Fermionic Operators and Anticommutation Relations

In our previous lesson, we explored why the first-quantized picture, with its cumbersome Slater determinants, is impractical for describing the vast number of electrons in a solid. We concluded that a more natural approach is the occupation number representation, where we simply specify which single-particle states are occupied. Today, we will build the mathematical tools to work in this new framework. We will define the operators that create and destroy particles in these states and uncover the fundamental algebraic rules they must obey.

This lesson addresses the second learning outcome of our foundational module: to define fermionic creation and annihilation operators and verify their canonical anticommutation relations. Mastering this operator algebra is the key to unlocking the power of second quantization.

Defining the Operators: Creation and Annihilation

In the occupation number picture, the state of our system is a vector in Fock space, like $∣ n_{1}, n_{2}, \dots, n_{α}, \dots ⟩$ , where $n_{α}$ is 0 or 1 for fermions. We need a way to move between these states—to add a particle to an empty orbital or remove one from an occupied orbital. This is precisely what creation and annihilation operators do.

The creation operator, denoted $c_{α}^{†}$ , adds a fermion to the single-particle state $α$ .
The annihilation operator, denoted $c_{α}$ , removes a fermion from the single-particle state $α$ .

It follows that $c_{α}$ is the Hermitian conjugate of $c_{α}^{†}$ . The Pauli exclusion principle is automatically built into their definitions: you cannot create a fermion in a state that is already occupied, nor can you annihilate a fermion from a state that is already empty. In both cases, the result of the operation is simply zero (or, more formally, the null vector in the Hilbert space).

The video below from the "Professor M does Science" channel provides a careful and clear definition of these operators and their actions on Fock states.

Fermion creation and annihilation operators

This video carefully introduces the creation and annihilation operators, explaining how they are defined by their action on the occupation number basis states.

First, watch the segment on the creation operator. Pay close attention to how applying c^\dagger_{u_i} to a state where orbital u_i is already occupied results in zero. Next, watch the definition of the annihilation operator, which is introduced as the adjoint of the creation operator. Notice how its action is to remove a particle, and applying it to an empty state also results in zero.

The Crucial Role of Ordering and Antisymmetry

In the first-quantized picture, the antisymmetry of the fermionic wavefunction is enforced by the Slater determinant, where swapping any two particle coordinates (exchanging two columns) introduces a minus sign. In second quantization, this fundamental property is not lost; it is transferred to the algebra of the operators themselves.

The key insight is that the order in which you create particles matters. Creating a particle in state $α$ and then state $β$ should yield the negative of the state created by first adding to $β$ and then to $α$ . That is, $c_{β}^{†} c_{α}^{†} ∣0 ⟩ = - c_{α}^{†} c_{β}^{†} ∣0 ⟩$ , where $∣0 ⟩$ is the vacuum state with no particles. This must hold true when acting on any state, not just the vacuum.

This leads to a new kind of algebraic relation. Instead of a commutator, $[A, B] = A B - B A$ , we need an anticommutator, defined as:

{A, B} = A B + B A

The antisymmetry requirement upon particle exchange is equivalent to the statement that any two creation operators must anticommute: ${c_{α}^{†}, c_{β}^{†}} = 0$ .

The following segment of the video you just watched explains this subtlety of ordering, which is essential for understanding the origin of the anticommutation relations.

Fermion creation and annihilation operators

This part of the video explains why the ordering of states in the Fock state ket is critical and how it leads to minus signs when applying operators.

Please watch the segment from the importance of order. The example of applying c_{u_2} to the states |u_1, u_2\rangle and |u_2, u_1\rangle makes this abstract point very concrete.

The Canonical Anticommutation Relations (CAR)

We are now ready to derive the complete set of algebraic rules that govern fermionic operators. These are known as the Canonical Anticommutation Relations (CAR). They are the fundamental axioms of the formalism, from which all other properties of many-fermion systems follow. There are three key relations to verify.

The video below provides a rigorous and step-by-step derivation of each one.

Fermion creation and annihilation operators

This video segment derives the three fundamental anticommutation relations that define the algebra of fermionic operators.

Watch the following three parts consecutively: Anticommutator of two creation operators: This part formally shows that \{c^\dagger_i, c^\dagger_j\} = 0, which also contains the Pauli principle ((c^\dagger_i)^2 = 0). Anticommutator of creation and annihilation operators: This is the most involved part, deriving that \{c_i, c^\dagger_j\} = \delta_{ij} by considering the cases i \neq j and i = j separately. Summary of the relations: A concise summary of the final results.

As derived in the video, the canonical anticommutation relations are:

The anticommutator of any two annihilation operators is zero: ${c_{α}, c_{β}} = c_{α} c_{β} + c_{β} c_{α} = 0$
The anticommutator of any two creation operators is zero: ${c_{α}^{†}, c_{β}^{†}} = c_{α}^{†} c_{β}^{†} + c_{β}^{†} c_{α}^{†} = 0$
The anticommutator of an annihilation operator and a creation operator is the Kronecker delta: ${c_{α}, c_{β}^{†}} = c_{α} c_{β}^{†} + c_{β}^{†} c_{α} = δ_{α β}$

These three simple-looking equations contain all the physics of identical, indistinguishable fermions obeying the Pauli exclusion principle. Any calculation in the second quantization formalism is ultimately an exercise in applying these rules.

To round out your understanding, let's look at how these concepts are presented in a standard set of lecture notes. The notation might be slightly different, but the content is the same.

Chapter 2 Second Quantisation

This excerpt from lecture notes by B.D. Simons at Cambridge University provides a concise, formal statement of the operator definitions and their algebraic relations.

Please read the section starting from the physical interpretation of the operators. Notice the use of the generalized bracket notation [A, B]_{-\zeta} where \zeta=-1 for fermions corresponds to the anticommutator \{A, B\}. This is a common and compact way to write the relations for both bosons and fermions simultaneously.

Conclusion

In this lesson, we have introduced the essential building blocks of second quantization for fermions: the creation operator $c_{α}^{†}$ and the annihilation operator $c_{α}$ . We defined them by their action of adding or removing a particle from a specific single-particle state $α$ within the Fock space.

The key takeaway is that the core principles of fermionic quantum mechanics—indistinguishability and the Pauli exclusion principle—are elegantly captured by the algebra of these operators. This algebra is completely defined by the canonical anticommutation relations (CAR). These three relations are the rules of the game, and we will use them extensively from now on.

In our next lesson, we will see how to use these operators to build properly antisymmetrized N-particle states from the vacuum. This will bridge the gap between the abstract occupation number states $∣ n_{1}, n_{2}, \dots ⟩$ and the more familiar Slater determinant wavefunctions from first quantization, giving us a concrete representation of many-body states within this powerful new formalism.

Recommended resources

Defining the Operators: Creation and Annihilation

The Crucial Role of Ordering and Antisymmetry

The Canonical Anticommutation Relations (CAR)

Conclusion