An Application of a New Limitation in Physical Theory

Abstract

IN attempts to understand the relation between the quantum theory and the theories of relativity and electro-magnetism, a certain limitation, applicable to the motion of a particle of electromagnetic mass m₀and of charge, is brought to light. It appears that the expression (m₀c²d e/c_mdx^m) can assume only those values which are multiples of Planck's constant¹. d is an element of proper time associated with the track of the particle, dx^m is a component of displacement and_m is a component of electromagnetic potential. An interesting result is obtained if this be applied to an electron in an electrostatic field of potential Ne/r. This is the case with an electron in an atomic orbit where N is the number of the atom. The above condition then implies (m₀c²² Ne²/r)dt ® h where = 12;/c, ½ being the velocity of the particle, and where account is taken of the fact that the nuclear and electronic charges are of opposite sign. If we consider the case of the K-level of an atom and make use of Sommerfeld's value for Ne²/r, which may be regarded as a sufficient approximation, we obtain m₀c² (^{2 2}/=²) dt ® h. In this case = Nα, where is the fine-structure constant 2e²/hc. We note that as approaches the value 1/2 the factor of dt approaches zero very rapidly, and the limitation states that the least possible value of dt is very large. We interpret this as an indication of the breakdown of the description of the charge as a particle in motion. This gives a clue to the nature of the limitation. It provides us at each point of space and time with a criterion for the dynamical description of an electric charge.