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I would like to ask about the process of measuring the Spin of a Dirac 4-spinor Ψ that is not in the rest frame.

Note that even though there is plenty of information about what a Dirac spinor is, what reasoning lead to its discovery and how it can be expressed in terms of particle and antiparticle solutions, there are very few examples of measuring Spin when the spinor particle is moving.

Let v

_{1}be the 3-vector representing the direction in which the spinor particle Ψ is moving and v

_{2}the direction in which Spin is being measured. The process that I think I would have to follow to get the probability amplitude of finding the particle in the +½ state would be:

1- Use a linear combination of gamma matrices γ

^{i}to build the 4x4 matrix M that measures Spin on the v

_{2}direction. For instance, if v

_{2}is proportional to (x,y,z)=(1,2,5) then M would be proportional to (γ

^{1},2⋅γ

^{2},5⋅γ

^{3}).

2- Obtain the eigenvectors of that matrix. Those eigenvectors would represent particles and antiparticles moving in the v

_{2}direction with definite spin (+½ or -½). Actually these solutions would be the so-called helicity eigenstates, their projection of spin onto vector v

_{2}is ±½.

3- Express the spinor Ψ as a linear combination of the eigenvectors in step 2.

4- The probability amplitude of measuring +½ would be |a|

^{2}+ |b|

^{2}, where a and b represent the complex factor multiplying the +½ particle and antiparticle helicity eigenstate respectevely.

Is what I explained above correct?

Many thanks in advance.