Measuring the spin of a moving Dirac spinor particle

[Alhaurin]

Hello,

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 v₁ be the 3-vector representing the direction in which the spinor particle Ψ is moving and v₂ 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 γⁱ to build the 4x4 matrix M that measures Spin on the v₂ direction. For instance, if v₂ is proportional to (x,y,z)=(1,2,5) then M would be proportional to (γ¹,2⋅γ²,5⋅γ³).

2- Obtain the eigenvectors of that matrix. Those eigenvectors would represent particles and antiparticles moving in the v₂ direction with definite spin (+½ or -½). Actually these solutions would be the so-called helicity eigenstates, their projection of spin onto vector v₂ 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.

 

[vanhees71]

I've no clue how to measure anyting of a Dirac spinor. It's a mathematical object. To be precise, it's a field operator with a well defined transformation behavior under Poincare transformations (including spatial reflections).

Historically Dirac came to his equation by trying to find (a) a wave equation of first order in the time derivative due to the fact that Schrödinger's non-relativistic wave equation was successful in describing non-relativistic particles and (b) find a wave function for particles with spin (the equivalent of the Pauli equation in non-relativistic QM). It's not the best way to understand the logic from this historical procedure. It's only an amazing demonstration of Dirac's ingenious intuition about physics.

Today we can explain relativistic QT starting from symmetry principles in a quite logical way. As it turns out, if you want a representation of the Poincare group for spin-1/2 particles including the possibility for spatial reflections within a local microcausal relativistic field theory one straight-forward way is the quantized Dirac field. The other possibility are Majorana spinors.

 

[Alhaurin]

Thank you for your explanation.

However I do not understand how it is not possible to measure something of a Dirac spinor in relativistic QT. In that theory, the 4 component Dirac spinor represents the wave function of the electron and it should be possible to measure spin (among other things) and calculate corresponding probability amplitudes. All of that independently of the fact that QFT is the ultimate description of particle physics and Dirac's theory just an approximation.

 

[vanhees71]

You can measure properties of quantum particles, e.g., electrons. You cannot measure spinors, but maybe that's just semantics.

In relativistic QT there's no way to properly interpret the unquantized Dirac field as a "wave function". Only the QFT formulation ("2nd quantization") leads to positive definite probabilities and an energy bounded from below.

You can circumvent these issues within the 1st-quantization formalism in a very complicated way, known as "Dirac's hole formulation", but this has been worked out only for QED, and it's so utterly more complicated than the "modern" formulation as QFT that nobody bothers to try to find a hole-theoretical formulation of the Standard Model.