---
title: "Quantizing the normal modes"
description: "Build the free scalar vacuum from harmonic oscillators and calculate a mode’s zero-point variance."
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language: "en"
page_type: "story"
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  zh-CN: "https://sandboxphysics.com/zh/learn/field-theory/quantizing-a-field/"
last_modified: "2026-10-27"
---

[Fields, scales and critical phenomena](https://sandboxphysics.com/learn/field-theory/)

FR01 · L2 · Chapter 3

# Quantizing the normal modes

Build the free scalar vacuum from harmonic oscillators and calculate a mode’s zero-point variance.

Before you start: Chapters 1–2 and basic quantum probability. The commutator and complex amplitude are explained here.

You will be able to: Calculate occupation energy and vacuum variance without confusing a sample with a real-time trajectory.

A free quantum field is a collection of quantum harmonic oscillators, one for each independent spatial mode. A particle is an excitation of a mode. The vacuum is the state in which every mode has its lowest allowed energy. Its field average can be zero even though repeated measurements have a nonzero spread.

We can build this picture without imagining tiny particles hidden inside a classical wave. Begin with the oscillator we already know, change the rules for its coordinates, and then repeat the construction for the normal modes.

## One oscillator needs two operators

Use a mass-normalized coordinate $Q$ and its conjugate momentum $P$. For an ordinary displacement $q$ and oscillator mass $M$, these are $Q=\sqrt Mq$ and $P=p/\sqrt M$. The Hamiltonian is

$$
H=\frac12P^2+\frac12\omega^2Q^2.
$$

In quantum mechanics, coordinates become operators acting on states. The order of two operations can matter. Their commutator is $[Q,P]=QP-PQ=i\hbar$, where $i^2=-1$ and $\hbar$ is Planck’s constant divided by a full turn in radians. Complex amplitudes encode both magnitude and phase; their absolute square gives a probability.

Combine the two operators into a lowering operator $b$ and a raising operator $b^\dagger$:

$$
Q=\sqrt{\frac{\hbar}{2\omega}}(b+b^\dagger),\qquad P=-i\sqrt{\frac{\hbar\omega}{2}}(b-b^\dagger),\qquad [b,b^\dagger]=1.
$$

Substitute these expressions into the Hamiltonian. The terms with two lowering or two raising operators cancel. Reordering the remaining product with $bb^\dagger=b^\dagger b+1$ gives

$$
H=\hbar\omega\left(b^\dagger b+\frac12\right),\qquad E_n=\hbar\omega\left(n+\frac12\right).
$$

The integer $n$ counts excitations. The ground state satisfies $b|0\rangle=0$. Its coordinate and momentum uncertainties have a nonzero minimum product fixed by the commutator.

## Zero average does not mean zero spread

In the ground state, the averages of $b$ and $b^\dagger$ vanish, so $\langle Q\rangle=0$. In $Q^2$, the product $bb^\dagger$ contributes one even though $b^\dagger b$ contributes zero. Therefore

$$
\langle Q^2\rangle_0=\frac{\hbar}{2\omega},\qquad \langle P^2\rangle_0=\frac{\hbar\omega}{2}.
$$

Each term supplies half of the zero-point energy. For an occupation state, the coordinate variance becomes $\hbar(n+1/2)/\omega$.

For example, choose reference units with $\hbar=1$ and $\omega=2$. The vacuum energy is $1$, and the coordinate variance is $0.25$. Adding three excitations raises the energy to $7$ and the coordinate variance to $1.75$. The spread grows because an excited state puts more weight on displacements far from the center.

These variances refer to repeated preparations and measurements of a state. The distribution describes repeated measurements in a stationary state. A trajectory would require a separately specified dynamics or measurement protocol.

## Repeat the construction for the field

After mass normalization, a finite chain’s quadratic potential can be written as a matrix. Choose orthonormal eigenvectors of that matrix: each eigenvector specifies a shape that oscillates without changing into another shape. Transforming to these coordinates separates the Hamiltonian into independent oscillators,

$$
H=\frac12\sum_\alpha\left(P_\alpha^2+\omega_\alpha^2Q_\alpha^2\right).
$$

Quantize each pair. Its excitation number can increase independently of the others. The energy of one added quantum in mode $\alpha$ is $\hbar\omega_\alpha$. In a translation-invariant system, traveling modes also have definite momentum. This is the free-field particle interpretation: a mode’s excitations have the energy and momentum carried by particles.

For a relativistic scalar field in natural units $\hbar=c=1$, a mode with spatial momentum $k$ has $\omega_k=\sqrt{k^2+m^2}$. The parameter $m$ is a rest-energy scale. The mechanical support frequency in Chapter 1 played the corresponding mathematical role; the mechanical chain itself remains a different physical system.

## Why the Euclidean experiment has a different covariance

The equal-time vacuum covariance of mass-normalized quantum modes is $1/(2\omega_\alpha)$. A Gaussian probability density with action $S_E=\phi^{\mathsf T}K\phi/2$ instead has covariance $K^{-1}$. These expressions answer different questions.

One bridge is imaginary time. For a single oscillator in its ground state, define the Euclidean correlation at separation $\tau$ by inserting the decaying evolution operator $e^{-H|\tau|}$, with the vacuum energy removed. Since $Q$ acting on the vacuum creates one excitation of energy $\omega$ in natural units, the result is

$$
G_E(\tau)=\langle0|Qe^{-(H-E_0)|\tau|}Q|0\rangle=\frac{e^{-\omega|\tau|}}{2\omega}.
$$

The same function is the inverse of the one-dimensional differential operator $-\partial_\tau^2+\omega^2$ on an infinite Euclidean line. Thus a one-dimensional Euclidean scalar path model can teach us about the oscillator’s imaginary-time correlations. A periodic finite interval corresponds to a finite inverse temperature; fixed endpoints condition the paths. The infinite-extent limit removes the periodic thermal corrections.

The Euclidean correlation Lab samples this one-dimensional Gaussian path distribution. Its horizontal coordinate is a Euclidean coordinate or a statistical-field position, according to the stated interpretation. Replay moves through sampled configurations. To model the equal-time vacuum of a scalar field with one spatial dimension, we would need the distinct mode covariance $1/(2\omega_k)$.

## A task about the range of correlations

Before opening the correlation Lab, predict what happens to the Euclidean decay when the mass parameter doubles. In the infinite-line continuum limit, the decay length halves. At finite spacing the lattice decay length is $\xi_a=a/[2\operatorname{arsinh}(am/2)]$, which approaches $1/m$ when $am$ is small. Compare masses $0.5$ and $1$ at spacing $0.125$ in reference units. Then increase the number of sites at fixed spacing to test the boundary influence. Use the analytic finite-domain reference for short intervals.

## Exercises

### How much energy does one more quantum add?

An oscillator in reference units has $\omega=3$ and $\hbar=1$. Find the energy difference between the second and fifth occupation states.

Answer and reasoning

The difference is $3\hbar\omega=9$. The zero-point contribution appears in both energies and cancels. Changing occupation counts changes energy in integer steps of the mode frequency.

### Can the average field identify the vacuum?

Both the vacuum and a first excited oscillator state have zero mean displacement. What additional measurement distinguishes them?

Answer and reasoning

Measure the displacement variance or the energy over repeated preparations. Their coordinate variances are $\hbar/(2\omega)$ and $3\hbar/(2\omega)$ respectively. A zero mean can hide very different probability distributions.

### Which covariance belongs in a new simulation?

A proposed program draws spatial Fourier amplitudes with variance $1/(k^2+m^2)$ and labels the result an equal-time vacuum of a scalar field with one spatial dimension. What must change?

Answer and reasoning

For mass-normalized canonical modes, the equal-time vacuum variance is $1/(2\sqrt{k^2+m^2})$. The proposed variance belongs to a Euclidean Gaussian statistical measure. The program must either change the distribution or describe the Euclidean model it actually samples, including its normalization and boundary conditions.

## Try the physics

[Euclidean Field Correlations →Move two probes apart on a sampled scalar field. Compare the correlation estimated from many field configurations with the lattice prediction and its continuum limit.](https://sandboxphysics.com/labs/vacuum-correlations/)

## Sources and reading scope

1. [David Tong, Quantum Field Theory, Chapter 2](https://davidtong.org/teaching/quantum-field-theory/qfthtml/S2)

   Sections 2.1–2.7: oscillator quantization, free scalar modes, vacuum two-point functions and spacelike commutators.

2. [David Tong, Statistical Field Theory, Chapter 2](https://davidtong.org/teaching/statistical-field-theory/sfthtml/S2)

   Gaussian integrals, inverse-kernel covariance, correlation length and the Euclidean quantum-mechanical interpretation. Finite-chain normalizations are derived for this course.

[← Symmetry tells us what flows](https://sandboxphysics.com/learn/field-theory/symmetry-and-causality/) [What a correlation function predicts →](https://sandboxphysics.com/learn/field-theory/correlators-to-scattering/)

Reading order

[1. How a chain of oscillators becomes a field](https://sandboxphysics.com/learn/field-theory/from-oscillators-to-fields/) [2. Symmetry tells us what flows](https://sandboxphysics.com/learn/field-theory/symmetry-and-causality/) [3. Quantizing the normal modes](https://sandboxphysics.com/learn/field-theory/quantizing-a-field/) [4. What a correlation function predicts](https://sandboxphysics.com/learn/field-theory/correlators-to-scattering/) [5. Putting a Euclidean field on a lattice](https://sandboxphysics.com/learn/field-theory/euclidean-lattice-and-errors/) [6. Changing resolution while keeping predictions](https://sandboxphysics.com/learn/field-theory/keeping-low-energy-physics/) [7. Testing a convincing data collapse](https://sandboxphysics.com/learn/field-theory/finite-size-scaling/) [8. Reading a Wilson–Fisher flow diagram](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/)