240 
THE QUANTUM [xvii. 5 
and by others.* It has been shown by Ehrenfest and Uhlenbeckf 
that a graphical representation of the phase waves of de Broglie 
may be obtained in the five-dimensional universe. 
A remarkable result which indicates an inner unity between 
the quantum theory and gravitational relativity has been 
announced by Wiener and Struik.J The connecting link is the 
wave theory of Schrodinger. If we define the gravitational field 
in the proper invariantive manner in terms of a wave equation, 
the quantization of this equation follows from the gravitational 
field equations. The equation also defines an electromagnetic 
potential, to which most of Weyl’s considerations apply. 
The fundamental equation of these authors can be rendered 
homogeneous by means of a suitable substitution, and a treat 
ment of their theory is then obtained analogous to that of Klein. 
“ The fifth dimension turns out to be a mere mathematical 
convention that can be compared to the introduction of homo 
geneous co-ordinates in other parts of mathematics.” 
One test of a scientific theory is its comprehensiveness, and 
the wide sweep of the new quantum mechanics is shown not only 
in the various ways of formulating it in mathematical language, 
but also in the physical ideas that may be associated with it. 
It is probable that the views of the quantum suggested by 
Whittaker and by the present writer, in which its magnetic 
aspects are emphasized, may be simply related to the new theory. 
In a recent paper Whittaker has described a simple light quantum 
in which a disembodied magnetic molecule, travelling with the 
speed of light, forms a singularity on the wave front and confers 
upon it the desired quantum properties. It may be suggested 
that such a quantum is related to a quantum magnetic tube on 
the one hand, and to Schròdinger’s wave mechanics on the other. 
5. Statistical Methods and the Quantum Theory 
The application of probability methods in quantum theory 
has attracted considerable attention since the publication of an 
important paper by Einstein § in 1917. He showed that Planck’s 
radiation law could be derived by considering the probability 
of transitions between different stationary states, assuming Bohr’s 
frequency condition to hold for the radiation emitted or absorbed 
in a transition. He considered an enclosure containing a gas, 
* Klein, Zeits. f. Physik, voi. 37, p. 895, 1926 ; Fock, Zeits. f. Physik, 
voi. 39, p. 226, 1926 ; Gamow and Iwanenko, Zeits. f. Physik, voi. 39, 
p. 865, 1926; Iwanenko and Landau, Zeits. f. Physik, voi. 40, p. 161, 1926. 
f Ehrenfest and Uhlenbeck, Zeits. f. Physik, voi. 39, p. 495, 1926. 
j Wiener and Struik, Nature, voi. 119, p. 853, 1927. 
§ Einstein, Phys. Zeits., voi. 18, p. 122, 1917. See Birtwistle, Quantum 
Theory, p. 36.