Exercise involving Dirac fields and Fermionic commutation relations

In summary, the equation for fermions can be derived by considering the effect of the field operator and its adjoint on both fermion and anti-fermion states.
  • #1
snypehype46
12
1
Homework Statement
Derive a similar equation for fermions
Relevant Equations
$$\psi(x) = \sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s) e^{iq \cdot x}\right)$$
I'm trying to the following exercise:
1620037584475.png

I've proven the first part and now I'm trying to do the same thing for fermions.
The formulas for the mode expansions are:
1620037669696.png


What I did was the following:

$$\begin{align*}
\sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s) e^{iq \cdot x}\right)|p,r\rangle = \\
\sum_s \int d\tilde{q} u(q,s) e^{-iq \cdot x} a_s(q) a^\dagger_r(p)|0\rangle +
\\ \sum_s \int d\tilde{q} v(q,s) e^{iq \cdot x} b_s^\dagger(q) b_r^\dagger(p) |0\rangle \\=
\sum_s \int d\tilde{q} u(q,s) e^{-iq \cdot x} \left( \delta_{rs} \delta(q-p) - a_r^\dagger(p)a_s(q)\right)|0\rangle + \\
\sum_s \int d\tilde{q} v(q,s) e^{iq \cdot x} b_s^\dagger(q) b_r^\dagger(p) |0\rangle
\end{align*}
$$

In the first the integral and the sum vanish because the kronecker delta and the delta function pick out a specific value of q and s.
Then taking multiplying from the left by ##\langle 0 |## we get:

$$u(p,s)e^{-ip\cdot x} \langle 0 | 0 \rangle + 0 = u(p,s)e^{-ip\cdot x}$$

where for the first term I've used the fact an annihilation operators gives zero acting on the vacuum and for the second term I've used the fact that:

$$\left(\langle 0 |b_s(q)\right)^\dagger = 0$$

Is this the correct way of proceeding? One thing I'm unsure is what I did for the writing of the state ##|p,r\rangle## is correct, because as you can see I "created" the state using two different operators in the same line: ##a_r^\dagger(p,r) |0 \rangle## and ##b_r^\dagger(p,r) |0 \rangle##.
 
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  • #2
snypehype46 said:
Homework Statement:: Derive a similar equation for fermions
Relevant Equations:: $$\psi(x) = \sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s) e^{iq \cdot x}\right)$$

I'm trying to the following exercise:
View attachment 282408
I've proven the first part and now I'm trying to do the same thing for fermions.
The formulas for the mode expansions are:
View attachment 282409

What I did was the following:

$$\begin{align*}
\sum_s \int d\tilde{q} \left(a_s(q) u(q,s) e^{-iq \cdot x}+ b_s^\dagger(q) v(q,s) e^{iq \cdot x}\right)|p,r\rangle = \\
\sum_s \int d\tilde{q} u(q,s) e^{-iq \cdot x} a_s(q) a^\dagger_r(p)|0\rangle +
\\ \sum_s \int d\tilde{q} v(q,s) e^{iq \cdot x} b_s^\dagger(q) b_r^\dagger(p) |0\rangle \\=
\sum_s \int d\tilde{q} u(q,s) e^{-iq \cdot x} \left( \delta_{rs} \delta(q-p) - a_r^\dagger(p)a_s(q)\right)|0\rangle + \\
\sum_s \int d\tilde{q} v(q,s) e^{iq \cdot x} b_s^\dagger(q) b_r^\dagger(p) |0\rangle
\end{align*}
$$

In the first the integral and the sum vanish because the kronecker delta and the delta function pick out a specific value of q and s.
Then taking multiplying from the left by ##\langle 0 |## we get:

$$u(p,s)e^{-ip\cdot x} \langle 0 | 0 \rangle + 0 = u(p,s)e^{-ip\cdot x}$$

where for the first term I've used the fact an annihilation operators gives zero acting on the vacuum and for the second term I've used the fact that:

$$\left(\langle 0 |b_s(q)\right)^\dagger = 0$$

Is this the correct way of proceeding? One thing I'm unsure is what I did for the writing of the state ##|p,r\rangle## is correct, because as you can see I "created" the state using two different operators in the same line: ##a_r^\dagger(p,r) |0 \rangle## and ##b_r^\dagger(p,r) |0 \rangle##.
It's not quite right. You have to consider the effect of ##\psi## and its adjoint on both a fermion and an anti-fermion state (so essentially, you need four calculations). The fermion state is ##|p,r \rangle = a_r^\dagger(p,r) |0\rangle ## whereas the antifermion state is ##|p,r \rangle = b_r^\dagger(p,r) |0\rangle ##
 

Related to Exercise involving Dirac fields and Fermionic commutation relations

1. What are Dirac fields?

Dirac fields are mathematical objects used in quantum field theory to describe the behavior of fermions, which are particles with half-integer spin. They were first introduced by physicist Paul Dirac in 1928 to describe the behavior of electrons in a relativistic framework.

2. What is the significance of Fermionic commutation relations in exercise involving Dirac fields?

Fermionic commutation relations are mathematical rules that describe how fermionic particles interact with each other. In the context of exercise involving Dirac fields, these relations are important because they allow us to calculate the behavior of fermions in a given system and make predictions about their properties and interactions.

3. How are Dirac fields related to the Standard Model of particle physics?

The Standard Model is a theory that describes the fundamental particles and their interactions. Dirac fields are an essential part of this model, as they describe the behavior of fermions, which are one of the building blocks of matter. They are also used to describe the interactions between fermions and other particles, such as bosons.

4. What are some real-world applications of exercise involving Dirac fields and Fermionic commutation relations?

Exercise involving Dirac fields and Fermionic commutation relations has many practical applications in fields such as particle physics, condensed matter physics, and quantum computing. For example, these concepts are used to study the behavior of electrons in materials, which is important for understanding the properties of semiconductors and developing new technologies.

5. Are there any current research developments or open questions related to exercise involving Dirac fields and Fermionic commutation relations?

Yes, there is ongoing research in this area, particularly in the field of quantum computing. Scientists are exploring how to use Dirac fields and Fermionic commutation relations to develop more efficient and powerful quantum algorithms. There are also open questions regarding the behavior of fermions in extreme conditions, such as in black holes or the early universe.

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