Do creation operators for different spins commute?

In summary, fermion creation and annihilation operators in a hamiltonian with momenta and spin up/down can be anticommuted, following the same rules as in a bosonic theory. This means that the operators will pick up a minus sign when swapped.
  • #1
metapuff
53
6
Say I have a hamiltonian with fermion creation / annihilation operators like this:

[tex] \sum_{k_1,k_2,k_3,k_4} c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} c_{k_3,\downarrow} c_{k_4,\uparrow} [/tex]

where the k's are momenta and the arrows indicate spin up / spin down. Can I commute operators for different spins? That is, does

[tex] c_{k_1,\uparrow}^{\dagger} c_{k_2,\downarrow}^{\dagger} = c_{k_2,\downarrow}^{\dagger} c_{k_1,\uparrow}^{\dagger} [/tex]

Or do I pick up a minus sign as usual? Thanks!
 
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  • #2
The operators anticommute. Specifically all of the commutation relations of the bosonic theory carry over to fermions provided that we replace commutators by anticommutators.
 
  • #3
Yay! That's what I'd hoped for, but was afraid there might be some subtlety with spin that I wasn't picking up on. Thanks!
 

1. What are creation operators for different spins?

Creation operators for different spins are mathematical operators used in quantum mechanics to represent the process of creating a particle with a specific spin. They are denoted by a '†' symbol and are typically used in conjunction with annihilation operators to describe the behavior of particles with different spins.

2. How do creation operators for different spins work?

Creation operators for different spins work by acting on a state vector to create a new state with a specific spin. They essentially add a particle with the desired spin to the existing state. This process is reversible, as the corresponding annihilation operator can be used to remove the particle with the same spin.

3. Do creation operators for different spins commute?

This is a highly debated question in quantum mechanics and there is no definitive answer. Some theories suggest that creation operators for different spins do commute, meaning they can be interchanged without affecting the outcome. However, other theories suggest that they do not commute and their order does matter.

4. What is the significance of commutation in quantum mechanics?

In quantum mechanics, commutation refers to the order in which operators are applied. The commutation of operators can affect the outcome of an operation and can also provide information about the state of a system. In the case of creation operators for different spins, their commutation may provide insight into the behavior of particles with different spins.

5. Are there any applications of creation operators for different spins?

Creation operators for different spins have various applications in quantum mechanics, particularly in the study of spin dynamics and magnetic properties of materials. They are also used in quantum computing and in the development of new technologies such as spintronic devices. Their understanding and manipulation are crucial in advancing our understanding of the quantum world and its potential applications.

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