Fock Space Construction and Manipulation

1. The Challenge of Many Identical Particles

In your previous studies, you've worked with quantum systems described by wavefunctions, like $\Psi (x_1,x_2,...,x_N,t)$. This is the "first quantization" picture. While perfectly valid, this approach becomes incredibly cumbersome when dealing with a large number of identical particles, which is the central problem in many-body physics.

The core issue stems from the principle of indistinguishability. The wavefunction of a system of identical particles must have a specific symmetry upon the exchange of any two particles:

• Bosons: Symmetric, $\Psi (...,x_i,...,x_j,...)=+\Psi (...,x_j,...,x_i,...)$
• Fermions: Anti-symmetric, $\Psi (...,x_i,...,x_j,...)=-\Psi (...,x_j,...,x_i,...)$

To understand the practical difficulties this creates, please read the following brief introduction.

As the text highlights, explicitly enforcing the symmetry and keeping track of which particle is in which coordinate becomes an algebraic nightmare. A much more natural description for identical particles is to simply state how many particles occupy each available single-particle quantum state, without trying to label the particles themselves. This is the central idea behind the occupation number representation.

2. The Occupation Number Basis and Fock Space

Instead of coordinates, we now label states by listing the number of particles in each single-particle state. Let's assume we have a complete, orthonormal basis of single-particle states, which we label with an index $\alpha$. For example, in a translationally invariant system, $\alpha$ could represent the momentum and spin, $(\vec{k},\sigma )$. We agree on a fixed order for these states (e.g., by increasing energy).

A many-body basis state is then written as a ket vector:
$|\Psi ⟩=|n_1,n_2,n_3,\dots ,n_{\alpha },\dots ⟩$
Here, $n_{\alpha }$ is the occupation number of the single-particle state $\alpha$. This ket contains all the necessary information about the many-body state.

The video below provides a clear introduction to this concept, contrasting it with the first quantization approach and defining the key terms.

Key Concepts

• Occupation Numbers: The set of integers $\{n_{\alpha }\}$ that define the state.
• Total Particle Number: For a state with a fixed number of particles $N$, we have the constraint $N={\sum }_{\alpha }{n}_{\alpha }$.
• The Vacuum State: A special, fundamental state in this formalism is the one with no particles at all. It is denoted by $|0⟩$ and is defined by having all occupation numbers equal to zero: $|0⟩\equiv |0,0,0,\dots ⟩$. The vacuum is a normalized state, $⟨0|0⟩=1$, and is distinct from the number 0.
• Fock Space: The collection of all possible occupation number states, for all possible total particle numbers (from $N=0$ to $N=\infty$), forms a complete vector space called Fock space. This is the arena for many-body physics, as it allows us to describe processes where the number of particles can change. A system with a fixed particle number $N$ is described by a state vector within a specific subspace of the full Fock space.

Bosons vs. Fermions

The nature of the particles imposes a fundamental constraint on the allowed occupation numbers:

• Bosons: Any number of bosons can occupy the same single-particle state. Thus, for bosons, $n_{\alpha }\in \{0,1,2,\dots \}$.
• Fermions: The Pauli exclusion principle dictates that no two identical fermions can occupy the same quantum state. Therefore, for fermions, $n_{\alpha }\in \{0,1\}$.

This fundamental difference is illustrated vividly in the following image.

3. Building States: Creation and Annihilation Operators

We now have a way to label our states, but how do we mathematically construct them and transition between them? We do this using a new set of operators. For each single-particle state $\alpha$, we introduce two operators:

• Creation Operator ($c_{\alpha }^{†}$): Adds one particle to the single-particle state $\alpha$.
• Annihilation Operator ($c_{\alpha }$): Removes one particle from the single-particle state $\alpha$.

These operators are the building blocks of second quantization. Any state in Fock space can be constructed by starting with the vacuum state $|0⟩$ and applying a sequence of creation operators.

A general Fock state is defined as:
$|n_1,n_2,\dots ⟩=\dots \frac{(c_2^{†}{)}^{n_2}}{\sqrt{n_2!}}\frac{(c_1^{†}{)}^{n_1}}{\sqrt{n_1!}}|0⟩$

Note: The normalization factors $\sqrt{n_{\alpha }!}$ are for bosons; for fermions, since $n_{\alpha }$ is 0 or 1, this factor is always 1.

The most crucial point is that the statistical nature of the particles is encoded in the algebraic properties of these operators.

• For Bosons: The order in which you create particles does not matter. The operators commute.
  $b_i^{†}{b}_j^{†}|0⟩=b_j^{†}{b}_i^{†}|0⟩⟹[b_i^{†},b_j^{†}]=0$
  (Here we use $b^{†}$ as is conventional for bosons.)

• For Fermions: Swapping the order of creation introduces a minus sign, directly reflecting the anti-symmetry of the wavefunction. The operators anti-commute.
  $a_i^{†}{a}_j^{†}|0⟩=-a_j^{†}{a}_i^{†}|0⟩⟹\{a_i^{†},a_j^{†}\}=a_i^{†}{a}_j^{†}+a_j^{†}{a}_i^{†}=0$
  (Here we use $a^{†}$ as is conventional for fermions.)

This property automatically enforces the Pauli principle. If you try to create two fermions in the same state $i$, you get:
$a_i^{†}{a}_i^{†}=-a_i^{†}{a}_i^{†}⟹2a_i^{†}{a}_i^{†}=0⟹a_i^{†}{a}_i^{†}=0$
Creating two fermions in the same state results in a null vector, a state of zero norm, which is not a physical state.

The following video segments provide concrete examples of how creation and annihilation operators act on bosonic and fermionic states.

Because the fermionic operators anti-commute, the sign of a fermionic state depends on the order in which the creation operators are written. For example, the state with one particle in state $i$ and one in state $j$ is $|...1_i,...,1_j,...⟩$. This state is typically defined by a fixed operator ordering, for instance, $a_j^{†}{a}_i^{†}|0⟩$, assuming $i$ comes before $j$ in our chosen basis ordering. Then, the state $a_i^{†}{a}_j^{†}|0⟩$ is equal to $-|...1_i,...,1_j,...⟩$. This sign-keeping is the primary 'bookkeeping' task when working with fermions.

Conclusion

In this lesson, we established the language for our study of many-body physics. We have moved from the cumbersome first quantization picture of particle coordinates to the elegant and efficient occupation number representation.

Here are the key takeaways:

• Many-body states are vectors in Fock space, labeled by the occupation numbers of single-particle states: $|n_1,n_2,\dots ⟩$.
• The vacuum state $|0⟩$ is the state with no particles and serves as the foundation from which all other states are built.
• Creation ($c_{\alpha }^{†}$) and annihilation ($c_{\alpha }$) operators are the tools used to build states and move between them.
• The fundamental statistics of particles are encoded in the algebra of these operators: bosonic operators commute, while fermionic operators anti-commute. This algebra automatically handles the symmetry requirements of the many-body system.

So far, we have introduced the creation and annihilation operators as tools for state construction. The real power of this formalism comes from their well-defined algebraic structure. In the next lesson, we will rigorously derive the canonical (anti)commutation relations that these operators obey. This algebra is the mathematical engine that will allow us to express and solve complex many-body Hamiltonians.