Editing Four-vector (section)

==Quantum theory==

===Four-probability current===

In [[quantum mechanics]], the four-[[probability current]] or probability four-current is analogous to the [[Four-current|electromagnetic four-current]]:<ref>Vladimir G. Ivancevic, Tijana T. Ivancevic (2008) ''Quantum leap: from Dirac and Feynman, across the universe, to human body and mind''. World Scientific Publishing Company, {{ISBN|978-981-281-927-7}}, [https://books.google.com/books?id=qyK95FevVbIC&pg=PA41 p. 41]</ref>
<math display="block">\mathbf{J} = (\rho c, \mathbf{j}) </math>
where {{math|''ρ''}} is the [[probability density function]] corresponding to the time component, and {{math|'''j'''}} is the [[probability current]] vector. In non-relativistic quantum mechanics, this current is always well defined because the expressions for density and current are positive definite and can admit a probability interpretation. In [[relativistic quantum mechanics]] and [[quantum field theory]], it is not always possible to find a current, particularly when interactions are involved.

Replacing the energy by the [[energy operator]] and the momentum by the [[momentum operator]] in the four-momentum, one obtains the [[four-momentum operator]], used in [[relativistic wave equation]]s.

===Four-spin===

The [[four-spin]] of a particle is defined in the rest frame of a particle to be
<math display="block">\mathbf{S} = (0, \mathbf{s})</math>
where {{math|'''s'''}} is the [[Spin (physics)|spin]] pseudovector. In quantum mechanics, not all three components of this vector are simultaneously measurable, only one component is. The timelike component is zero in the particle's rest frame, but not in any other frame. This component can be found from an appropriate Lorentz transformation.

The norm squared is the (negative of the) magnitude squared of the spin, and according to quantum mechanics we have
<math display="block">\|\mathbf{S}\|^2 = -|\mathbf{s}|^2 = -\hbar^2 s(s + 1)</math>

This value is observable and quantized, with {{math|''s''}} the [[spin quantum number]] (not the magnitude of the spin vector).