# Second Quantization: Explaining c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i

• Niles
In summary, the relation is that if you have two fermionic operators and you subtract their respective commutators, the result is the anti-commutator of the commutators.

#### Niles

Hi

Say I have the following two fermionic creation/annihilation operators

$$c^\dagger_ic_j$$

1) Yesterday, my lecturer said that the following is valid

$$c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i$$

Can you guys explain to me, where this formula comes from? I originally thought that it was one of the anti-commutator relations, but it cannot come from there.
2) Say I have en expression of the form

$$c_{k+q}^\dagger c_{k-q'} c^\dagger_{k'-q'}c_{k'}$$

If the operators are fermionic, then if I want to have all dagger-operators on the LHS and non-dagger operators on the RHS, then do I have to use anti-commutator relatations in order to rewrite the expression?

Likewise, if they were bosonic operators, then I would have to use commutator relations in order to rewrite the expression?Niles.

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2) Yes.

Thank you, it is kind of you to answer so quickly.

Niles said:
$$c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i$$

probably he meant c^*_ic_j = \delta_{i,j}+c_jc^*_i
which is the CCR for bosons.

Thanks. I have another question related to fermionic operators, so I'll just ask it here. It is regarding the relation

$$\delta (t - t')\left\langle {\left\{ {c_i (t),c_i^\dag (t')} \right\}} \right\rangle = \delta (t - t')$$

The curly brackets denote an anti-commutator. Is there an easy way of showing this? The way I would show this is to look at the case t = t' and the case t != t'.

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A. Neumaier said:
probably he meant c^*_ic_j = \delta_{i,j}+c_jc^*_i
which is the CCR for bosons.

Sorry, this is not quite true; $$c^*_ic_j = -\delta_{i,j}+c_jc^*_i$$

Niles said:
Thanks. I have another question related to fermionic operators, so I'll just ask it here. It is regarding the relation

$$\delta (t - t')\left\langle {\left\{ {c_i (t),c_i^\dag (t')} \right\}} \right\rangle = \delta (t - t')$$

The curly brackets denote an anti-commutator. Is there an easy way of showing this? The way I would show this is to look at the case t = t' and the case t != t'.

You cannot distinguish cases in this way since this is meant in the sense of distributions. Thus you need to multiply both sides by a function f(t,t'), integrate over t and t', and simplify before you can interpret the statement.

## What is second quantization?

Second quantization is a mathematical framework used in quantum mechanics to describe systems with an indefinite number of particles. It involves representing quantum states using creation and annihilation operators, and allows for the description of many-particle systems in a more efficient and compact way.

## What is the significance of c^\dagger_ic_j = \delta_{i,j}c_jc^\dagger_i in second quantization?

This equation, also known as the anticommutation relation, is a fundamental property of the creation and annihilation operators used in second quantization. It ensures that the operators behave correctly when applied to quantum states, and is essential for correctly describing the behavior of many-particle systems.

## How does second quantization differ from first quantization?

In first quantization, quantum states are described using wave functions and operators act on single-particle states. In second quantization, quantum states are described using creation and annihilation operators and operators act on multi-particle states. This allows for a more efficient and general description of many-particle systems.

## What are the advantages of using second quantization?

Second quantization allows for a more compact and elegant description of many-particle systems compared to first quantization. It also makes it easier to apply mathematical techniques, such as perturbation theory, to these systems. Additionally, it allows for the description of systems with an indefinite number of particles, which is not possible in first quantization.

## What are some applications of second quantization?

Second quantization is used in a variety of fields, including quantum chemistry, condensed matter physics, and nuclear physics. It is particularly useful in describing systems with many interacting particles, such as atoms in a gas or electrons in a solid. It has also been applied to problems in quantum computing and quantum information theory.