De Broglie relation

Four-vectors

Using four-vectors, the de Broglie relations form a single equation: $${\displaystyle \mathbf {P} =\hbar \mathbf {K} ,}$$ which is frame-independent. Likewise, the relation between group/particle velocity and phase velocity is given in frame-independent form by: $${\displaystyle \mathbf {K} =\left({\frac {\omega _{0}}{c^{2}}}\right)\mathbf {U} ,}$$ where