Interpreting the Dirac equation

[snoopies622]

Why does the $\psi$ of the Dirac equation return four complex numbers instead of one, as in the Schrödinger equation? I know it has something to do with spin, but I'm not finding a clear answer to this question in my sources. What do these four complex numbers represent?

 

[The_Duck]

The usual wave function that get by solving the Schrödinger equation tells us the amplitude to find a particle at a given position. With a spin-1/2 particle in a relativistic theory, though, you need four amplitudes at each position: the amplitude to find a spin-up electron, the amplitude to find a spin-down electron, the amplitude to find a spin-up positron, and the amplitude to find a spin-down positron.

 

[snoopies622]

Wow, how simple! Thanks The Duck.

 

[akhmeteli]

Why does the $\psi$ of the Dirac equation return four complex numbers instead of one, as in the Schrödinger equation? I know it has something to do with spin, but I'm not finding a clear answer to this question in my sources. What do these four complex numbers represent?

Actually, in a general case (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf ). Furthermore, this remaining component can be made real by a gauge transform. So you can replace the Dirac equation by an equivalent equation for just one complex or real function.

 

[vanhees71]

Well, the explanation of The_Duck was simple, but unfortunately not fully correct. The reason is that a single-particle wave-function interpretation of relativistic wave functions is only approximately possible, because for interacting particles at relativistic energies there's always the possibility that new particles get created or particles are annihilated leading to new other particles, etc. Thus the only correct interpretation is a many-body theory, and this is most conveniently described as a quantum field theory.

For (asymptotically) free single-particle states the interpretation is however correct. The Dirac field describes charged particles of spin 1/2 (2 field degrees of freedom) and their corresponding antiparticles of also spin 1/2 (2 field degrees of freedom).

 

[kith]

Also note that in non-relativistic QM, we have the Pauli equation if we want to include spin. There, ψ has two components.

 

[snoopies622]

So the Dirac equation is consistent with the Minkowski metric but says nothing about the creation and annihilation of particles?