Measuring the spin of a moving Dirac spinor particle

In summary, the process of measuring the spin of a Dirac 4-spinor Ψ that is not in the rest frame would involve measuring the eigenvectors of a matrix that measures Spin on the v2 direction.
  • #1
Alhaurin
7
0
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

1. What is a Dirac spinor particle?

A Dirac spinor particle is a type of elementary particle that has intrinsic angular momentum, or spin. It was first proposed by physicist Paul Dirac in the 1920s as a way to reconcile the principles of quantum mechanics with the theory of relativity.

2. How do you measure the spin of a moving Dirac spinor particle?

The spin of a moving Dirac spinor particle can be measured using a device called a Stern-Gerlach apparatus. This apparatus uses a magnetic field to split the particle's spin states into two distinct paths, which can then be detected and measured.

3. What is the significance of measuring the spin of a Dirac spinor particle?

Measuring the spin of a Dirac spinor particle can provide valuable information about its quantum state, such as its energy level and angular momentum. This information is crucial for understanding the behavior and properties of these particles.

4. Are there any challenges in measuring the spin of a moving Dirac spinor particle?

Yes, there are some challenges in measuring the spin of a moving Dirac spinor particle. One of the main challenges is ensuring that the particle's spin state is not disturbed or altered during the measurement process. Additionally, the sensitivity of the measurement equipment must be high enough to accurately detect the split spin states.

5. Can the spin of a moving Dirac spinor particle be changed or manipulated?

Yes, the spin of a moving Dirac spinor particle can be changed or manipulated using a variety of techniques, such as applying external electromagnetic fields or using quantum entanglement. This ability to manipulate spin states is an essential aspect of quantum computing and other advanced technologies.

Similar threads

Replies
2
Views
862
  • Quantum Physics
Replies
7
Views
2K
Replies
2
Views
924
Replies
26
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
508
Replies
1
Views
744
  • Quantum Physics
Replies
12
Views
1K
Replies
2
Views
1K
Replies
8
Views
2K
Replies
4
Views
1K
Back
Top