Commutation between spin-operator and creation operator(QFT)

In summary, the conversation revolves around the derivation of the commutation relation for the z-component of the spin operator using Noether's theorem for a transverse polarized photon. The mode expansion is mentioned and it is discovered that the commutator can be easily calculated using the defining commutation relations for creation and annihilation operators. The conversation also touches upon the definition of ##\hat{S}_z## in the book, which is the helicity operator for a photon with momentum in the z-direction.
  • #1
retardedgreensfunc
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TL;DR Summary
Want to find the commutation relation between the z-component spin-operator and creation operator for a transverse polarized photon.
Hi, so I'm currently reading the book "QFT for the gifted amateur", and doing the exercises. In exercise 14.2, which in itself is fine, the authors say that you can show using Noether's theorem that for a transverse polarized photon of momentum q, the z-component of the spin operator obeys the commutation relation:
1585393149065.png

Here, the epsilon is the polariztion vectors, and we assume the photon is traveling in the z-direction.
I really want to be able to derive this for myself, but I have tried for a while now, but without success. Anyone have any tips on how to do this?
 
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  • #2
How is ##\hat{S}_z## defined in the book, I don't have at hand right now?
 
  • #3
vanhees71 said:
How is ##\hat{S}_z## defined in the book, I don't have at hand right now?
I know the mode expansion can be written:
1585393961960.png
 
  • #4
Turns out I was overthinking it, and once I had the mode expansion it was very easy to just take the commutator by using the defining commutation relations for the creation and anihilation operators.
 
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  • #5
For a photon with ##\vec{p}## in ##z## direction that's the helicity operator (i.e., the projection of the total angular momentum to the direction of the photon momentum). That's the correct polarization quantity for a massless photon. A photon has only two helicity states with eigenvalues ##\pm 1##.
 

1. What is the commutation relation between spin-operator and creation operator in quantum field theory?

The commutation relation between spin-operator and creation operator in quantum field theory is given by [S_i, a^\dagger_j] = \frac{\hbar}{2}\delta_{ij}a^\dagger_j, where S_i is the spin operator, a^\dagger_j is the creation operator, and \hbar is the reduced Planck constant.

2. How does the commutation between spin-operator and creation operator affect the creation of particles in quantum field theory?

The commutation between spin-operator and creation operator affects the creation of particles in quantum field theory by determining the spin of the created particle. The spin operator measures the intrinsic angular momentum of a particle, while the creation operator creates a particle in a specific quantum state. The commutation relation ensures that the spin of the created particle is conserved.

3. What is the significance of the commutation between spin-operator and creation operator in quantum field theory?

The commutation between spin-operator and creation operator is significant in quantum field theory as it allows for the description of spin in terms of operators, which is crucial for understanding the behavior of particles at the quantum level. It also ensures that the spin of particles is conserved in interactions, which is a fundamental principle in quantum mechanics.

4. How does the commutation between spin-operator and creation operator relate to the Pauli exclusion principle?

The commutation between spin-operator and creation operator is related to the Pauli exclusion principle as it ensures that no two particles can occupy the same quantum state with the same spin. This is because the commutation relation shows that the creation of a particle with a certain spin is only possible if there is no particle with the same spin already in that state.

5. Can the commutation between spin-operator and creation operator be extended to other types of operators in quantum field theory?

Yes, the commutation between spin-operator and creation operator can be extended to other types of operators in quantum field theory, as long as they are compatible with the spin operator. This means that they must commute with the spin operator, preserving the spin of the particles. Examples of such operators include momentum, energy, and charge operators.

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