The wave a row of moving clocks leaves on an instant

Now is a choice of slicing found that two events simultaneous for one observer are not simultaneous for another moving past, and that the disagreement grows with the distance between the events: an observer moving at $v$ along a line finds that clocks synchronised in the laboratory are out of step by $vx/c^2$ over a distance $x$. Usually that fact is put to work on two events, or on a rod with a front and a back. It is worth putting to work on a whole row of clocks, because the row turns a statement about simultaneity into a wave — and the wave turns out to be one that physics had already named for a different reason.

The construction is Louis de Broglie’s, from the notes of 1923 that became his doctoral thesis the next year. The usual account of the thesis says that he took the relation between a photon’s energy and its frequency, applied it to an electron, and wrote down a wavelength. Everything has a wavelength followed that account. But de Broglie’s own route started from a clock and from relativity, and it shows more clearly than the usual one why the matter wave has the strange properties it has: a frequency that seems to contradict time dilation, and crests that move faster than light.

A row of clocks that do not agree #

Take a long row of identical clocks, each ticking at a frequency $\nu_0$ in its own rest frame, and synchronise them in that frame by any of the usual methods. Then let the whole row move past a laboratory at a steady speed $v$. The first figure draws the row’s history on the laboratory’s spacetime diagram. Each clock is a straight line sloping at the row’s speed; the dots on each line mark the moments at which that clock completes a tick.

Join the dots at which the clocks show the same reading — every clock reaching tick seven, say — and the result is a straight line across the diagram, the row’s own instant “everyone reads seven”. It is not horizontal. It slopes upward to the right with slope $v/c$, which is the tilt of a moving frame’s simultaneity that which came first, and who decides found. The laboratory’s instant, by contrast, is the horizontal line. On that line the clocks of the row do not agree: the one at the back has already reached seven when the one a little ahead is still at six.

How much they disagree is fixed by the Lorentz transformation. The clock at laboratory position $x$ reads $t' = \gamma(t - vx/c^2)$ at laboratory time $t$, where $\gamma = 1/\sqrt{1-v^2/c^2}$. Along a single laboratory instant, the reading therefore falls in proportion to distance, by $\gamma v/c^2$ for each unit of length. The readings fall by one full tick, a time $1/\nu_0$, over a distance

$$\lambda = \frac{c^2}{\gamma v \nu_0}.$$

The readings form a wave #

A clock’s reading is a phase: it goes round once every tick. Plot the cosine of each clock’s phase along one laboratory instant, as the second figure does for three speeds, and the result is a wave, with the wavelength just found. Nothing about waves went into it. It is a row of ordinary clocks, synchronised where they rest, read from a frame in which they move; the wave is the pattern that the tilt of simultaneity paints across the row.

The wavelength shrinks as the row speeds up. At $0.3c$ the readings repeat every 3.2 light-periods of a clock — the distance light travels in 3.2 ticks — and at $0.85c$ every 0.6. At low speeds it is enormous, because the tilt of the moving instant is tiny, and at speeds near light’s it falls as $1/\gamma$.

Now give each clock a particular frequency: its rest energy divided by Planck’s constant, $\nu_0 = mc^2/h$. For an electron that is $1.24\times10^{20}$ cycles a second, the frequency de Broglie supposed every particle carries with it, and which is now called the Compton frequency. With that choice the wavelength becomes

$$\lambda = \frac{c^2}{\gamma v}\cdot\frac{h}{mc^2} = \frac{h}{\gamma m v} = \frac{h}{p},$$

the de Broglie wavelength, Planck’s constant over the momentum. It is not a separate postulate in this route. Given a clock that ticks at its rest energy over $h$, the Lorentz transformation produces the wavelength by itself.

Two frequencies for one clock #

This is where de Broglie found a paradox, and resolved it. A particle’s energy rises with its speed as $\gamma mc^2$, and if frequency is energy over $h$, the frequency should rise too, to $\gamma\nu_0$. But a moving clock runs slow: by the clock that has to slow, and why no clock can refuse, a clock ticking at $\nu_0$ at rest ticks, by the laboratory’s time, at only $\nu_0/\gamma$ when it moves. One particle seems to need two frequencies that differ by a factor $\gamma^2$.

The third figure shows that both are present, in the same set of readings, sampled along two different lines. Stand at one fixed place in the laboratory and watch the clocks go by: the reading at that spot climbs at $\gamma\nu_0$, faster than any clock at rest ticks, because each passing clock is ahead of the last one in reading by a little more than the time between their arrivals would account for. Follow one clock instead: it ticks at $\nu_0/\gamma$, as time dilation requires. The two numbers are the frequency of the wave at a fixed point and the ticking of a source carried along with the wave, and they are related exactly as a moving source’s ticking and the field it is embedded in should be.

De Broglie called the agreement the harmony of phases, and stated it as a theorem: a clock carried along with the particle stays permanently in phase with the wave at its own position. In the readings it is almost a tautology. The pattern is the clocks’ readings, so whatever clock sits at a given place necessarily shows the pattern’s phase there. The surprise is only in the speeds: the clock moves at $v$, and the crests of the pattern move past it at a different speed, so the clock stays in step with a wave that is overtaking it.

Crests faster than light, carrying nothing #

How fast do the crests move? A crest is a line of constant reading — one of the row’s own instants — and on the spacetime diagram those lines have slope $v/c$, so they sweep along the laboratory’s $x$ axis at $c^2/v$. For a row moving at $0.6c$ the crests move at $1.67c$; for a slow row, faster still, without limit as the row comes to rest. That is the phase velocity of the de Broglie wave, $c^2/v$, the result that everything has a wavelength found had embarrassed the idea when it was first put forward.

In this construction there is nothing embarrassing about it. A crest of the reading pattern is not a thing; it is a set of events at which a moving frame’s clocks agree, and the speed at which that set slides along another frame’s axis is a property of the two frames, not of anything travelling. It is the same kind of speed as the spot of a searchlight swept across a distant screen in the spot that outruns light and carries nothing, and as the “speed” of the place where two tilted rulers cross. Superluminal phase velocity, in the de Broglie wave, is the relativity of simultaneity seen sideways.

The fourth figure adds what a real particle needs. A single infinite row with one exact speed fills all of space with its pattern, and a particle is somewhere. Add together the patterns of many rows with slightly different speeds, and they reinforce only where their readings agree, which happens in one region that moves; elsewhere they cancel. The result is a packet. Its peak travels at $0.600c$, the speed of the clocks, while the crests inside it run forward at $1.67c$, appear at the back of the packet and vanish at its front. The packet that moves at another speed than its own crests found the general rule — a packet moves at the group velocity $d\omega/dk$, and the crests at $\omega/k$ — and for the clock pattern the two are $v$ and $c^2/v$, whose product is $c^2$. The envelope is where the clocks are. The crests are where their frames’ instants are.

How long the wave of a real particle is #

The fifth figure puts the construction to work on real particles. A clock with the rest energy of a particle has a Compton wavelength, $h/mc$, the distance light travels in one of its ticks; and the reading pattern’s wavelength is that, divided by $\gamma v/c$. For an electron the Compton wavelength is 2.4 picometres, and an electron moving at a few per cent of light’s speed — accelerated through a hundred volts — has a pattern 0.123 nanometres long, comparable with the spacing of atoms in a crystal, which is why Clinton Davisson and Lester Germer saw electrons diffract off nickel in 1927. A neutron is 1,839 times heavier, so its clock ticks 1,839 times faster, and a neutron from a reactor, moving at a mere 2,200 metres a second, has a pattern of 0.18 nanometres, again a crystal spacing. The molecule that interferes until it glows followed fullerene molecules through an interferometer, and the first such experiment, in 1999, diffracted $\mathrm{C}_{60}$ molecules of 720 atomic mass units, moving at a couple of hundred metres per second, from a grating with a period of 100 nanometres. Their pattern is under three picometres long, some three hundred times smaller than the molecule itself.

The clocks themselves tick far too fast to observe. An electron’s Compton frequency is $10^{20}$ cycles a second, a neutron’s $2\times10^{23}$, and nothing can count either. What is observable is the slow drift of the reading along a moving beam, the thing a grating or a crystal compares at two places a few nanometres apart. The pattern is visible precisely because it is the small difference between two nearly equal and invisible things: the readings of clocks at neighbouring places.

One invariant, two relations #

The construction says something about the two relations that every introduction to quantum mechanics states separately: energy is Planck’s constant times frequency, and momentum is Planck’s constant over wavelength. In the clock picture they are not two facts but one.

A clock’s reading at an event is a number every observer agrees on. Whatever frame the row is described in, the clock that passes through a given event shows the reading it shows; the Lorentz transformation changes the coordinates of the event and leaves the reading alone. So the phase $2\pi\nu_0 t'$ is a scalar field on spacetime. How fast it changes with laboratory time at a fixed place is the wave’s angular frequency, $2\pi\gamma\nu_0$, and how fast it changes along laboratory space at a fixed time is its wavenumber, $2\pi\gamma v\nu_0/c^2$. Those two rates of change are the time and space components of one four-vector, the gradient of an invariant, and they transform together. Multiply both by $h/2\pi$ and they become $\gamma mc^2$ and $\gamma mv$: the energy and momentum, which speeds that refuse to add found to transform together for exactly the same reason. The statement “frequency is energy over $h$” and the statement “wavelength is $h$ over momentum” are the time part and the space part of the single statement that a particle’s phase is its rest frequency times its proper time.

That is also why the product of phase and group velocities is $c^2$ and not some other speed. For any four-vector of the form (energy, momentum), the ratio of energy to momentum times the derivative of energy with respect to momentum is $c^2$, by the relation $E^2 = p^2c^2 + m^2c^4$. Light has $m = 0$ and both velocities equal to $c$; it has no rest frame, no proper time and no clock, and its wave cannot be built this way.

Why no row of real clocks shows it #

The tilt of simultaneity is tiny at everyday speeds, so the pattern exists for every row of moving clocks and is invisible for any row that can be built. Put a line of lamps along a railway platform, synchronised to flash once a second, and ride past them on a train at thirty metres per second. On the train’s instants, lamps a kilometre apart flash out of step by $vx/c^2$, a third of a picosecond. The pattern of flashes is a wave of wavelength $c^2/v\nu_0$, which for these numbers is $3\times10^{15}$ metres — a third of a light-year. Nothing on the train could register it.

The wave becomes short only when the clocks tick enormously fast. The wavelength is the distance light travels in one tick divided by $\gamma v/c$, and to bring it down to the scale of anything measurable at modest speeds the tick has to be shorter than the time light takes to cross an atom. Rest-energy clocks are the only ones that fast. That is the real content of the construction: the de Broglie wave is the relativity of simultaneity made visible by the fastest clock there is.

Whether the clock is real #

The argument gives a wave from a clock, and the obvious question is whether the clock exists. De Broglie assumed it did, as an internal periodic process of every particle; quantum mechanics as it developed did not need it, because the Schrödinger equation’s solutions carry a phase that does the same job without being interpreted as a ticking thing. In standard quantum mechanics the absolute phase of a wavefunction is unobservable, and so is the Compton frequency that would set it — only differences of phase between paths show up.

The question was reopened in 2013, when Holger Müller and colleagues argued that atom interferometers, which split an atom into two wave packets that travel different paths and recombine them, measure the Compton frequency directly: the phase difference between the two paths is the atom’s Compton frequency multiplied by the difference in proper time along them. They used the interpretation to measure the mass of a caesium atom against its Compton frequency, a “clock based on a single massive particle”. The interpretation was disputed — in the standard analysis the Compton frequency cancels and what is measured is a recoil frequency, a difference that involves Planck’s constant and the mass separately. Both sides agree on the numbers; they disagree about which of them deserves the name of a clock. The orbit that ages less than a throw met the same proper-time phase in a different setting, and the fall that leaves the mass in the phase found it in neutrons falling through a gravitational field, where the mass enters the interference pattern because it sets the rate at which the phase accumulates.

Where the picture is exact and where it is not #

The construction is exact special relativity: a row of clocks, a Lorentz transformation, nothing else. Its two inputs beyond relativity are that a particle carries a clock at $mc^2/h$ and that the pattern of readings behaves like a wave — that patterns from different possible paths add, cancel and interfere. The first input gives the wavelength; the second is the content of quantum mechanics, and no amount of relativity supplies it. Clocks do not interfere; amplitudes do.

The picture also strains for a single particle. A row of clocks has a clock at every position, and the reading pattern is defined everywhere along the row. One electron is one clock, at one place, and the wave describes the amplitude for finding it at each place rather than a clock that is there. De Broglie’s picture is therefore a way of seeing why the matter wave has the frequency, wavelength and phase velocity it has, given that particles have waves at all. It does not explain why they do.

The argument is also strictly a free-particle one. In a potential, or in a gravitational field, the clock’s rate varies from place to place and the pattern is no longer a uniform wave; it bends, which is what lets the wave be focused by electric fields in an electron microscope and refracted by gravity in a neutron interferometer. The local wavelength remains $h/p$ with the local momentum, and the clock picture survives in the form of the proper time accumulated along each path, but the simple tilted row of the first figure does not.

A pattern of readings that does not interfere #

The figures show a classical row of clocks and a pattern of their readings. They do not show interference of the pattern with itself, which is what a diffraction experiment detects, and they do not show the probabilistic reading of the wave, in which its intensity is the chance of finding the particle. The packet of the fourth figure is built from rows of clocks with a smooth spread of speeds, which mimics a quantum wave packet’s spread of momenta; it does not show the packet spreading out over time, which the same construction would give if it were run long enough, because the rows with different speeds drift apart.

Still open: whether a particle’s proper time can be made to interfere #

If a particle carries a clock, then sending one particle along two paths with different proper times should leave the two parts of its wave carrying different readings, and the difference should be visible in the interference — not as the familiar phase shift, which every interferometer sees, but as a loss of contrast, because the two parts’ clocks would show which path was taken. Proposals to see this with atoms carrying an internal clock, such as a superposition of two electronic levels whose difference ticks at an optical frequency, have been made since 2011; the time-dilation difference between paths at different heights in the Earth’s gravity is small, and no experiment has yet seen the predicted loss of contrast clearly. Whether it appears exactly as proper time predicts, for a quantum clock in a superposition of two worldlines, is a test of how relativity’s time and quantum mechanics’ superposition combine.

The habit worth carrying away is to look for the instants hidden in a wave. A row of clocks synchronised in its own frame shows, on any laboratory instant, readings that fall along the row by one tick every $c^2/\gamma v\nu_0$; with $\nu_0 = mc^2/h$ that is $h/p$, 0.123 nanometres for a 100 eV electron, and the crests, being the row’s own instants, sweep along at $c^2/v$ while a packet of such rows moves at $v$. De Broglie’s wavelength is the relativity of simultaneity applied to a particle’s clock; what quantum mechanics adds is that the readings interfere.