124a) proves to be considerably smaller than the cross section of reaction (124b). The experimental data known at present do not permit the validity of assertion (123), and hence of the Hamiltonian (120), to be regarded as fully established. One can only say with confidence that the Hamiltonian for the interaction of \(\pi^{\pm}\)-mesons with nucleons is invariant under rotation

axis 3 in isotopic space by \(180^\circ\). In this case the interaction Hamiltonian is written in the form

\[ H' = g \sum_{k=\pm 1} \Phi_\mu^k \left(U \tau_{-k} F_\mu U\right). \tag{125} \]

The concept of isotopic spin for meson and nucleon fields makes it possible, for calculating the relative probability of certain effects\(^*\), to use the results of the theory of representations of the group of three-dimensional rotations. The data obtained in this way are not connected with the use of perturbation theory and do not depend on the specific form of \(\Phi_\mu^k\) and \(F_\mu\) in the Hamiltonian (120), which makes it possible in a number of cases reliably to predict consequences following from the isotopic-spin hypothesis. A detailed consideration of this circle of questions, however, lies beyond the scope of the present article.

6. ADDITIONAL SELECTION RULES IN PERTURBATION THEORY

When perturbation theory is used to calculate the probabilities of decays of elementary particles, certain additional selection rules arise, first considered by Furry\(^{52}\) (see also \(^{53-56}\)). A rigorous derivation of these selection rules is given in Appendix IV. Here, in order to clarify the essence of the matter, we shall confine ourselves to a not quite rigorous, but physically transparent exposition.

Let us consider the decay of a neutral boson \(B_0\) into \(N\) other neutral bosons \(B_1 \ldots B_N\). Let us assume, moreover, that all the bosons participating in the process interact directly with some one and the same spinor-particle field, for example the nucleon field. Then, in the lowest order of perturbation theory, the decay process is represented as a chain of virtual transitions of the following, for example, type: production by the boson \(B_0\) of a nucleon–antinucleon pair, successive production by the nucleon of bosons \(B_1 \ldots B_{N-1}\), and annihilation of the nucleon and antinucleon with production of the boson \(B_N\). The matrix element for a transition of the indicated type may be represented in the form

\[ M_{\text{nucl}} = g^{(0)} g^{(1)} \ldots g^{(N)} M, \tag{126} \]

where \(g^{(0)}, g^{(1)}, \ldots, g^{(N)}\) are the constants of interaction of the bosons with the nucleon field. Since the theory is symmetric with respect to the nucle-

\(^*\) For example, reactions (124a) and (124b), the processes \(p + p \to p + n + \pi^+\) and \(p + p \to p + p + \pi^0\), different decay paths of the charged \(V^+\)-particle (see \(^{51}\)), etc.

ons and antinucleons, and moreover in the final state there are neither the former nor the latter, then we shall obtain exactly the same matrix element if we make the replacement nucleon \(\rightleftarrows\) antinucleon. Denoting this matrix element by \(M_{\text{ant}}\), we may write:

\[ M_{\text{ant}}=g^{(0)\prime}g^{(1)\prime}\ldots g^{(N)\prime}M \tag{127} \]

(\(g^{(0)\prime}, g^{(1)\prime}\ldots g^{(N)\prime}\) are the constants of the interaction of bosons with antinucleons). Since

\[ M_{\text{ant}}=M_{\text{nucl}}, \tag{128} \]

then

\[ g^{(0)\prime}g^{(1)\prime}\ldots g^{(N)\prime}=g^{(0)}g^{(1)}\ldots g^{(N)}, \tag{129} \]

provided only that \(M\) in (126) and (127) is not equal to zero. It is not difficult to show, using Table II and the results of § 4, that the interaction constant of bosons with nucleons changes sign in passing to antinucleons in the case of vector and tensor coupling (for the vector case this is especially clear, since vector coupling occurs in the interaction of charged particles with the electromagnetic field):

\[ \left. \begin{aligned} g'_V&=-g_V,\\ g'_T&=-g_T \end{aligned} \right\} \tag{130} \]

Fig. 3. Decay of a boson into \(N\) bosons through an intermediate nucleon field.

If among the bosons participating in the process there is an odd number of particles whose interaction with nucleons is characterized by vector or tensor couplings, then equality (129) will not be satisfied. Then, for (128) to hold, it is necessary that \(M=M_{\text{ant}}=M_{\text{nucl}}=0\). In other words, with an odd number of bosons with vector and tensor couplings, the process in the order of perturbation theory under consideration will be forbidden. Of course, this prohibition is not absolute—it may fail to hold in a higher order of perturbation theory. To determine for which particular chains of virtual transitions the indicated prohibition does not occur, it is convenient to depict these chains graphically in the form of diagrams (see \(^{57\text{—}59}\)). The diagram corresponding to the matrix element (126) has the form of a loop and is shown in Fig. 3, where the world lines of the bosons are conventionally shown by dashed lines, and the world line of the nucleon by a solid line; moreover the antinucleon is represented as a nucleon moving “backward in time.” To each vertex of the diagram (intersection of the nucleon and boson

lines) there corresponds one of the constants \(g^{(i)}\). If one of the constants (130) is associated with a given vertex, then we shall call it odd. The selection rule obtained above for the decay of a neutral boson \(B_0\) into neutral bosons \(B_1, B_2, \ldots, B_N\) may now be formulated as follows: the matrix elements for closed nucleon loops with an odd number of odd vertices are equal to zero. From what has been said it is clear that diagrams differing from Fig. 3 by the presence of internal virtual boson lines (Fig. 4, \(a\)), although they correspond to a higher order of perturbation theory, do not remove the prohibitions considered above, since the parity of the number of odd vertices remains unchanged. On the other hand, if the diagram in Fig. 4, \(a\) corresponds to a matrix element equal to zero, then diagram 4, \(b\) gives, generally speaking, a nonzero matrix element for the same process, but in a higher order of perturbation theory (diagram 4, \(b\) corresponds to the following chain of virtual transitions: absorption of \(B_0\) and creation of a nucleon–antinucleon pair, creation of the bosons \(B_1, B'\) by the nucleon, annihilation of the proton and antiproton with creation of the boson \(B_2\); absorption by the boson \(B_1\) of the boson \(B'\), which is thus virtual).

Fig. 4. Decay into two bosons:
\(a)\) the matrix element is equal to zero;
\(b)\) the matrix element is different from zero.

In the considerations developed above, symmetry in isotopic-spin space was not taken into account. If it is assumed to exist, then it is most convenient to consider separately two cases:

a) all the bosons participating in the process are neutral (this case, without taking isotopic symmetry into account, was considered above);

b) among the bosons participating in the process there are charged ones.

Let us first consider case a). Taking into account that neutral bosons can be created and absorbed by both protons and neutrons, we must write for the matrix elements of any chain of virtual transitions:
\[ M_{\text{nucl}}=M_{\mathrm p}+M_{\mathrm n}, \tag{131} \]
where \(M_{\mathrm p}\) and \(M_{\mathrm n}\) are given by equation (126), in which by \(g^{(0)},\ldots,g^{(N)}\) one should now understand the constants of interaction of the bosons with protons and neutrons, respectively. From equation (131) we

we immediately obtain a new result if we assume that the interaction of some of the bosons is described by the Hamiltonian (120), while for the remaining bosons the equality \(g_{\rho}=g_{n}\) holds. Since for bosons of type (120) \(g_{\rho}=-g_{n}\), then, for an odd number of them, their matrix element (131) vanishes (by virtue of isotopic symmetry \(M_{\rho}\) and \(M_{n}\) differ only by the constants \(g_{\rho}^{(i)}\) and \(g_{n}^{(i)}\)). Thus, in case a) we obtain the following selection rules: the matrix element is equal to zero if the closed nucleon loop contains an odd number of odd vertices or an odd number of external lines of bosons of type (120). Let us now pass to the consideration of case b). For definiteness let us suppose that the boson \(B_{0}\) is negatively charged and decays into the charged boson \(B_{1}^{-}\) and the neutral bosons \(B_{2}, B_{3},\ldots, B_{N}\).

Then the first virtual transition may be represented as follows: a proton in a state of negative energy absorbs \(B_{0}\), thereby turning into a neutron of positive energy (the “hole” remaining in the states of negative energy symbolically represents an antiproton). After this, a transformation of the neutron into a proton with emission of \(B_{1}^{-}\) is possible, and then the production by the proton of the bosons \(B_{2},\ldots, B_{N}\), one of these neutral particles being produced in the annihilation of the proton and antiproton (i.e. upon the return of the proton to the initial state of negative energy). Let us now note that the matrix element of the transition will not, in the case under consideration, be invariant under the transformation of charge conjugation for the nucleon field, since in the version of the theory used (the isotopic spin of the nucleon is equal to \(1/2\)) the “antiproton vacuum” cannot absorb a negatively charged particle.* The operation of charge conjugation for nucleons must in this case be combined with the replacement proton \(\rightleftarrows\) neutron. Denoting the corresponding matrix elements by \(M_{\rho}\) and \(\overline{M}_{\rho}\) (\(\overline{M}_{\rho}\) is the matrix element obtained from \(M_{\rho}\) by the two indicated transformations), we can write for each of them equation (126) and, proceeding from charge and isotopic symmetry, require:

\[ M_{\rho}=\overline{M}_{\rho}. \tag{132} \]

Comparing (126) and (132), we immediately find that \(M_{\rho}=\overline{M}_{\rho}=0\), if:

1) the closed nucleon loop contains an odd number of external lines of neutral bosons of type (120) and an even number of odd vertices;

*) If this were possible, then in the virtual transitions there would figure a “nucleon” with doubled elementary charge, i.e. \(e|m_{\tau}|=\dfrac{3}{2},\ \tau=\dfrac{3}{2}\).

2) a closed nucleon loop contains an even number of external lines of neutral bosons of type (120) and an odd number of odd vertices.

In other words, if by \(N(\tau_3)\) we denote the number of neutral bosons of type (120) participating in the process, and by \(N(V)\) and \(N(T)\) the numbers of vertices with vector and tensor couplings, then, in the order of perturbation theory under consideration, the process will be forbidden when the sum \(N(\tau_3)+N(V)+N(T)\) for the given nucleon loop is odd. This selection rule will, of course, be valid not only for the special type of decay considered by us \((B_0^- \to B_1^- + B_2 + \cdots + B_N)\), but in general for any processes involving charged bosons.

The selection rules obtained in this paragraph are given in Table V*). Using Table V, let us consider as examples

Table V

Selection rules for closed loops in perturbation theory

Type of process	\(N(\tau_3)+N(V)+N(T)=2n\)	\(N(\tau_3)+N(V)+N(T)=2n+1\)	\(N(\tau_3)\) or \(N(V)+N(T)=2n\)	\(N(\tau_3)\) or \(N(V)+N(T)=2n+1\)
With the participation of charged bosons	allowed	forbidden
Without the participation of charged bosons			allowed	forbidden
Note. \(N(\tau_3)\) is the number of neutral mesons of type (120) (see § 5); \(N(V)\) is the number of vector couplings; \(N(T)\) is the number of tensor couplings.

the decays of the \(V_2^0\)-particle and of the \(\tau\)-meson. According to the results of § 3, the \(V_2^0\)-particle may be either vector or scalar. If \(V_2^0\) is a vector particle of type (120), then its decay into two charged \(\pi\)-mesons is allowed in the lowest conceivable order of perturbation theory, since in this case

\[ N(\tau_3)=1,\quad N(V)+N(T)=1,\quad N(\tau_3)+N(V)+N(T)=2^{**}). \]

If, however, for the \(V_2^0\)-particle one has \(g_n=g_p\), then the decay into two

\[ \underline{\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}} \]

*) In connection with the selection rules placed in Table V, it is necessary to point out that in a paper published by the author earlier, article \(5^0\) obtained an incorrect result because of an error that crept into the calculation.

If, however, instead of the scalar neutral mesons with scalar coupling considered in the cited article, one uses, for example, scalar mesons with vector coupling, then the prohibitions listed in Table V can be avoided.

**) We regard the \(\pi\)-mesons as pseudoscalar particles.

charged \(\pi\)-meson is forbidden in the lowest possible order of perturbation theory. These same selection rules will be valid in the case when the \(V_2^0\)-particle is scalar, but is characterized by a vector coupling with nucleons. Using Table V, it is likewise easy to find that the decay of the \(V_2^0\)-particle according to the scheme

\[ V_2^0 \to 2\pi^0 \]

is forbidden in the lowest order if the interaction of the \(V_2^0\)-particle with the nucleon field is described by the Hamiltonian (120). The decay

Fig. 5. Decay of a neutral boson into a pair of particles with half-integer spin \((f^+, f^-)\):
a) decay into the pair \(f^+, f^-\) is forbidden; b) decay into the pair \(f^+, f^-\) and the boson \(B\) is allowed; c) the prohibition on decay into the pair \(f^+, f^-\) is lifted in a higher order of perturbation theory.

of a \(\tau\)-meson into three charged \(\pi\)-mesons is allowed in the lowest possible order of perturbation theory only in the case when the \(\tau\)-meson is a pseudoscalar or pseudovector particle. Under this condition, moreover, the decay\({}^{60}\)

\[ \tau^\pm \to \pi^\pm + 2\pi^0 \]

will be allowed independently of whether the \(\pi^0\)-mesons are described by the Hamiltonian (120) or not. The selection rules given in Table V can also be used to judge the degree of forbiddenness of processes of decay of a neutral boson through an intermediate nucleon field into a pair of particles with half-integer spin. The diagram corresponding to such a process is shown in Fig. 5a. If the boson \(B_0\) is, for example, a \(\pi^0\)-meson, and the boson \(B'\) is a photon, then the decay into the electron–positron pair in the order of perturbation theory indicated in diagram 5a will be forbidden. This prohibition is removed, however, when passing to higher orders (see Figs. 5b, 5c).

7. Conclusion

The symmetry properties considered in this article may, on the basis of the preceding discussion, be divided into two classes. The first includes the rules of exact symmetry—spatial and charge symmetry; the second includes the symmetry in the space of isotopic spin, which is in principle inexact and problematic in the sense of the very fact of its existence. This latter kind of symmetry is at present insufficiently studied not only from the experimental, but also from the theoretical point of view, especially for particles with integral isotopic spin.

With regard to the symmetry properties of space-time, we note here that investigation of the question of the role of time reflections in the theory has in fact only just begun.

At the same time, the experimental and theoretical material on the various symmetry properties that has now been accumulated already makes it possible to draw valuable conclusions concerning the nature of certain elementary particles and the course of a number of nuclear processes. The special significance of the conclusions obtained in this way consists in the fact that, for the most part, they are not connected with the use of computational schemes whose validity, in the present state of the theory, might be open to doubt.

Appendix I

On the Wave Functions of States with a Given Angular Momentum

As indicated in the text of § 3, the relativistic wave function is reducible with respect to the group of three-dimensional rotations. However, in order to find the number of possible states with a complete angular momentum, it is sufficient to consider all possibilities for the irreducible part of the wave function that has the largest number of components \((2\sigma+1\), if \(\sigma\) is the spin of the particle). Let us first consider states of particles with integral spin. From the representation theory of the group of three-dimensional rotations and reflections it follows that the irreducible representation \(D_j^\xi\), according to which the principal part of the wave function of a state with total angular momentum \(j\) and spatial parity \(\xi=(-1)^{\xi'}\) is transformed, is contained in the direct product of the irreducible representations \(D_l^{+}\), \(D_\sigma^{\zeta'}\), where \(\zeta'=(-1)^\sigma \zeta\) (\(\zeta\) is the intrinsic parity of the particle), and \(l\) is some integer. Indeed, according to the Clebsch–Gordan theorem we have:

\[ D_l^{+}\times D_\sigma^{\zeta'}=D_{l+\sigma}^{\zeta'}+D_{l+\sigma-1}^{-\zeta'}+\cdots \]

\[ \cdots +D_{l+\sigma-k}^{\zeta'(-)^k}+\cdots +D_{|l-\sigma|}^{\zeta'(-)^n}, \tag{1a} \]

where

\[ \left. \begin{array}{ll} n=2l, & 0 \leq k \leq 2l \quad \text{for } \sigma \geq l,\\ n=2\sigma, & 0 \leq k \leq 2\sigma \quad \text{for } \sigma \leq l. \end{array} \right\} \tag{16} \]

Choosing \(l\) and \(k\) so that

\[ l+\sigma-k=j, \tag{1в} \]

we obtain (51). The spatial parity of a function transforming according to the representation \(D'{}^{\,k}_{l+\sigma-k=j}\) will be

\[ \xi=(-1)^{l+\sigma}\zeta' = (-1)^l \zeta . \tag{1г} \]

Thus, for a given \(\zeta\), \(\xi\) is completely determined by the parity or oddness of the number \(l\).

In the case of half-integral spin, the concept of parity has a relative character. Therefore (1a) is replaced by the formula

\[ D_l^+ \times D_\sigma = D_{l+\sigma}^+ + D_{l+\sigma-1}^- +\cdots \]
\[ \cdots + D_{l+\sigma-k}^{(-)^k} +\cdots + D_{|l-\sigma|}, \tag{1д} \]

where the symbols \((-)^k\) attached to \(D\) indicate the relative spatial parity of the functions transforming according to the representations of the rotation group \(D_{l+\sigma}\) and \(D_{l+\sigma-k}\).

It should be noted that, in constructing a relativistic wave function from parts irreducible with respect to spatial reflections and rotations, the internal parities of these irreducible parts may prove to be different. In this case, instead of (1д), for example, one should consider direct products \(D_l^+ \times D_\sigma^\xi\), \(D_l^+ \times D_{\sigma'}^{\xi'}\), \(D_l^+ \times D_{\sigma''}^{\xi''}\), etc., where \(\xi'\), \(\xi''\) are relative internal parities of spins of dimensions \(2\sigma+1\), \(2\sigma'+1\), \(2\sigma''+1\), etc., entering into the composition of the relativistic wave function.

Let us consider in somewhat greater detail the important case \(\sigma=\tfrac{1}{2}\). The relativistic wave function \(\Psi\) will be four-component and reducible with respect to the group of three-dimensional rotations. It decomposes into two two-component spinors \(\varphi\) and \(\dot{\varphi}\) \(\left(\Psi=\left\{\begin{array}{c}\varphi\\ \dot{\varphi}\end{array}\right\}\right)\), transforming according to an irreducible representation \(D_j\) of this group. It is well known (see, for example, \(^{2,12}\)) that, for a certain form of the matrices \(h_\alpha\) (or \(h'_\alpha\)) considered in § 2, the matrix \(T\) (or \(T'\)), corresponding to inversion of the spatial axes, can be given the form

\[ T= \begin{pmatrix} 0 & \hat{\tau}\\ \hat{\tau} & 0 \end{pmatrix}, \]

where \(\hat{\tau}\) is a two-row matrix. Thus, under inversion of spa-

spatial axes \(\varphi \to \tau\varphi,\ \dot\varphi \to \tau\dot\varphi\). If we pass to a new equivalent representation by putting

\[ \Phi=P\Psi= \begin{pmatrix} \Psi_{\mathrm I}\\ \Psi_{\mathrm{II}} \end{pmatrix},\quad P= \begin{pmatrix} \hat{1} & \hat{1}\\ \hat{1} & -\hat{1} \end{pmatrix},\quad \hat{1}= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}, \]

\[ \varphi=\frac12(\Psi_{\mathrm I}+\Psi_{\mathrm{II}}),\quad \dot\varphi=\frac12(\Psi_{\mathrm I}-\Psi_{\mathrm{II}}), \]

then for the transformation of the spinors \(\Psi_{\mathrm I}\) and \(\Psi_{\mathrm{II}}\) under inversion of the spatial axes we obtain:

\[ \left. \begin{aligned} \Psi'_{\mathrm I}&=\hat{\tau}\dot\varphi+\hat{\tau}\varphi=\hat{\tau}\Psi_{\mathrm I},\\ \Psi'_{\mathrm{II}}&=\hat{\tau}\dot\varphi-\hat{\tau}\varphi=-\hat{\tau}\Psi_{\mathrm{II}}. \end{aligned} \right\} \tag{1e} \]

It follows from equality (1e) that the Dirac wave function is reducible not only with respect to three-dimensional rotations, but also with respect to spatial reflections, and the spinors \(\Psi_{\mathrm I}\) and \(\Psi_{\mathrm{II}}\) have different intrinsic parity. If under rotations and reflections \(\Psi_{\mathrm I}\) (or \(\Psi_{\mathrm{II}}\)) transforms according to the representation \(D_j^{+}\), contained in the direct product \(D_{j-\frac12}^{+}\times D_{\frac12}^{1}\), then \(\Psi_{\mathrm{II}}\) (or \(\Psi_{\mathrm I}\)) transforms according to the representation \(D_j^{-\zeta}\), entering into the composition of the direct product

\[ D_{j+\frac12}^{+}\times D_{\frac12}^{\zeta}=D_{j+1}^{\zeta}+D_j^{-\zeta}, \]

where \(\zeta\) is the relative intrinsic parity of \(\Psi_{\mathrm I}\) and \(\Psi_{\mathrm{II}}\). Since \(\zeta=-1\), ultimately both spinors transform according to one and the same representation \(D_j\).

Appendix II

DECAY OF A BOSON WITH ZERO SPIN

We first prove the validity of the selection rules (62). In all cases of decay into two particles, the wave function of the decaying particle must transform under three-dimensional rotations and reflections according to an irreducible representation \(D_\sigma^{\zeta'}\), contained in the direct product

\[ D_t^{+}\times D_{\sigma_1}^{\zeta_1'}\times D_{\sigma_2}^{\zeta_2'}, \]

where \(t\) is the orbital angular momentum of the relative motion of the decay products, \(\sigma,\sigma_1,\sigma_2\) are the spins of the particles participating in the process, and \(\zeta',\zeta_1',\zeta_2'\) are the quantities indicated in Appendix I; in our case

\[ \sigma=\sigma_2=0,\quad \sigma_1=1,\quad \zeta'=\zeta,\quad \zeta_2'=\zeta_2,\quad \zeta_1'=-\zeta_1. \]

Expanding, according to formula (Ia), the direct product \(D_l^+ \times D_1^{-\zeta_1}\), we see that, since \(\sigma=0\), \(l\) must be equal to 1. Therefore

\[ D_l^+ \times D_1^{-\zeta_1}=D_1^+ \times D_1^{-\zeta_1}=D_2^{-\zeta_1}+D_1^{\zeta_1}D_0^{-\zeta_1}, \]

\[ D_l^+ \times D_1^{-\zeta_1}\times D_0^{\zeta}=D_2^{-\zeta_1\zeta}+D_1^{\zeta_1\zeta}+D_0^{-\zeta_1\zeta}. \]

The decay will be allowed if

\[ D_0^{\zeta}=D_0^{-\zeta_1\zeta_2} \]

or

\[ \zeta=-\zeta_1\zeta_2, \]

as was required.

In the decay of a boson with spin 0 into three bosons with zero spins, the selection rule (76) holds. To verify this, let us decompose the direct product

\[ D_{l_1}^+ \times D_{l_2}^+ \times D_0^{\zeta_1}\times D_0^{\zeta_2}\times D_0^{\zeta_3} = D_{l_1}^+ \times D_{l_2}^+ \times D_0^{\zeta_1\zeta_2\zeta_3} \]

into irreducible parts (\(l_1, l_2\) are the orbital angular momenta of particles 1 and 2 with respect to particle 3). By conservation of angular momentum one must have

\[ l_1=l_2=l. \]

According to Appendix I,

\[ D_l^+ \times D_l^+ = D_{2l}^+ + \cdots + D_0^+. \]

Therefore

\[ D_{l_1}^+ \times D_{l_2}^+ \times D_0^{\zeta_1\zeta_2\zeta_3} = D_{2l}^{\zeta_1\zeta_2\zeta_3}+\cdots+D_0^{\zeta_1\zeta_2\zeta_3}. \]

Since the wave function of the decaying particle transforms according to the irreducible representation \(D_0^\zeta\), the condition for the possibility of the decay will be:

\[ D_0^\zeta=D_0^{\zeta_1\zeta_2\zeta_3}, \]

whence (76) follows.

Appendix III

TRANSFORMATION PROPERTIES OF THE WAVE FUNCTION OF A SYSTEM OF TWO SPINOR PARTICLES WITH ZERO ORBITAL ANGULAR MOMENTA

As indicated in Appendix I, the relativistic wave function of a particle with spin \(1/2\) can be decomposed into two irreducible, with respect to spatial rotations and reflections, two-component spinors transforming according to the representation \(D_j^+\).

Therefore the wave function of a system of two particles with the same relative intrinsic parity will, for a fixed angular momentum, transform according to one of the representations contained in the direct product \(D_{j_1}^{+}\times D_{j_2}^{+}\). If \(j_1=j_2=\frac{1}{2}\) and the spinor particles under consideration belong to the type \(T, T_0\) or \(T, T_0'\) (see Table I), then

\[ D_{1/2}^{+}\times D_{1/2}^{+}=D_1^{+}+D_0^{-}. \tag{IIIa} \]

Thus, for zero orbital angular momentum and total angular momentum \(I=1\) we have, using the notation of Appendix I,

\[ \xi'=(-1)^I\xi=-\xi=-\zeta=1 \]

or

\[ \zeta=-1, \]

i.e. the wave function is a vector. If the total angular momentum of the system is equal to zero, then according to (IIIa) the wave function transforms according to the representation \(D_0^{-}\) and, consequently, is a pseudoscalar.

If the spinor particles belong to the type \(T'T_0'\) or \(T'T_0\), then the decomposition \(D_{1/2}^{\prime +}\times D_{1/2}^{\prime +}\) will have the form

\[ D_{1/2}^{\prime +}\times D_{1/2}^{\prime +}=D_1^{-}+D_0^{+}. \]

In this case, for \(I=1\) we obtain a pseudovector, and for \(I=0\)—a scalar.

Appendix IV

CHOICE OF FURRY SELECTION RULES

The transition matrix element for the diagram shown in Fig. 3 can be written in the form*)

\[ M=M_1+M_2, \]

\[ \left. \begin{aligned} M_1&=\int \operatorname{Sp}\{Q_0K(0,N)Q_N\ldots Q_1K(1,0)\}\,d^4p,\\ M_2&=\int \operatorname{Sp}\{K(0,1)Q_1\ldots Q_NK(N,0)Q_0\}\,d^4p. \end{aligned} \right\} \tag{IVa} \]

The two terms in equation (IVa) arise in connection with the two possible directions of traversing the loop in the diagram of Fig. 3

*) In (IVa) all constants inessential for the present derivation have been omitted.

(\(M_1\) corresponds to the direction of traversal shown in Fig. 3). The letters \(Q_i\) denote the Fourier components, multiplied by the matrix \(\gamma_0\), of the operators entering into the interaction Hamiltonian (120), and for our derivation only that part of them is essential which consists of the product of the matrices \(\tau\) and \(F\), so that in what follows we shall write:

\[ Q_i=\tau^i f^i,\qquad \tau^i=\tau_{r_i},\quad f^i=\gamma_0 F^i,\quad r_i=0,\pm1, \tag{IVб} \]

omitting, for brevity, the lower index on the matrices (cf. (120)). By the symbols \(K(l,m)\) we denote the matrices

\[ \begin{gathered} K(1,0)=\left(\hat p+m\right)\left(\hat p^{\,2}-m^2\right)^{-1}\ldots\\ \ldots K(N,0)=\left(\hat p_N+m\right)\left(\hat p_N^{\,2}-m^2\right)^{-1},\\ K(0,1)=\left(-\hat p+m\right)\left(\hat p^{\,2}-m^2\right)^{-1}\ldots\\ \ldots K(0,N)=\left(-\hat p_N+m\right)\left(\hat p_N^{\,2}-m^2\right)^{-1},\\ \hat p_1=\hat p-\hat k_1,\quad \hat p_{i+1}=\hat p_i-\hat k_{i+1},\\ \hat p=i\gamma_\alpha p_\alpha,\quad \hat k_j=i\gamma_\alpha k^j_\alpha, \end{gathered} \tag{IVв} \]

where \(k^j_\alpha\) is the four-dimensional momentum of the \(j\)-th boson.

Let us now compare \(M_1\) and \(M_2\). Taking into account that the \(\tau^i\) commute with all \(f^i\) and \(K(l,m)\), we write:

\[ \begin{aligned} M_1&=a\int \operatorname{Sp}\{f^0K(0,N)\ldots f^1K(1,0)\}\,d^4p,\\ M_2&=a'\int \operatorname{Sp}\{K(0,1)f^1\ldots K(N,0)f^0\}\,d^4p, \end{aligned} \tag{IVг} \]

where

\[ a=\operatorname{Sp}(\tau^0\tau^N\ldots\tau^1),\qquad a'=\operatorname{Sp}(\tau^1\ldots\tau^N\tau^0). \tag{IVд} \]

If charged bosons do not participate in the process, then the operators under the trace sign in (IVд) contain only the matrices \(\tau_0=2\tau_3\). For an odd number of them \(a=a'=0\). Thus, if

\[ N(\tau_{\pm1})=0,\qquad N(\tau_3)=2n+1, \tag{IVе} \]

then

\[ M_1=M_2=0. \tag{IVж} \]

Suppose that \(N(\tau_{\pm 1})\ne 0\). Then, as is easy to show,

\[ a'=(-1)^{N(\tau_3)}a. \tag{IVз} \]

Noting further that \(\operatorname{Sp} A=\operatorname{Sp}\widetilde A\), we may write:

\[ \operatorname{Sp}\{K(0,1)f^1\ldots K(N,0)f^0\}= \]

\[ =\operatorname{Sp}\{\widetilde f^{\,0}\widetilde K(N,0)\ldots \widetilde f^{\,1}\widetilde K(0,1)\}. \tag{IVи} \]

Using (IVв), Table II, and equation (83) of § 4, we find:

\[ \left. \begin{aligned} -\widetilde\gamma_\mu&=C\gamma_\mu C^{-1},\qquad \overline C=\gamma_0 C,\\ f&=\gamma_\alpha\gamma_\beta\ldots f_\chi;\qquad \widetilde f=\gamma_\chi\ldots\gamma_\beta\gamma_\alpha,\\ \widetilde f&=\gamma_\chi\ldots\gamma_\beta\gamma_\alpha=-\gamma_5 f\gamma_5,\\ \widetilde f&=\overline C f\,\overline C^{-1},\\ \widetilde K(l,m)&=\overline C K(l,m)\overline C^{-1}. \end{aligned} \right\} \tag{IVк} \]

Substituting IVк into IVи, we find:

\[ \operatorname{Sp}\{K(0,1)f^1\ldots K(N,0)f^0\}= \]

\[ =\operatorname{Sp}\{\overline C\overline f^{\,0}\overline C^{-1}\overline C K(0,N)\overline C^{-1}\ldots \overline C\overline f^{\,1}\overline C^{-1}\overline C K(1,0)\overline C^{-1}\}. \tag{IVл} \]

Taking into account that always \(\operatorname{Sp}SAS^{-1}=\operatorname{Sp}A\), we obtain:

\[ M_2=\int \operatorname{Sp}\{\overline f^{\,0}K(0,N)\ldots \overline f^{\,1}K(1,0)\}\,d^4p. \tag{IVм} \]

Using (IVб) and Table II, we immediately obtain:

\[ \overline f=-f \tag{IVн} \]

for vector and tensor interactions, and

\[ \overline f=f \tag{IVо} \]

for scalar, pseudoscalar, and pseudovector couplings. On the basis of (IVз), (IVм)—(IVо), we may write

\[ M_1=(-1)^{N(\tau_3)+N(V)+N(T)}M_2, \]

as was required.

If \(N(\tau_{\pm 1})=0\), but \(N(\tau_3)=2n\), then, using (IVм)—(IVо), we obtain

\[ M_1=(-1)^{N(\hat V)+N(T)}M_2 \]

in agreement with Table V.

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