On Some Forgotten Formulas of L. de Broglie and the Nature of Thermal Time

From 1948 until around 1965, Louis de Broglie, awarded the Nobel Prize for Physics in 1929 for his fundamental contributions to quantum theory, pursued a systematic study of the formal analogies between wave mechanics and the thermomechanics of Boltzmann and Helmholtz. As part of this line of research, he produced several interesting observations, which were, however, published only in French, and, therefore, had a very limited diffusion. Here, we reconsider, in particular, a result of his relating to the analogy between the internal clock (de Broglie phase) of a free particle and a cyclic isothermal process in a thermomechanical system. We show that the fundamental equivalence obtained by him can be derived under more convenient hypotheses than the original ones, essentially tied to the quantization of the action exchanged by the particle with a suitable thermostat. In this emended formulation, the relations proposed by de Broglie describe the emergence of the particle proper time from a thermal background. They also suggest a specific physical meaning of the Wick rotation, often used in quantum mechanical calculations, and the thermal time that appears in it.