Fermion creation and annihilation operators

In summary, we discussed the properties of fermion annihilation and creation operators c and c^\dagger, including their commutation and anti-commutation relations. We saw that the commutator of c and c^\dagger is equal to negative c, and that this does not imply that the commutator and anti-commutator are equal to unity. This is because the operator c is nilpotent and not always invertible, requiring caution in certain manipulations.
  • #1
daudaudaudau
302
0
Hi.

If [itex]c[/itex] and [itex]c^\dagger[/itex] are fermion annihilation and creation operators, respectively, we know that [itex]cc^\dagger+c^\dagger c=1[/itex] and [itex]cc=0[/itex] and [itex]c^\dagger c^\dagger=0[/itex]. I can use this to show the following
[tex]
[c^\dagger c,c]=c^\dagger cc-c c^\dagger c=-cc^\dagger c=-c(1-cc^\dagger)=-c
[/tex]

But on the other hand I have
[tex]
[c^\dagger c,c]=c^\dagger[c,c]+[c^\dagger,c]c=[c^\dagger,c]c
[/tex]

Does this not imply that [itex][c^\dagger,c]=-1[/itex] and consequently that BOTH the commutator and anti-commutator of [itex]c[/itex] and [itex]c^\dagger[/itex] is equal to unity?
 
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  • #2
No, it does not imply that. Namely, the commutator is:

[tex][c^\dag , c] = c^\dag c - cc^\dag = c^\dag c - 1 + c^\dag c = 2c^\dag c - 1[/tex]

As you can see, the first term gives zero when you act with this commutator on the operator [tex]c[/tex]. In other words,

[tex][c^\dag , c]c = (2c^\dag c - 1)c = -c[/tex]

Which is your result. But for operators you cannot use that if AB = CB, then C = A. The reason is that the operator B is not always invertible (as in this case), but is nilpotent instead. So you have to be careful with these types of manipulations.
 
  • #3
I see. Thanks.
 

FAQ: Fermion creation and annihilation operators

1. What are fermion creation and annihilation operators?

Fermion creation and annihilation operators are mathematical operators used in quantum field theory to describe the creation and annihilation of fermions, which are particles that follow the Fermi-Dirac statistics. These operators are used to create or destroy fermions at specific positions in space and time.

2. How do fermion creation and annihilation operators work?

Fermion creation operators are denoted by a hat symbol (^) and are used to create a fermion at a specific position in space and time. Annihilation operators are denoted by a dagger symbol (†) and are used to destroy a fermion at a specific position in space and time. These operators are used in quantum field theory to describe the behavior of fermions in a mathematical framework.

3. What is the commutation relation for fermion creation and annihilation operators?

The commutation relation for fermion creation and annihilation operators is given by {ai,aj} = {ai,aj} = 0, where i and j represent different fermion states. This means that these operators do not commute, and the order in which they are applied matters.

4. What is the significance of fermion creation and annihilation operators?

Fermion creation and annihilation operators are important in quantum field theory as they allow scientists to describe and understand the behavior of fermions in a mathematical framework. They are also used in many mathematical models and calculations in particle physics and quantum mechanics.

5. How do fermion creation and annihilation operators relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle states that the position and momentum of a particle cannot be simultaneously known with absolute certainty. Fermion creation and annihilation operators are used to describe the position and momentum of fermions, and their commutation relation is related to the uncertainty principle. This shows that there is inherent uncertainty in the behavior of fermions, which is a fundamental concept in quantum mechanics.

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