Jordan ’ s resolution of the conundrum of the wave - particle duality of light . ⋆

In 1909, Einstein derived a formula for the mean square energy fluctuation in a subvolume of a box filled with black-body radiation. This formula is the sum of a wave term and a particle term. Einstein concluded that a satisfactory theory of light would have to combine aspects of a wave theory and a particle theory. In a key contribution to the 1925 Dreimännerarbeit with Born and Heisenberg, Jordan used Heisenberg’s new Umdeutung procedure to quantize the modes of a string and argued that in a small segment of the string, a simple model for a subvolume of a box with black-body radiation, the mean square energy fluctuation is described by a formula of the form given by Einstein. The two terms thus no longer require separate mechanisms, one involving particles and one involving waves. Both terms arise from a single consistent dynamical framework. Jordan’s derivation was later criticized by Heisenberg and others and the lingering impression in the subsequent literature is that infinities of one kind or another largely invalidate Jordan’s conclusions. In this paper, we carefully reconstruct Jordan’s derivation and reexamine some of the objections raised against it. Jordan could certainly have presented his argument more clearly. His notation, for instance, fails to bring out that various sums over modes of the string need to be restricted to a finite frequency range for the final result to be finite. His derivation is also incomplete. In modern terms, Jordan only calculated the quantum uncertainty in the energy of a segment of the string in an eigenstate of the Hamiltonian for the string as a whole, while what is needed is the spread of this quantity in a thermal ensemble of such states. These problems, however, are easily fixed. Our overall conclusion then is that Jordan’s argument is basically sound and that he deserves credit for resolving a major conundrum in the development of quantum physics.