Quantizing the Scalar Field: Operators and Commutation Relations

1. From Field Operators to Ladder Operators

Since we know the field behaves like a collection of harmonic oscillators, we can quantize it using the same tools you've mastered in your Advanced Quantum Mechanics course: creation ($a^{†}$) and annihilation ($a$) operators.

For a single harmonic oscillator with coordinate $q$, momentum $p$, and frequency $\omega$, you'll recall the definitions:
$a=\sqrt{\frac{\omega }{2}}q+\frac{i}{\sqrt{2\omega }}p,a^{†}=\sqrt{\frac{\omega }{2}}q-\frac{i}{\sqrt{2\omega }}p$
which satisfy the commutation relation $[a,a^{†}]=1$.

We will now apply this logic to the field. Each momentum mode $\vec{p}$ of our field is an independent harmonic oscillator with frequency ${\omega }_{\vec{p}}=\sqrt{|\vec{p}{|}^2+m^2}$. We therefore associate a pair of ladder operators, ${\hat{a}}_{\vec{p}}$ and ${\hat{a}}_{\vec{p}}^{†}$, with each mode.

Just as $q$ and $p$ can be expressed as linear combinations of $a$ and $a^{†}$, we can express the field operator $\hat{\varphi }(\vec{x})$ and its momentum $\hat{\pi }(\vec{x})$ as a sum (or rather, an integral) over all the ladder operators for all momentum modes. In the Schrödinger picture (at a fixed time, say $t=0$), this expansion takes the form:

$\hat{\varphi }(\vec{x})=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\vec{p}}}}\left({\hat{a}}_{\vec{p}}{e}^{i\vec{p}\cdot \vec{x}}+{\hat{a}}_{\vec{p}}^{†}{e}^{-i\vec{p}\cdot \vec{x}}\right)$
$\hat{\pi }(\vec{x})=\int \frac{d^{3}p}{(2\pi {)}^3}(-i)\sqrt{\frac{{\omega }_{\vec{p}}}{2}}\left({\hat{a}}_{\vec{p}}{e}^{i\vec{p}\cdot \vec{x}}-{\hat{a}}_{\vec{p}}^{†}{e}^{-i\vec{p}\cdot \vec{x}}\right)$

Let's break this down:

• The integral $\int d^{3}p/(2\pi {)}^3$ sums over all the independent harmonic oscillators (momentum modes).
• The plane waves $e^{\pm i\vec{p}\cdot \vec{x}}$ are the mode functions.
• The operators ${\hat{a}}_{\vec{p}}$ and ${\hat{a}}_{\vec{p}}^{†}$ are the annihilation and creation operators for the mode $\vec{p}$.
• The normalization factor $1/\sqrt{2{\omega }_{\vec{p}}}$ is a convention, but it is chosen carefully to simplify later expressions and ensure they are Lorentz invariant.

The following reading from David Tong's lecture notes provides a clear and concise presentation of this expansion.

2. Commutation Relations for Creation and Annihilation Operators

We have two sets of commutation relations: the ETCRs for $\hat{\varphi }$ and $\hat{\pi }$, and the (yet unknown) relations for ${\hat{a}}_{\vec{p}}$ and ${\hat{a}}_{\vec{q}}^{†}$. A consistent quantum theory requires that these two sets be equivalent.

We postulate that the ladder operators for the field modes satisfy relations analogous to the single harmonic oscillator, but adjusted for a continuum of modes:
$[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}]=0$
$[{\hat{a}}_{\vec{p}}^{†},{\hat{a}}_{\vec{q}}^{†}]=0$
$[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}^{†}]=(2\pi {)}^3{\delta }^{(3)}(\vec{p}-\vec{q})$

The Dirac delta function ${\delta }^{(3)}(\vec{p}-\vec{q})$ appears instead of a Kronecker delta because $\vec{p}$ and $\vec{q}$ are continuous labels. It ensures that the creation and annihilation operators for different momentum modes are independent.

The main task is to show that these commutation relations for $\hat{a}$ and ${\hat{a}}^{†}$ are precisely what is needed to reproduce the ETCRs for $\hat{\varphi }$ and $\hat{\pi }$. The derivation involves substituting the field expansions into the commutator $[\hat{\varphi }(\vec{x}),\hat{\pi }(\vec{y})]$ and using the properties of the ladder operators.

The same section of David Tong's notes you just looked at contains a concise proof of this equivalence.

The following video provides an alternative and very detailed walkthrough of the same concepts. It starts by defining $\hat{a}$ and ${\hat{a}}^{†}$ in terms of $\hat{\varphi }$ and $\hat{\pi }$ (the inverse of our expansion) and then derives their commutation relations. This complementary perspective is very valuable for a complete understanding.

3. Time Evolution and the Heisenberg Picture

So far, we have worked in the Schrödinger picture, where operators are time-independent. However, our initial Lagrangian was Lorentz invariant, a symmetry that is obscured in this formalism. The Heisenberg picture, where operators evolve in time, makes the relativistic nature of the theory more apparent.

An operator $\hat{O}$ in the Heisenberg picture evolves according to ${\hat{O}}_H(t)=e^{i\hat{H}t}{\hat{O}}_S{e}^{-i\hat{H}t}$. Applying this to our ladder operators, and using the (soon to be derived) commutation relations $[\hat{H},{\hat{a}}_{\vec{p}}]=-{\omega }_{\vec{p}}{\hat{a}}_{\vec{p}}$ and $[\hat{H},{\hat{a}}_{\vec{p}}^{†}]=+{\omega }_{\vec{p}}{\hat{a}}_{\vec{p}}^{†}$, we find their time evolution is remarkably simple:
${\hat{a}}_{\vec{p}}(t)={\hat{a}}_{\vec{p}}{e}^{-i{\omega }_{\vec{p}}t},{\hat{a}}_{\vec{p}}^{†}(t)={\hat{a}}_{\vec{p}}^{†}{e}^{+i{\omega }_{\vec{p}}t}$

Substituting this back into our expansion for $\hat{\varphi }(\vec{x})$, we get the Heisenberg field operator $\hat{\varphi }(x)=\hat{\varphi }(\vec{x},t)$:
$\hat{\varphi }(\vec{x},t)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\vec{p}}}}\left({\hat{a}}_{\vec{p}}{e}^{-i{\omega }_{\vec{p}}t+i\vec{p}\cdot \vec{x}}+{\hat{a}}_{\vec{p}}^{†}{e}^{+i{\omega }_{\vec{p}}t-i\vec{p}\cdot \vec{x}}\right)$
Using the four-vector notation $p=({\omega }_{\vec{p}},\vec{p})$ and $x=(t,\vec{x})$, with the metric signature $(+,-,-,-)$, the exponent becomes $p\cdot x={\omega }_{\vec{p}}t-\vec{p}\cdot \vec{x}$. The expansion then takes the beautifully compact and manifestly Lorentz-covariant form:
$\hat{\varphi }(x)=\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2E_{\vec{p}}}}\left({\hat{a}}_{\vec{p}}{e}^{-ip\cdot x}+{\hat{a}}_{\vec{p}}^{†}{e}^{+ip\cdot x}\right)$
(Here we've switched to the more common notation $E_{\vec{p}}$ for the energy ${\omega }_{\vec{p}}$). This expression is the foundation of free quantum field theory.

4. A Bridge to Computation

Given your background in computational physics, it's worth seeing how these continuous-field expressions relate to a system that could be simulated on a computer. In numerical simulations, like lattice field theory, spacetime is discretized. In a 1D space of length $L$ with $N$ lattice sites, the continuous integral over momenta is replaced by a discrete sum over allowed momentum modes $p_k$, and the Dirac delta becomes a Kronecker delta.

The field expansion and commutation relations become:

• Field Expansion: ${\hat{\varphi }}_n\propto {\sum }_{k=0}^{N-1}\frac{1}{\sqrt{{\omega }_k}}\left({\hat{a}}_k{e}^{i{p}_k{x}_n}+{\hat{a}}_k^{†}{e}^{-i{p}_k{x}_n}\right)$
• Commutation Relation: $[{\hat{a}}_k,{\hat{a}}_l^{†}]\propto {\delta }_{kl}$

This discretized form is the starting point for implementing QFT on a computer. The following resource shows these exact expressions, providing a direct link between the theory we're learning and its numerical application.

Conclusion

Today, we have made a significant leap. By combining the canonical quantization formalism with the harmonic oscillator analogy, we have re-expressed the fundamental quantum field operators in terms of a more intuitive set of building blocks.

Key Takeaways:

• The quantum scalar field $\hat{\varphi }(x)$ and its conjugate momentum $\hat{\pi }(x)$ can be expanded as a Fourier integral over creation (${\hat{a}}_{\vec{p}}^{†}$) and annihilation (${\hat{a}}_{\vec{p}}$) operators.
• These ladder operators obey the commutation relations: $[{\hat{a}}_{\vec{p}},{\hat{a}}_{\vec{q}}^{†}]=(2\pi {)}^3{\delta }^{(3)}(\vec{p}-\vec{q})$, with other commutators being zero. These are the fundamental dynamical relations in momentum space.
• In the Heisenberg picture, the field expansion takes on a manifestly Lorentz-covariant form, confirming that our quantum theory respects special relativity.
• The operator ${\hat{a}}_{\vec{p}}^{†}$ is interpreted as creating a quantum of excitation with momentum $\vec{p}$, while ${\hat{a}}_{\vec{p}}$ annihilates one. These quanta are what we will soon identify as particles.

Preview of the Next Lesson:

We now have the operators that create and destroy field quanta. The next logical step is to define the states on which these operators act. In the next lesson, we will construct the Fock space for scalar particles and define the vacuum state. We will start with the state of lowest energy, the vacuum, which is defined as the state annihilated by all ${\hat{a}}_{\vec{p}}$, and then build a tower of multi-particle states by acting on it with creation operators.