Proving the Spin Half Nature of Dirac Quanta

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Discussion Overview

The discussion revolves around the proof of the spin half nature of Dirac quanta when quantized, exploring various methods and concepts in quantum field theory (QFT) and relativistic quantum mechanics (QM). Participants express confusion and seek clarity on the derivation and implications of spin in different fields.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about proving that the Dirac field describes spin half quanta when quantized and requests references for derivations.
  • Another participant suggests taking the nonrelativistic limit of the Dirac equation to recover the spin-term in the Hamiltonian, indicating that fermions must have half-integer spin due to the Spin-Statistics Theorem.
  • A question is raised about how to demonstrate that fields with higher spins (spin 1, spin 3/2, spin 2) describe quanta with corresponding intrinsic angular momenta.
  • It is proposed that counting the number of polarization states can indicate how many components a field has, linking this to the transformation properties under Lorentz transformations.
  • A correction is made regarding the transformation properties of spinor-vector fields, with a participant asserting that it transforms as a spin-3/2 object, not 5/2.
  • Another participant seeks clarification on demonstrating spin using quantum field theory in terms of operators acting on Fock space.
  • A later reply acknowledges a previous error regarding the transformation of spinor-vector fields and suggests computing angular momentum by studying the action of the spin operator on the field.

Areas of Agreement / Disagreement

Participants express differing views on the transformation properties of fields and the methods to demonstrate spin characteristics, indicating that multiple competing views remain and the discussion is unresolved.

Contextual Notes

Some participants reference specific texts for further reading, but there is no consensus on a singular method or derivation for proving the spin nature of Dirac quanta or higher spin fields.

quantumfireball
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but I am confused
how do you proof that the dirac field describes spin half quanta when quantized?
please refer me to a link on the net where this derivation is shown if possible
i can't find it in any of the books on quantized field theory
 
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One way you can do it is to take the nonrelativistic limit of the Dirac equation and recover the spin-term in the hamiltonian. then you'll see explicitly that it's s=1/2. see any text on relativistic QM.

In QFT, you usually go the other way around: assume that it's a fermion and you get the Dirac equation. Fermions have to have half-integer spin thanks to the famous "Spin-Statistics Theorem". I believe this was proved first by Pauli, but I'm not sure. I'm sure Weinberg does it in his QFT tome.
 
well what about spin a1 fields,spin 3-2 fields,spin 2 fields
how does one go about showing that they describe quanta of that much amount of intrinsic angular momenta
 
count the number of polarization states (this follows from the field equations); that tells you how many components the field has. You know that a field with N components must transform under Lorentz transformations the same way as an object with spin (N-1)/2 - this is just ordinary quantum mechanics.
 
blechman said:
You know that a field with N components must transform under Lorentz transformations the same way as an object with spin (N-1)/2

Not correct! spinor-vector has six components, it transforms as spin-3/2 object not 5/2.

regards

sam
 
but how to show it using quantum field theory?
that is in terms of an operator which acts on a fock space having eigenvalue=sqrt(s(s+1))
times the number of quanta in the fock space
im totally confused
 
samalkhaiat said:
blechman said:
Not correct! spinor-vector has six components, it transforms as spin-3/2 object not 5/2.

regards

sam

you're right. i was too sloppy with this comment. i retract it.

QFB: you can compute the angular momentum by studying the action of the spin operator on the field. Weinberg Vol 1 talks about this.
 
thanks Dr Belchman
 

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