The most important and most simple crucial example for a nucleus with a different parity of M and Z is the heavy hydrogen or deuteron with M = 2 and Z = 1 which has symmetrical states and the spin I = 1, as could be proved by the investigation of the band spectra of a molecule with two deuterons19. From the spin value I of the deuteron can be concluded that the neutron must have a half-integer spin. The simplest possible assumption that this spin of the neutron is equal to , just as the spin of the proton and of the electron, turned out to be correct.
There is hope, that further experiments with light nuclei, especially with protons, neutrons, and deuterons will give us further information about the nature of the forces between the constituents of the nuclei, which, at present, is not yet sufficiently clear. Already now we can say, however, that these interactions are fundamentally different from electromagnetic interactions. The comparison between neutron-proton scattering and proton-proton scattering even showed that the forces between these particles are in good approximation the same, that means independent of their electric charge. If one had only to take into account the magnitude of the interaction energy, one should therefore expect a stable di-proton or (M = 2, Z = 2) with nearly the same binding energy as the deuteron. Such a state is, however, forbidden by the exclusion principle in accordance with experience, because this state would acquire a wave function symmetric with respect to the two protons. This is only the simplest example of the application of the exclusion principle to the structure of compound nuclei, for the understanding of which this principle is indispensable, because the constituents of these heavier nuclei, the protons and the neutrons, fullfil it.
In order to prepare for the discussion of more fundamental questions, we want to stress here a law of Nature which is generally valid, namely, the connection between spin and symmetry class. A half-integer value of the spin quantum number is always connected with antisymmetrical states (exclusion principle), an integer spin with symmetrical states. This law holds not only for protons and neutrons but also for protons and electrons. Moreover, it can easily be seen that it holds for compound systems, if it holds for all of its constituents. If we search for a theoretical explanation of this law, we must pass to the discussion of relativistic wave mechanics, since we saw that it can certainly not be explained by non-relativistic wave mechanics.
We first consider classical fields20, which, like scalars, vectors, and tensors transform with respect to rotations in the ordinary space according to a one-valued representation of the rotation group. We may, in the following, call such fields briefly “one-valued” fields. So long as interactions of different kinds of field are not taken into account, we can assume that all field components will satisfy a second-order wave equation, permitting a superposition of plane waves as a general solution. Frequency and wave number of these plane waves are connected by a law which, in accordance with De Broglie’s fundamental assumption, can be obtained from the relation between energy and momentum of a particle claimed in relativistic mechanics by division with the constant factor equal to Planck’s constant divided by 2π. Therefore, there will appear in the classical field equations, in general, a new constant µ with the dimension of a reciprocal length, with which the rest-mass m in the particle picture is connected by m = h µ/c, where c is the vacuum-velocity of light. From the assumed property of one-valuedness of the field it can be concluded, that the number of possible plane waves for a given frequency, wave number and direction of propagation, is for a non-vanishing µ always odd. Without going into details of the general definition of spin, we can consider this property of the polarization of plane waves as characteristic for fields which, as a result of their quantization, give rise to integer spin values.
The simplest cases of one-valued fields are the scalar field and a field consisting of a four-vector and an antisymmetric tensor like the potentials and field strengths in Maxwell’s theory. While the scalar field is simply fulfilling the usual wave equation of the second order in which the term proportional to µ2 has to be included, the other field has to fulfill equations due to Proca which are a generalization of Maxwell’s equations which become in the particular case µ = 0. It is satisfactory that for these simplest cases of one-valued fields the energy density is a positive definite quadratic form of the field-quantities and their first derivatives at a certain point. For the general case of one-valued fields it can at least be achieved that the total energy after integration over space is always positive.
The field components can be assumed to be either real or complex. For a complex field, in addition to energy and momentum of the field, a four-vector can be defined which satisfies the continuity equation and can be interpreted as the four-vector of the electric current. Its fourth component determines the electric charge density and can assume both positive and negative values. It is possible that the charged mesons observed in cosmic rays have integral spins and thus can be described by such a complex field. In the particular case of real fields this four-vector of current vanishes identically.
Especially in view of the properties of the radiation in the thermodynamical equilibrium in which specific properties of the field sources do not play any role, it seemed to be justified first to disregard in the formal process of field quantization the interaction of the field with the sources. Dealing with this problem, one tried indeed to apply the same mathematical method of passing from a classical system to a corresponding system governed by the laws of quantum mechanics which has been so successful in passing from classical point mechanics to wave mechanics. It should not be forgotten, however, that a field can only be observed with help of its interaction with test bodies which are themselves again sources of the field.
The result of the formal process of field quantization were partly very encouraging. The quantized wave fields can be characterized by a wave function which depends on an infinite sequence of (nonnegative) integers as variables. As the total energy and the total momentum of the field and, in case of complex fields, also its total electric charge turn out to be linear functions of these numbers, they can be interpreted as the number of particles present in a specified state of a single particle. By using a sequence of configuration spaces with a different number of dimensions corresponding to the different possible values of the total number of particles present, it could easily be shown that this description of our system by a wave function depending on integers is equivalent to an ensemble of particles with wave functions symmetrical in their configuration spaces.
Moreover Bohr and Rosenfeld21 proved in the case of the electromagnetic field that the uncertainty relations which result for the average values of the field strengths over finite space-time regions from the formal commutation rules of this theory have a direct physical meaning so long as the sources can be treated classically and their atomistic structure can be disregarded. We emphasize the following property of these commutation rules: All physical quantities in two world points, for which the four-vector of their joining straight line is spacelike commute with each other. This is indeed necessary for physical reasons because any disturbance by measurements in a world point Pν can only reach such points P2, for which the vector P1P2, is timelike, that is, for which c (t1 − t2) > r12. The points P2 with a spacelike vector P1P2 for which c (t1 − t2) < r12 cannot be reached by this disturbance and measurements in P1 and P2 can then never influence each other.
This consequence made it possible to investigate the logical possibility of particles with integer spin which would obey the exclusion principle. Such particles could be described by a sequence of configuration spaces with different dimensions and wave functions antisymmetrical in the coordinates of these spaces or also by a wave function depending on integers again to be interpreted as the number of particles present in specified states which now can only assume the values 0 or 1. Wigner and Jordan22 proved that also in this case operators can be defined which are functions of the ordinary spacetime coordinates and which can be applied to such a wave function. These op
erators do not fulfil any longer commutation rules: instead of the difference, the sum of the two possible products of two operators, which are distinguished by the different order of its factors, is now fixed by the mathematical conditions the operators have to satisfy. The simple change of the sign in these conditions changes entirely the physical meaning of the formalism. In the case of the exclusion principle there can never exist a limiting case where such operators can be replaced by a classical field. Using this formalism of Wigner and Jordan I could prove under very general assumptions that a relativistic invariant theory describing systems of like particles with integer spin obeying the exclusion principle would always lead to the non-commutability of physical quantities joined by a spacelike vector23. This would violate a reasonable physical principle which holds good for particles with symmetrical states. In this way, by combination of the claims of relativistic invariance and the properties of field quantization, one step in the direction of an understanding of the connection of spin and symmetry class could be made.
The quantization of one-valued complex fields with a non-vanishing four-vector of the electric current gives the further result that particles both with positive and negative electric charge should exist and that they can be annihilated and generated in external electromagnetic field22. This pair-generation and annihilation claimed by the theory makes it necessary to distinguish clearly the concept of charge density and of particle density. The latter concept does not occur in a relativistic wave theory either for fields carrying an electric charge or for neutral fields. This is satisfactory since the use of the particle picture and the uncertainty relations (for instance by analyzing imaginative experiments of the type of the γ -ray microscope) gives also the result that a localization of the particle is only possible with limited accuracy24. This holds both for the particles with integer and with half-integer spins. In a state with a mean value E of its energy, described by a wave packet with a mean frequency ν = E/h, a particle can only be localized with an error Δ x > hc/E or Δ x > c/v. For photons, it follows that the limit for the localization is the wavelength; for a particle with a finite rest mass m and a characteristic length µ−1 = ħ/mc , this limit is in the rest system of the center of the wave packet that describes the state of the particles given by Δx > ħ/mc or Δx > µ−1.
Until now I have mentioned only those results of the application of quantum mechanics to classical fields which are satisfactory. We saw that the statements of this theory about averages of field strength over finite spacetime regions have a direct meaning while this is not so for the values of the field strength at a certain point. Unfortunately in the classical expression of the energy of the field there enter averages of the squares of the field strengths over such regions which cannot be expressed by the averages of the field strengths themselves. This has the consequence that the zero-point energy of the vacuum derived from the quantized field becomes infinite, a result which is directly connected with the fact that the system considered has an infinite number of degrees of freedom. It is clear that this zero-point energy has no physical reality, for instance it is not the source of a gravitational field. Formally it is easy to subtract constant infinite terms which are independent of the state considered and never change; nevertheless it seems to me that already this result is an indication that a fundamental change in the concepts underlying the present theory of quantized fields will be necessary.
In order to clarify certain aspects of relativistic quantum theory I have discussed here, different from the historical order of events, the one-valued fields first. Already earlier Dirac25 had formulated his relativistic wave equations corresponding to material particles with spin using a pair of so-called spinors with two components each. He applied these equations to the problem of one electron in an electromagnetic field. In spite of the great success of this theory in the quantitative explanation of the fine structure of the energy levels of the hydrogen atom and in the computation of the scattering cross section of one photon by a free electron, there was one consequence of this theory which was obviously in contradiction with experience. The energy of the electron can have, according to the theory, both positive and negative values, and, in external electromagnetic fields, transitions should occur from states with one sign of energy to states with the other sign. On the other hand there exists in this theory a four-vector satisfying the continuity equation with a fourth component corresponding to a density which is definitely positive.
It can be shown that there is a similar situation for all fields, which, like the spinors, transform for rotations in ordinary space according to two-valued representations, thus changing their sign for a full rotation. We shall call briefly such quantities “two-valued”. From the relativistic wave equations of such quantities one can always derive a four-vector bilinear in the field components which satisfies the continuity equation and for which the fourth component, at least after integration over the space, gives an essentially positive quantity. On the other hand, the expression for the total energy can have both the positive and the negative sign.
Is there any means to shift the minus sign from the energy back to the density of the four-vector? Then the latter could again be interpreted as charge density in contrast to particle density and the energy would become positive as it ought to be. You know that Dirac’s answer was that this could actually be achieved by application of the exclusion principle. In his lecture delivered here in Stockholm10 he himself explained his proposal of a new interpretation of his theory, according to which in the actual vacuum all the states of negative energy should be occupied and only deviations of this state of smallest energy, namely holes in the sea of these occupied states are assumed to be observable. It is the exclusion principle which guarantees the stability of the vacuum, in which all states of negative energy are occupied. Furthermore the holes have all properties of particles with positive energy and positive electric charge, which in external electromagnetic fields can be produced and annihilated in pairs. These predicted positrons, the exact mirror images of the electrons, have been actually discovered experimentally.
The new interpretation of the theory obviously abandons in principle the standpoint of the one-body problem and considers a many-body problem from the beginning. It cannot any longer be claimed that Dirac’s relativistic wave equations are the only possible ones but if one wants to have relativistic field equations corresponding to particles, for which the value of their spin is known, one has certainly to assume the Dirac equations. Although it is logically possible to quantize these equations like classical fields, which would give symmetrical states of a system consisting of many such particles, this would be in contradiction with the postulate that the energy of the system has actually to be positive. This postulate is fulfilled on the other hand if we apply the exclusion principle and Dirac’s interpretation of the vacuum and the holes, which at the same time substitutes the physical concept of charge density with values of both signs for the mathematical fiction of a positive particle density. A similar conclusion holds for all relativsitic wave equations with two-valued quantities as field components. This is the other step (historically the earlier one) in the direction of an understanding of the connection between spin and symmetry class.
I can only shortly note that Dirac’s new interpretation of empty and occupied states of negative energy can be formulated very elegantly with the help of the formalism of Jordan and Wigner mentioned before. The transition from the old to the new interpretation of the theory can indeed be carried through simply by interchanging the meaning of one of the operators with that of its hermitian conjugate if they are applied to states originally of negative energy. The infinite “zero charge” of the occupied states of negative energy is then formally analogous to the infinite zero-point energy of the quantized one-valued fields. The former has no physicial reality either and is not the source of an electromagnetic field.
In spite of the formal analogy between the quantization of the one-valued fields leading
to ensembles of like particles with symmetrical states and to particles fulfilling the exclusion principle described by two-valued operator quantities, depending on space and time coordinates, there is of course the fundamental difference that for the latter there is no limiting case, where the mathematical operators can be treated like classical fields. On the other hand we can expect that the possibilities and the limitations for the applications of the concepts of space and time, which find their expression in the different concepts of charge density and particle density, will be the same for charged particles with integer and with half-integer spins.
The difficulties of the present theory become much worse, if the interaction of the electromagnetic field with matter is taken into consideration, since the well-known infinities regarding the energy of an electron in its own field, the so-called self-energy, then occur as a result of the application of the usual perturbation formalism to this problem. The root of this difficulty seems to be the circumstance that the formalism of field quantization has only a direct meaning so long as the sources of the field can be treated as continuously distributed, obeying the laws of classical physics, and so long as only averages of field quantities over finite space-time regions are used. The electrons themselves, however, are essentially non-classical field sources.
The Dreams That Stuff is Made of Page 41