The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed sufficiently to form a fairly complete theory of dynamics. One can treat mathematically the problem of any dynamical system composed of a number of particles with instantaneous forces acting between them, provided it is describable by a Hamiltonian function, and one can interpret the mathematics physically by a quite definite general method. On the other hand, hardly anything has been done up to the present on quantum electrodynamics. The questions of the correct treatment of a system in which the forces are propagated with the velocity of light instead of instantaneously, of the production of an electromagnetic field by a moving electron, and of the reaction of this field on the electron have not yet been touched. In addition, there is a serious difficulty in making the theory satisfy all the requirements of the restricted　principle of relativity, since a Hamiltonian function can no longer be used. This relativity question is, of course, connected with the previous ones, and it will be impossible to answer any one question completely without at the same time answering them all. However, it appears to be possible to build up a fairly satisfactory theory of the emission of radiation and of the reaction of the radiation field on the emitting system on the basis of a kinematics and dynamics which are not strictly relativistic. This is the main object of the present paper. The theory is noil-relativistic only on account of the time being counted throughout as a c-number, instead of being treated symmetrically with the space co-ordinates. The relativity variation of mass with velocity is taken into account without difficulty.

The underlying ideas of the theory are very simple. Consider an atom interacting with a field of radiation, which we may suppose for definiteness to be confined in an enclosure so as to have only a discrete set of degrees of freedom. Resolving the radiation into its Fourier components, we can consider the energy and phase of each of the components to be dynamical variables describing the radiation field. Thus if E r is the energy of a component labelled r and 0r is the corresponding phase (defined as the time since the wave was in a standard phase), we can suppose each Er and 0,. to form a pair of canonically conjugate variables. In the absence of any interaction between the field and the atom, the whole system of field plus atom will be describable by the HamiltonianH S A + H 0 (1)equal to the total energy, I I 0 being the Hamiltonian for the atom alone, since the variables Er, 0,. obviously satisfy their canonical equations of motion3H0E,When there is interaction between the field and the atom, it could be taken into account on the classical theory by the addition of an interaction term to the Hamiltonian (1), which would be a function of the variables of the atom and of the variables En 0(. that describe the field. This interaction term would give the effect of the radiation on the atom, and also the reaction of the atom on the radiation field.In order that an analogous method may be used on the quantum theory, it is necessary to assume th at the variables Er, 0r are q-numbers satisfying the standard quantum conditions 0rE,. — E,.0f = ih, etc., where h is (27r)-i times the usual Planck’s constant, like the other dynamical variables of the problem.

For if vr is the frequency of the component r, 2-irvr0r is an angle variable, so th at its canonical conjugate Er/2nvr can only assume a discrete set of values differing by multiples of h, which means th at Er can change only by integral multiples of the quantum (2nh) vr. If we now add an interaction term (taken over from the clasical theory) to the Hamiltonian (1), the problem can be solved according to the rules of quantum mechanics, and we would expect to obtain the correct results for the action of the radiation and the atom on one another. It will be shown that we actually get the correct laws for the emission and absorption of radiation, and the correct values for Einstein’s A’s and B’s. In the author’s previous theory, where the energies and phases of the components of radiation were c-numbers, only the B’s could be obtained, and the reaction of the atom on the radiation could not be taken into account.

It will also be shown th at the Hamiltonian which describes the interaction of the atom and the electromagnetic waves can be made identical with the Hamiltonian for the problem of the interaction of the atom with an assembly of particles moving with the velocity of light and satisfying the Einstein-Bose statistics, by a suitable choice of the interaction energy for the particles. The number of particles having any specified direction of motion and energy, which can be used as a dynamical variable in the Hamiltonian for the particles, is equal to the number of quanta of energy in the corresponding wave in the Hamiltonian for the waves. There is thus a complete harmony between the wave and light-quantum descriptions of the interaction. We shall actually build up the theory from the light-quantum point of view, and show that the Hamiltonian transforms naturally into a form which resembles th at for the waves.

I t should be observed th at there is a difference between a light-wave and the de Broglie or Schrodinger wave associated with the light-quanta. Firstly, the light-wave is always real, while the de Broglie wave associated with a light- quantum moving in a definite direction must be taken to involve an imaginary exponential. A more important difference is that their intensities are to be interpreted in different ways. The number of light-quanta per unit volume associated with a monochromatic light-wave equals the energy per unit volume of the wave divided by the energy (2tth)vof a single light-quantum. On the other hand a monochromatic de Broglie wave of amplitude a (multiplied into the imaginary exponential factor) must be interpreted as representing 2 light- quanta per unit volume for all frequencies. This is a special case of the general rule for interpreting the matrix analysis,* according to which, if (£'/a') or <]v Cik) is the eigenfunction in the variables of the state a' of an atomic system (or simple particle), | <jv ( | fc')|2 is the probability of each having the value [or | 4 'ad^k)12 dZf••• is the probability of each lying betweenthe values Ef and Ek -j~ dEDk, when the have continuous ranges of characteristic values] on the assumption that all phases of the system are equally probable. The wave whose intensity is to be interpreted in the first of these two ways appears in the theory only when one is dealing with an assembly of the associated particles satisfying the Einstein-Bose statistics. There is thus no such wave associated with electrons.

We shall now consider the interaction of an atom and radiation from the wave point of view. We resolve the radiation into its Fourier components, and suppose that their number is very large but finite. Let each component be labelled by a suffix r, and suppose there are ar components associated with the Tadiation of a definite type of polarisation per unit solid angle per unit frequency range about the component r. Each component can be described by a vector potential tcT chosen so as to make the scalar potential zero. The perturbation term to be added to the Hamiltonian will now be, according to the classical theory with neglect of relativity mechanics, c-1 S r kt X f, where Xr is the component of the total polarisation of the atom in the direction of y , which is the direction of the electric vector of the component r.

The interaction of an atom with electromagnetic waves is then considered, and it is shown that if one takes the energies and phases of the waves to be q-numbers satisfying the proper quantum conditions instead of c-numbers, the Hamiltonian function takes the same form as in the light-quantum treatment. The theory leads to the correct expressions for Einstein’s A’s and B’s.

I would like to express my thanks to Prof. Niels Bohr for his interest in this work and for much friendly discussion about it.