# The commutations relations for left/right handed fermions

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• decerto
In summary, the problem is that you don't know how to commute the left handed operators past the psi's. You can use the full commutator to combine the projections, or you can change the indices with the delta function and use P_L^{aa}P_L^{ib} = P_L^{ab} to combine the projections.

#### decerto

I have a problem where I have to know the commutation relations for left handed fermions. I know
##\psi_L=\frac{1}{2}(1-\gamma^5)\psi##

##\psi^\dagger_L=\psi^\dagger_L\frac{1}{2}(1-\gamma^5)##

and

## \left\{ \psi(x) , \psi^\dagger(y)\right\} = \delta(x-y)##

So writing

## \left\{P_L\psi(x) , \psi^\dagger(y)P_L\right\} ##

##=P_L\psi(x) \psi^\dagger(y)P_L + \psi^\dagger(y)P_LP_L\psi(x)##

And I don't see how I can go any further.

it looks fine up to here...

ChrisVer said:
it looks fine up to here...

I need to write the left handed commutator in terms of the unprojected commutator I don't see a nice way to do this? How can I commute the P_L operators past the psi's?

I want to say its the same, and will just be ##P_L \delta(x-y)##, but I'm not sure.

Maybe what you can do is write out the spinor indices explicitly. Normally you have :
$$\left\{\psi^a , \psi^{\dagger b} \right\} = \psi^a \psi^{\dagger b}+\psi^{\dagger b} \psi^a = \delta(x-y) \delta^{ab}$$
now we want to understand
$$\left\{\psi_L^a , \psi_L^{\dagger b} \right\} = P_L^{ai} \psi^i \psi^{\dagger j}P_L^{jb} + \psi^{\dagger j} P_L^{jb}P_L^{ai} \psi^i$$

we can combine and use the full commutator:

$$\left\{\psi_L^a , \psi_L^{\dagger b} \right\} = P_L^{ai}P_L^{jb} (\psi^i \psi^{\dagger j} + \psi^{\dagger j} \psi^i)\\ \left\{\psi_L^a , \psi_L^{\dagger b} \right\} = P_L^{ai}P_L^{jb} \delta(x-y) \delta^{ij}$$
you can change the indices with the delta function and use ##P_L^{ai}P_L^{ib} = P_L^{ab}## to combine the projections and get
$$\left\{\psi_L^a , \psi_L^{\dagger b} \right\} = P_L^{ab} \delta(x-y)$$
If ##a=b## then you get(in D then 4 dimensions)
$$P_L^{aa} = \frac{D}{2} = 2$$
This is because by definition the trace of gamma_5 is zero, and the trace of the identity matrix is D(4).

I think this is right...

Thanks, that makes sense, the whole dropping spinor indexes for simplicity makes things less clear sometimes.

## 1. What are the commutation relations for left/right handed fermions?

The commutation relations for left/right handed fermions are a set of mathematical equations that describe how the operators for left and right handed fermions interact with each other. These relations are fundamental in quantum field theory and play a crucial role in understanding the behavior of fermions.

## 2. How do the commutation relations differ for left and right handed fermions?

The commutation relations for left/right handed fermions are different because they represent two distinct types of fermions. Left handed fermions have a spin that is aligned opposite to their momentum, while right handed fermions have a spin that is aligned in the same direction as their momentum. This fundamental difference in spin leads to different commutation relations for the two types of fermions.

## 3. Why are the commutation relations important in quantum field theory?

The commutation relations for left/right handed fermions are important in quantum field theory because they govern how these particles interact with each other. They are used to calculate important quantities such as scattering amplitudes and Feynman diagrams, which are essential in understanding the behavior of particles at the subatomic level.

## 4. How do the commutation relations for fermions differ from those for bosons?

The commutation relations for fermions differ from those for bosons because they represent two different types of particles. Fermions have half-integer spin and obey the Pauli exclusion principle, while bosons have integer spin and do not obey this principle. This fundamental difference leads to different commutation relations for the two types of particles.

## 5. Are there any real-world applications of the commutation relations for left/right handed fermions?

Yes, the commutation relations for left/right handed fermions have many real-world applications, particularly in the fields of particle physics and quantum computing. They are used to calculate the properties of subatomic particles and to understand the behavior of quantum systems. They also play a crucial role in the development of quantum technologies such as quantum computers.