Galilean Covariance Problem of Matter Waves

Antonio Díaz Miranda

Gustavo Rigolin

We show that the Schr\"odinger equation can be derived assuming the Galilean covariance of a generic wave equation and the validity of the de Broglie's wave-particle duality hypothesis. We also obtain from this set of assumptions the transformation law for the wave function under a Galilean boost and prove that complex wave functions are unavoidable for a consistent description of a physical system. The extension to the relativistic domain of the above analysis is also provided. We show that Lorentz covariance and wave-particle duality are consistent with two different transformation laws for the wave function under a Lorentz boost. This leads to two different wave equations, namely, the Klein-Gordon equation and the Lorentz covariant Schr\"odinger equation.

Morton Hamermesh

Georges Lochak, Regis Dutheil

M. de Montigny, F. C. Khanna, Ademir Eugenio de Santana +2 more

We use a five-dimensional approach to Galilean covariance to investigate the non-relativistic Duffin-Kemmer-Petiau first-order wave equations for spinless particles. The corresponding representation is generated by five 6×6 matrices. We consider the harmonic oscillator as an example.

Paweł Caban, Kordian A. Smoliński, Zbigniew Walczak

We consider a system of two spin-s particles. We show that the reduced density matrix of this system obtained by tracing out over momentum is invariant under Galilean boosts and translations, and covariant under rotations. It implies that the spin entanglement of the system is the same for all Galilean observers.