They hadn’t drifted far in among the branches, but they’d be within reach of any determined allies. If he descended to the floor of the cavern, crossed through the undergrowth and climbed up the neighboring tree, there’d be no guarantee that the arborines would still be waiting for him.
Carlo looked up toward the ceiling, wondering if he should go back and fetch Lucia. But even that might take too long.
He dragged himself out along the branch he was holding, then grabbed another one and pulled the two together to the test the way they flexed. They were loose and springy; maybe an arborine could judge exactly how they’d recoil, but the task was beyond him.
Then again, if he aimed low he might face a long climb to his target, but he probably wouldn’t find himself stranded.
Carlo glanced down at his torn skin. He’d come too far to give up on the chase now. He crawled to the end of the swaying branch, holding it only with his lower hands, then pushed himself away into the air.
33
“We’ve hit a dead end,” Romolo confessed. “Just when the Rule of Two was starting to look plausible, we checked it against the second set of spectra and it fell apart.”
Carla glanced at Patrizia, but she appeared equally dispirited. They had been toiling over the spectra from the optical solid for more than a stint, but the last time they’d reported to her they had seemed to be close to a breakthrough.
“Don’t give up now!” Carla urged them. “It’s almost making sense.” She had hoped that the problem would yield to a mixture of focus, persistence and brute-force arithmetic—and it was easier to free her two best students from other commitments than to achieve that state herself. Someone had to supervise the experiments the Council had actually approved.
“Making sense?” Patrizia hummed softly and pressed a fist into her gut, giving Carla a pang of empathetic hunger. When things were going well there was no better distraction than work, but the frustration of reaching an impasse had the opposite effect.
“Why should the Rule of Two depend on the polarization of the beams?” Romolo demanded.
“And why the Rule of Two in the first place?” Patrizia added. “Why not the Rule of Three, or the Rule of One?”
Carla tried to take a step back from the problem. “The first set of spectra does make sense if every energy level can only hold two luxagens. Right?”
“Yes,” Romolo agreed. “But why? Once they’re this close together, luxagens simply attract each other. So how does a pair of luxagens get the power to push any newcomers away?”
“I don’t know,” Carla admitted. “But it would solve Ivo’s stability problem.” If each energy level could hold at most two luxagens, then beyond a certain point it would be impossible to squeeze more of the particles into each energy valley. That would be enough to prevent every world in the cosmos from collapsing down to the size of a dust grain.
Patrizia said, “For the first set of spectra, we made the field in the optical solid as simple as possible—using light polarized in the direction of travel for all three beams. With that kind of field, each luxagen’s energy only depends on its position in the valley. For the second set, we changed the polarization of one of the beams, so the luxagen’s energy depends on the way it’s moving as well as its position. But the strangest thing is that it looks as if there are more energy levels than there are solutions to the wave equation!”
Carla said, “I don’t see how that’s possible.” Two solutions—two different shapes for the luxagen wave—might turn out to have the same energy, but the converse was nonsensical. The luxagen’s energy couldn’t change without changing the shape of its wave.
Patrizia pulled a roll of paper from a pocket and spread it across Carla’s desk. The depth of the valleys in the optical solid had been chosen to ensure that they only had ten energy levels—limiting the possible transitions between them to a manageable number. But the data showed clearly that when one of the three beams was polarized so its field pointed at right angles to the direction of the light, the spectra split into so many lines that it took more than ten levels to explain them all.
“What if the luxagen has its own polarization?” Carla suggested. She’d ignored that possibility when first deriving the wave equation, largely for the sake of simplicity. “Depending on the precise geometry of the light field, the luxagen’s polarization could start affecting the energy—adding new levels.”
“Then it’s a shame we didn’t find a Rule of Three!” Romolo replied. “We could have said that the true rule was the Rule of One: in every valley, you can have at most one luxagen with a given energy and a given polarization. The Rule of Three would only hold for the simplest fields—where you couldn’t tell that the three luxagens were different, because their polarization had no effect on their energy.”
Patrizia turned to him. “But what if luxagens could only have two polarizations?”
Romolo was bemused. “Isn’t that like asking for space to have one less dimension?”
Carla wasn’t so sure; it could be subtler than that. She said, “Let’s make a list. If we’ve been working from false assumptions, what exactly would we need to have been wrong about in order to make things right?”
Patrizia warmed to the idea. “Luxagens have no polarization—wrong! Polarizations only come in threes—wrong! Any number of luxagens can share the same state—wrong again! I think that would cover it.”
Carla said, “The first one’s just an empirical question, but the second one’s going to take some thought.” She glanced at the clock on the wall; she’d told Carlo she’d meet him in his apartment by the sixth bell, but he knew better than to expect her to be on time. “Why do we assume that polarizations come in threes? For light, you have two vectors in four-space: the direction of the light field itself, and the direction of the light’s future. If I see a bit of light over here, and you see a bit of light over there, then I ought to be able to grab the two vectors that describe my light and rotate them together in four-space so they agree with those describing your light. That’s the absolute core of rotational physics: if we couldn’t do that, your light and my light wouldn’t deserve to be called by the same name.”
Patrizia said, “If the vectors are constrained to be perpendicular, they’ll look perpendicular to everyone. Fix the direction of the light’s history through four-space, and that leaves you with three perpendicular choices for the field—three polarizations.”
“You can imagine a case where they’re parallel instead,” she added. “Everyone would agree on that too. But you could never rotate one kind of light into the other, so there’d be no reason to classify them as the same thing at all.”
“So what are the choices?” Romolo said. “Light has three polarizations, but the alternative where the vectors are parallel only has one.”
“A luxagen wave takes complex values,” Carla reminded him. “So it has a kind of two-dimensional character to it already, if you think of real and imaginary numbers as pointing in perpendicular directions. But that doesn’t double the possibilities for polarization. You can rotate a luxagen wave by any angle at all in the complex plane without changing the physical state it describes.”
“So it halves the possibilities,” Patrizia said. “A complex wave looks two dimensional, but it really only has one dimension.”
“Half four is two,” Romolo noted. “Half the size of an ordinary four-vector gives us the number of polarizations we’re seeing. Does that help?”