Carla turned to Patrizia; the theory was hers to define and defend. Patrizia said, “That’s right.”
“Then why not complete the pattern?” Onesto suggested. “If you have reason to believe that the light field can manifest as a particle, why should Nereo’s own particle be different? Shouldn’t the luxagen be associated with waves in a field of its own?”
Patrizia looked confused, so Carla stepped in.
“There would be an appealing symmetry to that,” she said. “To every wave, its particle; to every particle, its wave. But I think it would complicate the theory unnecessarily. Without any evidence for a ‘luxagen field’, it’s hard to see what could be gained by including it.”
Onesto inclined his head politely. “Thank you for listening. I’ll leave you in peace now.”
He was halfway to his desk when Patrizia said, “You want us to treat the luxagen as a standing wave?”
Onesto turned. “I wasn’t thinking of anything so specific,” he admitted. “It just seems odd to treat the two particles so differently.”
At this response Patrizia’s confidence wavered, but then she persisted with her line of thought. “Suppose the luxagen follows the same kind of rules as the photon,” she said. “It has its own waves—and just like light waves, their frequency is proportional to the particle’s energy.”
Carla said, “All right. But…”
“If the luxagen is trapped in an energy valley,” Patrizia said, “its wave must be trapped as well. A trapped wave, a standing wave, can only take on certain shapes—each one with a different number of peaks.”
Carla felt the scowl vanish from her face. Unlike Patrizia’s last suggestion, this wasn’t hunger-addled nonsense. Onesto’s proposal had sounded naïve—but now Carla could see where her infuriating, erratically brilliant student was taking it.
For each shape it could adopt, the luxagen’s standing wave would oscillate with a specific frequency. The same kind of principle governed the harmonics of a drum: the geometry of the resonant modes of the drumhead dictated the particular sounds it could make, each one with its characteristic pitch.
But Patrizia’s rule linked frequencies to energies—so a trapped luxagen would only be able to possess certain energies. The energy closest to the top of the valley would set the gap that needed to be jumped in order for tarnishing to occur, and there would be no doing it by halves: a luxagen couldn’t accumulate five photons’ worth of energy and then wait around for a sixth. Once you reached the highest energy level there were no more resting places; it was an all-or-nothing trip. You either made the total number of photons you needed, all at once, and escaped the valley… or you didn’t.
As they talked it over, Patrizia sketched the general idea. Onesto looked on, pleased that his suggestion had proved helpful but a little daunted by the strange outcome.
“I still don’t understand the details of the timing,” Carla confessed, “but if you don’t get to make the photons separately, one by one, there’s no reason to expect the time it takes to be proportional to the number of photons.”
“Can we quantify any of this?” Patrizia asked.
Carla said, “We could try to write an equation for the luxagen wave. Whatever we know about the luxagen’s energy, we translate into the wave’s frequency; whatever we know about the luxagen’s momentum, we translate into the wave’s spatial frequency.”
The idea seemed straightforward, but they struck a problem almost at once. Taking the rate of change of an oscillating wave multiplied it by a factor proportional to its frequency, but also shifted the wave by a quarter-cycle: at every peak of the original wave the rate of change crossed zero, heading downwards, while at every such zero of the original wave the rate of change was at a minimum, the bottom of a trough. When Yalda had devised her light equation she had been able to go one step further: the second rate of change was shifted by another quarter-cycle, putting it a half-cycle away from the original—yielding the original wave turned upside down and multiplied by the frequency squared.
Multiples of the original wave were easy to combine. The geometrical relationship Yalda had sought to express—that the sum of the squares of the wave’s frequencies in all four dimensions was a constant—could be encoded in the wave equation simply by multiplying every term in that relationship by the strength of the wave, then re-expressing the squared frequencies as second rates of change.
But with a luxagen in a solid, the relationship between its energy and momentum included its potential energy, which depended on its position in the energy valley. It was impossible to write this relationship purely in terms of the energy squared—so it was impossible to talk only of frequencies squared. To go halfway and include the frequency itself meant taking the square root of the operation that turned the wave upside down—putting the square root of minus one into the wave equation.
“It looks as if we’re stuck with complex numbers,” Carla declared. “What does that mean? That our premises are wrong?”
Patrizia seemed to share her sense of trepidation, but she wasn’t ready to give up. “Let’s follow the mathematics,” she suggested. “We should see what the final answers are before deciding whether or not it all makes sense.”
To make the calculations easier they chose a field described by a single number—albeit complex—rather than a vector like light, with its different polarizations. They also assumed that the luxagen would be moving slowly. For a parabolic energy valley—the easiest idealization to work with—it was possible to solve the equation exactly.
As Patrizia had guessed from the start, there was a sequence of solutions with distinct shapes. Those shapes could be described with real numbers alone, though the wave’s variation over time swept out a circle in the complex number plane with a frequency corresponding to its energy.
Some solutions shared the same energy, though that was just a consequence of the idealized shape of the valley. Carla pushed on further and managed to calculate the effect of switching to a more realistic valley, closer to the kind that was actually produced by Nereo’s force in a solid.
For the parabolic case all the energies were governed by the natural frequency at which a luxagen—as a particle—would be expected to vibrate in such a valley. The gaps between the allowed energies all corresponded precisely to that frequency, while the lowest energy sat one and a half times higher above the valley floor. For a more realistic valley, all the energies were reduced slightly, and the perfect agreement between the multiple higher-energy solutions broke down, splitting the idealized single energy levels into closely spaced sets.
Onesto said, “Suppose the natural frequency for the valley is greater than the maximum frequency of light. That’s the assumption at the core of the original theory of solids. But what does it mean, in your terms?”
Carla thought for a moment. “It means the energy gap exceeds the mass of a photon—so creating a single photon can never give you enough energy to jump the gap.”
“And if the valley’s not a perfect parabola,” Onesto observed, “that doesn’t really change the significance of the main energy gaps, does it? There’ll be smaller gaps as well, but if the main ones are large enough there’ll still be energy levels where you need to make more than one photon in order to rise any higher.”