This is important for probability interpretations, exemplified below. This equation is also true in Relativistic quantum mechanics pdf, provided the Heisenberg operators are modified to be consistent with SR. The derivative operators, and hence the energy and 3-momentum operators, are also non-invariant and change under Lorentz transformations.
Schrödinger equation gives a non-relativistic QM equation for the wavefunction: the procedure is a straightforward substitution of a simple expression. Again, there is the problem of the non-invariance of the energy operator, equated to the square root which is also not invariant. There is also the problem of incorporating spin in the Hamiltonian, which isn’t a prediction of the non-relativistic Schrödinger theory. Schrödinger in 1925 before he found the non-relativistic equation named after him, and by Klein and Gordon in 1927, who included electromagnetic interactions in the equation. Each factor is relativistically invariant.
The positive mass equation can continue to be used without loss of continuity. KG equation, but must instead be a four-component entity. Probability density and current also satisfy a continuity equation because probability is conserved, however this is only possible in the absence of interactions. Including interactions in RWEs is generally difficult.
In the case where the charge is zero, the equation reduces trivially to the free KG equation so nonzero charge is assumed below. Such solutions cannot in general describe particles with nonzero spin since spin components are not independent. Other constraint will have to be imposed for that, e. It does, however, correctly describe charged spinless bosons in the absence of other interactions. The positive energies do account accurately for the fine structure. This is, however, still only an approximation, and the Hamiltonian includes numerous long and complicated sums. Helicity indicates the orientations of the spin and translational momentum vectors.
Helicity is frame-dependent because of the 3-momentum in the definition, and is quantized due to spin quantization, which has discrete positive values for parallel alignment, and negative values for antiparallel alignment. Pauli rederived the same equation. 1948 using Lorentz group theory, applicable for all free particles with any spin. According to the relativistic energy-momentum relation, all massless particles travel at the speed of light, so particles traveling at the speed of light are also described by two-component spinors. 1913, prior to the spinors revealed in the RWEs following the year 1927. For equations describing higher-spin particles, the inclusion of interactions is nowhere near as simple minimal coupling, they lead to incorrect predictions and self-inconsistencies. 1 particles magnetic quadrupoles and electric dipoles are also possible.