Fock Space Construction and Vacuum State

Hello! Welcome to your fifth lesson on Quantum Field Theory.

In our last lesson, we took the crucial step of expressing the quantum scalar field $ϕ^(x)$ and its conjugate momentum $π^(x)$ as an integral over creation and annihilation operators, $a^_{p}^{†}$ and $a^_{p}$ . We established that the field behaves like a continuum of harmonic oscillators and derived the fundamental commutation relation:
$[a^_{p}, a^_{q}^{†}] = (2 π)^{3} δ^{(3)} (p - q)$
These operators are the fundamental building blocks of our theory. Today, we'll use them to construct the world they operate on. Our learning outcome is to construct the Fock space for scalar particles and define the vacuum state. This is where we will finally see particles emerge from the quantized field.

1. The Vacuum: The Ground State of a Field

In your study of the quantum harmonic oscillator, you know that there is a state of lowest energy, the ground state $∣0 ⟩$ , which is defined by the property that it is annihilated by the lowering operator: $a ∣0 ⟩ = 0$ .

We apply the same logic to our field. Since the free field is a collection of infinitely many independent harmonic oscillators (one for each momentum mode $p$ ), its ground state must be the state that is simultaneously the ground state for all of these oscillators. This lowest-energy state of the entire system is called the vacuum.

Formally, we define the vacuum state, denoted $∣0 ⟩$ , as the unique state that is annihilated by all annihilation operators:

a^_{p} ∣0 ⟩ = 0 for all p

The vacuum is not just empty space; it's a well-defined quantum state with rich properties. It is normalized, $⟨ 0∣0 ⟩ = 1$ , and serves as the foundation upon which we will build all other states.

2. Building Particles: One-Particle States

If the vacuum is the state with nothing in it, how do we create something? We use the creation operators. Acting on the vacuum with $a^_{p}^{†}$ "excites" the harmonic oscillator corresponding to the mode $p$ , taking it from its ground state to its first excited state. We interpret this new state as a one-particle state with momentum $p$ .

Let's define the state $∣ p ⟩$ as:

∣ p ⟩ \equiv a^_{p}^{†} ∣0 ⟩

What are the physical properties of this state? In the next lesson, we will formally derive the Hamiltonian ( $H^$ ) and total momentum ( $P^$ ) operators in terms of $a^$ and $a^^{†}$ . For now, let's state the results (after a procedure called normal ordering, which we'll also discuss next):
$H^= \int \frac{d ^{3} k}{( 2 π ) ^{3}} E_{k} a^_{k}^{†} a^_{k}$
$P^= \int \frac{d ^{3} k}{( 2 π ) ^{3}} k a^_{k}^{†} a^_{k}$
where $E_{k} = ∣ k ∣^{2} + m^{2}$ .

Using these expressions and the commutation relations, one can show that $∣ p ⟩$ is an eigenstate of both energy and momentum:

$H^∣ p ⟩ = E_{p} ∣ p ⟩$
$P^∣ p ⟩ = p ∣ p ⟩$

This is a remarkable result. The excitation created by $a^_{p}^{†}$ has energy $E_{p}$ and momentum $p$ , exactly satisfying the relativistic energy-momentum relation. This confirms our interpretation: $a^_{p}^{†}$ creates a relativistic particle of mass $m$ and momentum $p$ .

The following video provides a detailed discussion of this process, starting from the definition of the vacuum and moving to the interpretation of these excited states as particles.

Lecture 4: Canonical Quantization of a Free Scalar Field Theory

This lecture from MIT OpenCourseWare clearly explains how to define the vacuum state and then construct excited states representing particles. It explicitly calculates their energy and momentum eigenvalues, providing the physical interpretation.

Watch the segment from 34:54 to 45:12. Focus on how the vacuum is defined and how single-particle states are constructed. Pay close attention to the calculation showing that these states have the correct relativistic energy and momentum.

3. Multi-Particle States and Fock Space

We can continue this process by applying more creation operators to the vacuum to create states with multiple particles. A two-particle state, for example, is constructed as:

∣ p_{1}, p_{2} ⟩ = a^_{p_{1}}^{†} a^_{p_{2}}^{†} ∣0 ⟩

This state is an eigenstate of the Hamiltonian with eigenvalue $E_{p_{1}} + E_{p_{2}}$ and the momentum operator with eigenvalue $p_{1} + p_{2}$ , representing two non-interacting particles.

Bosonic Statistics

A crucial feature arises from the commutation relations. Since creation operators for different (or same) momenta commute, $[a^_{p_{1}}^{†}, a^_{p_{2}}^{†}] = 0$ , the order of creation does not matter:

∣ p_{1}, p_{2} ⟩ = a^_{p_{1}}^{†} a^_{p_{2}}^{†} ∣0 ⟩ = a^_{p_{2}}^{†} a^_{p_{1}}^{†} ∣0 ⟩ = ∣ p_{2}, p_{1} ⟩

The state is symmetric under the exchange of particles. This means that the quanta of the scalar field are bosons. This is a profound conclusion: the quantization of a classical scalar field naturally leads to particles that obey Bose-Einstein statistics.

Fock Space

The full Hilbert space of our theory is formed by the collection of all possible $n$ -particle states, for $n = 0, 1, 2, \dots$ . This space, which is a direct sum of the Hilbert spaces for each particle number sector, is called Fock space.

F = H_{0} \oplus H_{1} \oplus H_{2} \oplus \dots

where $H_{0}$ is the one-dimensional space spanned by the vacuum $∣0 ⟩$ , $H_{1}$ is the space of all one-particle states, and so on.

The following reading from David Tong's notes provides a concise and clear summary of multi-particle states, bosonic statistics, and the definition of Fock space.

2. Free Fields

Please read this section from David Tong's lecture notes, which formalizes the construction of multi-particle states and introduces the concept of Fock space.

Please read the subsection titled 'Multi-Particle States, Bosonic Statistics and Fock Space' on page 30. This short section succinctly covers all the key concepts we've just discussed.

To connect these ideas to your advanced quantum mechanics background, let's consider a visualization. A state with a definite number of particles, like the ones we are constructing, is called a Fock state or a number state.

Phase Space Representations of Quantum States — This image from the QuTiP documentation shows phase-space visualizations of three quantum states. The Fock state, on the right, is the type of state we are building. Its ring-like structure illustrates that the state has a definite energy (proportional to the radius squared) but a completely uncertain phase (the angle), a manifestation of the number-phase uncertainty principle. This is distinct from a classical-like coherent state (left) or a statistical thermal state (center).

4. Particles in Position Space

The states we have built, $∣ p ⟩$ , are momentum eigenstates. Like plane waves in ordinary quantum mechanics, they are completely delocalized in space. How do we get a more intuitive picture of a particle at a specific location?

We can create localized wave packets by taking superpositions of our momentum eigenstates:

∣ ψ ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3}} f (p) ∣ p ⟩

where $f (p)$ is some function that defines the shape of the wave packet in momentum space.

An even more direct, though somewhat formal, way to think about this comes from considering the action of the field operator $ϕ^(x)$ itself on the vacuum. In the Schrödinger picture at $t = 0$ , we have:

ϕ^(x) ∣0 ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} (a^_{p} e^{i p \cdot x} + a^_{p}^{†} e^{- i p \cdot x}) ∣0 ⟩ = \int \frac{d ^{3} p}{( 2 π ) ^{3}} \frac{1}{2 E _{p}} e^{- i p \cdot x} ∣ p ⟩

This shows that $ϕ^(x) ∣0 ⟩$ is a state that is a specific superposition of all momentum eigenstates. This superposition turns out to be localized at the position $x$ . Therefore, we gain the powerful interpretation that the field operator $ϕ^(x)$ creates a particle at spacetime point $x$ .

This beautiful interpretation is explained very clearly in the following video.

Quantum Field Theory I - Lecture 8

This segment from the IFT-UNESP lecture series demonstrates how the field operator itself can be interpreted as a particle creator in position space, solidifying the particle-field connection.

Watch the clip from 1:21:55 to 1:28:40. The lecturer shows that applying the field operator \hat{\phi}(\vec{x}) to the vacuum creates a state localized at \vec{x}. The calculation at the end reinforces this by showing it has the correct plane-wave projection in momentum space.

Test your understanding!

Why does quantizing a real scalar field lead to particles that are their own antiparticles, while a complex scalar field (which we will study later) leads to distinct particles and antiparticles? (Hint: Think about how many independent sets of creation/annihilation operators you have in each case).

Show answer

For a real scalar field, the field operator $ϕ^(x)$ must be Hermitian ( $ϕ^= ϕ^^{†}$ ). This condition links the coefficient of the $e^{- i p \cdot x}$ term to the coefficient of the $e^{+ i p \cdot x}$ term in the Fourier expansion. Specifically, it forces the operator that multiplies $e^{- i p \cdot x}$ (the annihilation operator $a^_{p}$ ) and the operator that multiplies $e^{+ i p \cdot x}$ (the creation operator $a^_{p}^{†}$ ) to be Hermitian conjugates of each other. We have only one set of ladder operators. The single type of creation operator $a^_{p}^{†}$ creates the only type of particle in the theory.

For a complex scalar field $ψ^(x)$ , the field is not Hermitian ( $ψ^\neq = ψ^^{†}$ ). We can decompose it into two independent real fields, or, more conveniently, we expand $ψ^$ and $ψ^^{†}$ separately. This leads to two independent sets of ladder operators, typically called $a^_{p}, a^_{p}^{†}$ and $b^_{p}, b^_{p}^{†}$ . $a^_{p}^{†}$ creates a "particle" and $b^_{p}^{†}$ creates a distinct "antiparticle".

Conclusion

In this lesson, we have populated our quantum field theory with its inhabitants: particles. By building on the harmonic oscillator analogy, we have given a concrete physical meaning to the creation and annihilation operators.

Key Takeaways:

The vacuum state $∣0 ⟩$ is the ground state of the field, defined as the state annihilated by all annihilation operators: $a^_{p} ∣0 ⟩ = 0$ .
Particle states are created by acting on the vacuum with creation operators. The state $a^_{p}^{†} ∣0 ⟩$ represents a single particle with relativistic energy $E_{p}$ and momentum $p$ .
The commutation relations of the creation operators, $[a^_{p}^{†}, a^_{q}^{†}] = 0$ , dictate that the particles of the scalar field are bosons.
The entire Hilbert space of the theory, composed of the vacuum and all possible multi-particle states, is known as Fock space.

Preview of the Next Lesson:

Now that we have both the operators and the states, we are ready to analyze the theory's dynamics and observables more formally. In the next lesson, we will calculate the Hamiltonian and momentum operators in terms of creation and annihilation operators. This will involve tackling the famous problem of the infinite zero-point energy of the vacuum and introducing the crucial technique of normal ordering to extract physically meaningful results.

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