Symmetry tells us what flows

Energy moves along a vibrating string because neighboring pieces do work on one another. A field equation can tell us how much energy leaves a small region as well as whether the total stays constant. Noether’s theorem connects that local flow to a symmetry: the laws do not change when the clock’s origin is shifted.

We will first calculate the flow directly. That gives the abstract word “current” something concrete to refer to.

Follow the energy of a short interval

Keep the displacement field and units from Chapter 1. The mass density is $\mu$, the tension parameter is $\kappa$, and the on-site frequency is $\Omega$. The energy per unit length is

Differentiate each term with respect to time. In the first term, replace $\mu\partial_t^2\phi$ using the field equation from Chapter 1. The term containing $\Omega^2$ cancels the time derivative of the support’s potential energy. The remaining terms combine by the product rule:

Define the energy current $j_E=-\kappa(\partial_t\phi)(\partial_x\phi)$. Its units are watts. A positive value means energy crossing a point toward increasing $x$. We obtain a local conservation equation,

Integrate from a left endpoint $x_1$ to a right endpoint $x_2$. The spatial derivative becomes the current at the two ends:

An interval loses energy when more flows out at the right than enters at the left. Its own energy need not be constant even when the whole isolated string conserves energy. Boundary conditions decide whether the endpoint currents vanish, cancel, or supply energy from outside.

Check the sign on a traveling wave

For $\phi=A\cos(kx-\omega t)$ with positive $k$ and $\omega$, the time derivative is $A\omega\sin(kx-\omega t)$ and the spatial derivative is $-Ak\sin(kx-\omega t)$. Their product is negative, so the energy current points right. Its average over a period is

Take a string without on-site restoring force, with $\mu=0.500\,\mathrm{kg\,m^{-1}}$ and $\kappa=2.00\,\mathrm{N}$. Then $c=2.00\,\mathrm{m\,s^{-1}}$. A wave with $A=0.0100\,\mathrm{m}$ and wavelength $2.00\,\mathrm{m}$ has $k=\pi\,\mathrm{m^{-1}}$ and $\omega=2\pi\,\mathrm{s^{-1}}$. The mean transmitted power is $1.97\,\mathrm{mW}$. Reversing the travel direction reverses the current; doubling the amplitude multiplies the power by four.

Why a symmetry produces a current

A symmetry changes the description while leaving the action unchanged, apart from an endpoint contribution. Shifting every time label by the same small amount leaves an autonomous string’s action unchanged. No external mechanism distinguishes “start the clock now” from “start it later.” The conserved quantity associated with this time-translation symmetry is energy.

For a general field, write $\pi^\mu=\partial\mathcal L/\partial(\partial_\mu\phi)$. An infinitesimal field change $\delta\phi$ that changes the density by a total derivative, $\delta\mathcal L=\partial_\mu F^\mu$, gives the current

Repeated indices denote a sum over the chosen coordinates. The proof is a short rearrangement: vary the density, integrate its derivative term by parts, and use the field equation to eliminate the remaining coefficient of $\delta\phi$. What survives is the divergence written above. For spacetime translations one also includes the coordinate shift; its time component gives the energy density and flux we already calculated.

A spatial translation gives momentum conservation in a uniform system. Noether currents arise from continuous symmetries. Reversing the sign of a real scalar field offers only two choices; the infinitesimal derivation above requires a continuously adjustable transformation.

Local coupling and causal propagation

The continuum field equation contains derivatives at one position. For the massless string it admits right- and left-moving waves $f(x-ct)+g(x+ct)$. A disturbance initially confined to a small interval reaches a point outside that interval only after a travel time set by $c$. Adding the local on-site restoring term changes the waveform but retains the continuum equation’s characteristic propagation speed.

The discrete chain has a different mathematical detail. Its wave packets have group velocity $d\omega/dk$ bounded by $c$, but a local perturbation can generate very small tails outside that packet’s main front. The continuum model’s exact domain of dependence should not be read straight into a finite lattice animation.

Relativistic quantum fields express causality through local observables. Their commutator compares the results of swapping two operations; Chapter 3 defines it precisely. Spacelike separation means the spatial distance exceeds the distance light can travel during the time difference. The commutator vanishes there, so the two operations cannot be used to signal between those events. A two-point correlation can still be nonzero. Correlation asks what values tend to occur together; response asks what a controlled intervention can change. We return to this distinction after quantization.

A standing wave can hide the flow

In the field-modes Lab, identify a fixed-end standing-wave shape. Use that shape as a classical displacement in this chapter’s equations. Over a cycle, the classical energy trades between motion and deformation. Decompose it into opposite traveling waves with equal mean power and predict the mean net current: the result is zero even though both components carry energy. The Lab’s occupation number is the separate quantum interpretation developed next.

Exercises

Does constant total energy prove local conservation?

Imagine a rule that removes energy at one point and adds the same amount at a distant point instantly. Would it conserve the total energy? Would the string’s local conservation law describe it?

Where does a driven interval get its energy?

An interval has zero current at its right endpoint and an incoming mean current of $2.00\,\mathrm{mW}$ at its left. Neglect dissipation. How much energy does it gain in $3.00\,\mathrm{s}$?

What if the tension is deliberately changed in time?

Suppose an external mechanism changes $\kappa(t)$ throughout the string. Which step of the conservation argument changes, and what physical quantity accounts for the difference?