The Eternal Flame

“But light, undeniably, is a wave. Giorgio and Nereo showed us how to measure its wavelength from the interference pattern it makes when it passes through two or more slits. Carla and Patrizia have never sought to deny that, but they do ask us to accept that this wave is always accompanied by a suitable entourage of particles —not so much driven along with the wave by any explicable force, as bound to it by an axiom too profound and opaque to yield to any further reflection or inquiry.”

Carla tried not to grow angry; in truth, this was the weakest part of their theory. But no amount of sarcasm directed at that awkward hybrid ontology could change the evidence: light’s granular nature was every bit as plain as its wavelike properties.

“What are we to make of this?” Assunto continued. “I propose a solution that builds on the success of Patrizia’s principle. The equation governing a particle trapped in an energy valley is transformed, by that principle, into an equation for a standing wave in the same valley. Such a wave can only take on certain distinct shapes, each with its own characteristic energy.

“But if our ideas about the mechanics of a simple particle require such a radical new approach, surely we shouldn’t apply it in a piecemeal fashion? Suppose we can identify another system that appears to be governed by the very same equations that we once thought adequate to account for the motion of a particle in a valley. Shouldn’t that system be treated in exactly the same way?”

Carla had no idea what example he had in mind, but on the face of it this sounded like a reasonable proposal.

Assunto said, “Consider a light wave with a single, pure frequency, traveling in a certain direction and possessing a definite polarization. In the real world we never encounter anything so simple—but the actual waves we do find traveling through the void can always be constructed by adding together a multitude of those idealized waves.

“Because this wave has a single frequency, we can capture everything about the way it changes over time by picking one location and measuring the strength, the amplitude, of the light field at that point. This amplitude behaves very simply: it oscillates back and forth at the frequency shared by the entire wave.

“Does that remind you of anything? Such as… a particle rolling back and forth in an energy valley?”

Assunto paused, as if expecting objections, but the room was silent. Carla wanted to leap ahead of him—to complete the analogy, grasp its implications and find some fatal flaw that he had missed—but her mind seized up and the opportunity passed.

“The parallels can be made precise,” Assunto claimed. “The amplitude of our idealized light wave corresponds to the distance of a particle from the center of a one-dimensional, infinitely high, parabolic energy valley. The energy of the light wave can be broken down into two parts: one analogous to the particle’s potential energy, due only to its position in the valley, and the other analogous to its kinetic energy, which depends only on its speed.

“Carla and her team have already shown us what happens when you apply Patrizia’s principle to a particle trapped in an energy valley in a solid. Our system is actually simpler, since the valley in a solid is three-dimensional, and it’s not exactly parabolic. The simpler version of all the same calculations yields an infinite sequence of energy levels, all spaced the same distance apart.

“What determines the spacing of those energy levels? In a solid, it comes from the natural frequency of a particle rolling in the valley—so in our case, it comes from the frequency at which the amplitude of the light field oscillates back and forth. So the light wave must have an energy that belongs to a set of discrete values, and the gap between each level and the next will be equal to Patrizia’s constant times the frequency of the light.”

Carla knew exactly where he was going now—and exactly what his belittling title meant.

“But that gap is precisely the energy attributed to each photon associated with this light wave!” he proclaimed. “So the fact that the wave can only change its energy in discrete steps no longer requires the peculiar fiction of a swarm of particles following the wave around, like mites caught on a breeze.”

Assunto sketched two examples on his chest.

“Are there particles of light in this picture?” he mused. “Not if you think a ‘particle of light’ is something like a tiny grain of sand. The number of photons associated with each wave is really just a label for its energy level, found by counting the steps up from the lowest level. It’s not a count of things you could hold in your hand.”

“The lowest level, with zero photons, doesn’t have zero energy,” Carla protested. “That’s…”

“Strange?” Assunto suggested. “I agree. But the same kind of thing is true of your luxagen in a solid: you can’t make it lie still at the bottom of the valley.”

“Yes, but at least there’s something there in the valley,” Carla replied. “You’re claiming that the void itself has energy—with a contribution from every possible mode of the light field!” Every frequency, every direction, every polarization that any light wave could possess would each leave the vacuum with a trace of energy—without the need for the light waves themselves to be present at all.

Assunto said, “Only changes in energy are detectable. The actual value is a meaningless concept: if you redefine every energy level by adding or subtracting the same amount, that won’t change anything you can measure. So it doesn’t bother me at all if a theory gives a non-zero value for empty space… but if you prefer to subtract that value from everything in sight, bringing the vacuum energy down to zero, go ahead and do that. It won’t make any difference.”

Carla fell silent. The result still struck her as preposterous, but she couldn’t yet see how to argue against it.

“Where has this taken us?” Assunto continued. “We started with Yalda’s light field, which has a precise value at every point in space and every moment in time. But now Patrizia’s principle has given us a theory where we can no longer think that way. Just as a luxagen in a solid lacks a precise location and is spread out across its valley, the amplitude of a light wave must also be spread out across a range of values. In a strong enough light wave, the spread of values can be much less than the peak amplitude of the light, so this need not contradict the way we use waves in conventional optics. But when a single luxagen ‘scatters a photon’—when it lowers by one the photon count for light of a certain frequency and direction, and then raises by one the photon count for light of a different frequency and direction—we should neither expect conventional optics to apply, nor assume that the failure of the old laws means that we’re describing something akin to colliding grains of sand.”

Assunto spread his arms in a gesture of finality. There’d be more details in the paper itself, but his presentation was finished. “Questions?”

Most people in the room looked as if they were still struggling to absorb what they’d heard, but Onesto responded immediately.

“What about luxagens?” he asked Assunto.

“What about them?”

“Can they fit into the same framework? If photons are really just steps in the energy levels of a light wave, can you account for luxagens the same way?”

Assunto said, “When a luxagen wave in a solid rises to a higher energy level, that doesn’t amount to making a new luxagen. It just means the original luxagen has more energy than before.”

“I understand that,” Onesto replied. “But I’m not talking about the energy levels in a solid. You took a light wave traveling through empty space, and showed that the energy levels of each mode amounted to what Carla and Patrizia would have called the number of photons in the wave. So why can’t you do the same thing with a luxagen wave in empty space, finding energy levels for each mode of that wave that correspond to the number of luxagens?”

“Because they’re completely different kinds of waves!” Assunto said. “A light wave isn’t all that different from a wave on a string: the higher its peaks, the more energy it carries. Given that relationship between energy and wave size, we can come along and apply Patrizia’s principle, which forces the energy to take on discrete values.

“But to get luxagen waves in the first place, we’ve already applied Patrizia’s