The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplicationi, 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 systemn in which the forces are propa- gated with the velocity of light instead of instanitaneously, of the produLction 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 miaking 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 non-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 inter- acting 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.
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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 Er, Or 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 that the variables Er, Or are q-numbers satisfying the standard quantum conditions OrEr- ErOr = ,h, etc., where h is (27)- times the usual Planck's constant, like the other dynamical variables of the problem. This assumption immediately gives light-quantum properties to
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It will also be shown that 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 Hiamiltonian for the waves. There is thus a complete harmony between the wave and liglit-quantum descriptions of the interaction. We shall actually build up the theory fromrl the light-quantum point of view, and show that the Hamiltonian transforms naturally into a form which resembles that for the waves.
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