Quantizing the normal modes

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

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 $b^\dagger$:

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

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

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,

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

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.

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?

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?