Einstein’s special relativity and the continuation of the theory beyond the speed of light predict a symmetry between the sub-luminal (particle) and superluminal (wave) worlds such that each is the mirror image of the other. One consequence of this fact is that basic physical parameters might depend on a combination of both particle and wave characteristics. A particular implication is the existence of a consistent extension of the Planck–de Broglie energy–momentum relations for the particle–wave duality of matter involving a second fundamental constant \(h'\) which is additional to the Planck constant h. Here we show that we may generalise the Planck–de Broglie relations for energy \(e = h\nu \) and momentum \(p =h/\lambda \) in an entirely consistent manner, where \(\nu \) is the wave frequency and \(\lambda \) is the wavelength by the expressions \(e = h\nu + h^{\prime }c/\lambda \) and \(p =h/\lambda + h^{\prime }\nu /c\) , where \(h'\) is a new fundamental constant. The new terms come into play when the wave is not travelling at the speed of light since in that case \(\lambda \nu = c\) and the new terms merge with the existing expressions. We show that the ratio \(\kappa = h^{\prime }/h\) connects with a Lorentz invariant particle energy expression which is a generalisation of Einstein’s energy expression.
