by Greg Egan
“Yes.” Carla looked down at her final result again. Two half-turns left a vector unchanged; two half-turns left a leftor multiplied by minus one. But the probabilities that could be extracted from a luxagen wave involved the square of the absolute value of some component of the wave. Multiplying the entire wave by minus one wouldn’t change any of those probabilities.
Romolo said, “So when you rotate this system all the way back to its starting point, the wave changes sign. But we can’t actually measure that… so it doesn’t matter?”
“It’s strange,” Carla agreed. “But what troubles me more is treating the rotation’s left vector differently from the right. All it takes to swap the role of those two vectors is to view the system in a mirror. Should physics look different, viewed in a mirror? Have we seen any evidence of that?”
Patrizia took the criticism seriously. “What if we tried to balance it, then? Could we throw in a ‘rightor’ as well as a leftor, for symmetry’s sake?” She wrote the transformation rule for this new geometrical object, a mirror image of her previous invention.
“Throw it in where?” Romolo asked.
“Into the luxagen wave,” Patrizia replied. “Two more complex numbers, but these ones transform by the rightor rule. If you look at the system in a mirror, the leftor and rightor change places.”
“That sounds very elegant,” Romolo said, “but haven’t you just doubled the number of polarizations from two to four?”
“Hmm.” Patrizia grimaced. “That would defeat the whole point.”
Carla pondered the new proposal. “The light field is a four-dimensional vector—but we don’t get four polarizations, because of the relationship between the field vector and the energy-momentum vector. What if there’s a relationship between the luxagen field—the leftor and the rightor—and the luxagen’s energy-momentum vector? Something that brings the number of polarizations back down to two.”
Romolo said, “What kind of relationship? Setting a leftor or a rightor perpendicular to an ordinary vector won’t work—when you rotate all three of them, they’ll change in different ways, so the relationship won’t be maintained.”
“That’s true,” Patrizia conceded. She drove her fist into her gut; the glorious distraction was losing its power again. “Maybe we should tear this up and start again.”
Carla said, “No. The relationship’s simple.”
She wrote:
“That’s it,” she said. “Just look at how these three things transform when we rotate them.”
“A leftor divided by a rightor changes in exactly the same way as an ordinary vector. So if we demand that the energy-momentum vector of a luxagen wave is proportional to the wave’s leftor divided by its rightor, rotation won’t break the relationship—and any free luxagen wave that meets this condition could be rotated into agreement with any other.”
Romolo said, “And the rightor is completely fixed by the leftor and the energy-momentum vector. There are no extra polarizations.”
Patrizia looked dazed. She said, “Follow the geometry and everything falls into place.” She exchanged a glance with Carla; this was not the first time they’d seen it happen, but the sheer power of the approach was indisputable now. “Two polarizations, to fit the Rule of Two. But what do they mean, physically?”
Carla said, “Let’s work with a stationary luxagen, to keep things simple. Then its energy-momentum vector points straight into our future. Suppose the luxagen field has a leftor of Up; its rightor will be the same, because Up divided by Up is Future.
“Suppose we rotate this luxagen in the horizontal plane: the North-East plane. Any such rotation will come from multiplying on the left and dividing on the right by a vector in the Future-Up plane—which will move our leftor and rightor from Up to some new position in the Future-Up plane. But the Future-Up plane is one we’re treating as a single complex number, so if the luxagen field remains within that plane, it hasn’t really undergone any physical change. And if you can rotate a luxagen in the horizontal plane without changing it, it must be vertically polarized.”
“So how do the same rotations affect the other polarization?” Patrizia wondered. “Pick any leftor in the other complex plane: the North-East plane. Say we choose North. If you multiply North on the left by a vector in the Future-Up plane, the result still lies in the North-East plane. So again, rotating the luxagen in the horizontal plane won’t change anything.”
“Two vertical polarizations?” Romolo hummed softly in confusion, but then he tried to work through the contradiction. “It’s meaningless to talk about two vertical polarizations of light—‘up’ as opposed to ‘down’—because the wave changes sign as it oscillates; if the light field points up at one instant it will point down a moment later. But when a leftor is multiplied by a complex number that oscillates over time, that oscillation will never move it from one complex plane to the other. So these two vertical polarizations really are separate possibilities.”
“But how could we turn one polarization into the other?” Carla pressed him. “Say, turn a leftor of North into a leftor of Up?”
“East times North is Up,” Romolo replied. “That’s the leftor, getting a quarter-turn. But the rotation of vectors that involves left-multiplication by East is a half-turn in the North-Up plane—which exchanges Up and Down. So when you flip a luxagen upside down, you swap the two vertical polarizations. That means they really do deserve to be called ‘up’ and ‘down’: the whole Future-Up plane for leftors describes a vertical polarization of ‘up’, and the whole North-East plane describes a vertical polarization of ‘down’.” He sketched the details, to satisfy himself that the rotation really did swap the planes as he’d claimed.
Patrizia said, “So the luxagen has a kind of axis in space that you can distinguish from its opposite. Like the two ways an object can spin around the same axis.”
Carla had been struggling to think of a suitable analogy herself, but Patrizia’s choice was weirdly evocative. “We should see if the new wave equation conserves the direction of this axis—if it really does stay fixed like the axis of a gyroscope.”
She converted the relationship between the field’s leftor and rightor and the energy-momentum vector into a more traditional form, where the energy and momentum came from the rates of change of the wave in time and in space. From there, they could work out the rate of change of the polarization axis—and it wasn’t necessarily zero. For some luxagen waves, the axis would shift over time.
“So it’s not like a gyroscope,” Patrizia said.
“Hmm.” Carla puzzled over the results. “The axis of a rotating object won’t always stay fixed. If the object is in motion—like a planet orbiting a star—and there’s some mechanism that allows angular momentum to flow back and forth between orbital motion and spin, you wouldn’t expect either one to be conserved individually. Only the total angular momentum will stay the same.”
Patrizia said warily, “So if we give the luxagen some angular momentum in its own right—as if it really were spinning around its polarization axis—then any change in that should be balanced by an equal and opposite change in orbital angular momentum?”
“Yes. If the analogy really does hold up that far.” Carla was exhausted, but she couldn’t leave the idea untested. As she ploughed on through the calculations she kept making small, stupid mistakes, but Romolo soon lost his shyness about correcting her.
The final result showed that the luxagen’s orbital angular momentum would not be conserved on its own. But by attributing half a unit of angular momentum to the luxagen itself—fixing the amount, but allowing its direction to vary with the polarization axis—the rate of change of the two combined came out to zero, and total angular momentum was conserved.
Patrizia’s chirp was half disbelief, half delight. “What would Nereo say? First his particles have spread out into waves, and now they’re spinning at the same time.”
Romolo gazed down at the spectra he’d brought. “So when we arrang
e the light field in the optical solid so the luxagen’s energy depends on its motion… it makes sense that it also depends on its spin.” The mystery that had spurred the night’s calculations had all but yielded. He looked up at Carla. “We can quantify the way the energy depends on the spin now, can’t we? The new wave equation will let us do that!”
Carla said, “Tomorrow.”
The three of them left the office together. The corridors of the precinct were empty, the rooms they passed dimmed to moss-light. “Your cos don’t mind how late you work?” Carla inquired.
“I moved out a few stints ago,” Patrizia said. “It’s easier.”
“I’ll probably do the same,” Romolo decided. “I don’t want to end up with children in the middle of this project!” He spoke without a trace of self-consciousness, but then added, “My co’s not ready either. We’ll both be happier without the risk.”
They parted, and Carla made her way up the axis to Carlo’s apartment. He was still awake, waiting for her in the front room.
“You’re looking better,” she said, gesturing to him to turn around so she could check that he wasn’t just relocating his wounds.
“I’m fine now,” Carlo assured her.
“So have the arborines bred yet?” Carla found the new project grotesque, but she didn’t want his ordeal in the forest to have been for nothing.
“Give them time.”
“How’s the influence peddling?” she asked.
“Some progress,” Carlo said cautiously. “We’ve managed to get tapes from a few people with infectious conditions—and they’re definitely putting out infrared.”
“And you let that tainted light fall on your own skin?”
“We make the recordings from behind a screen,” Carlo assured her. “We’re as careful as we can be. But these things are probably all over the mountain; I’m sure you’ve been exposed to all the same influences without even knowing it.”
“And now you need a volunteer who’ll let you play these tapes back to them, to see if they catch the disease?” It sounded like one of the tales from the sagas, where tracing the words of a forbidden poem on someone’s skin could strike them dead.
“We’re still working on the player,” he said. “But that would be the next step.”
Carlo extinguished the lamp and they moved into the bedroom. “You’re not skipping meals again, are you?” he asked sternly.
“No!” Carla helped him straighten the tarpaulin. “I’ll wait a few more stints, to be sure I’m having no more problems with my vision.” He didn’t reply, but she could see that he wasn’t happy. “It has to be done,” she said. “I’ll take it more slowly this time, but I can’t put it off forever.”
Carlo said, “I want you to wait another year before you risk your sight again. Wait and see what your choices are.”
“Another year?” Carla drew herself into the bed and lay in the resin-coated sand. He really thought he had a chance to compose his magic light poem by then, to spare her from Silvana’s fate? “But what if something happens first?” She looked up at him in the moss-light. “Before I’m ready?”
Carlo reached across her and pulled an object out of a storage nook on her side of the bed. It was a long, triangular hardstone blade, with three sharp edges tapering to a point.
“If I ever wake you in the night and start trying to change our plans,” he said, “show me this. That should bring me to my senses.”
Carla examined his face. He was serious. “And what if I’m the one who wakes you?”
He returned the knife to its hiding place and produced a second one from the other side of the bed.
34
Carlo arrived a few chimes early to take over from Macaria. Having done the arborine night shift himself for a stint he knew how tiring it was: the less active the animals were, the harder it had been to keep watching them closely without his mind wandering. It was only by constantly reminding himself of what a few lapses of inattentiveness might cost him that he’d managed to stay awake to the end of each shift.
“Anything unusual?” he asked.
“Zosimo was up for about a bell, leaping around the cage,” Macaria recounted. “At one point he woke his co; I was sure something was going to happen. But in the end all they did was exchange a few calls. Benigna and Benigno slept through all the drama.”
“Hmm.” Carlo had read old reports, written on the home world, claiming that arborines in the forests had been seen waking at night in order to breed. But he had doubts about the veracity of those accounts, let alone their relevance to these captives twice removed from their ancestors’ original habitat.
“There ought to be something we can do to encourage them,” Macaria said wearily. “Both couples are reproductively mature, so what are they waiting for? There must be some environmental change that would clinch it. Maybe a dietary signal—”
“If we increase the food supply any more, we risk them becoming quadraparous,” Carlo replied.
“Would that be so terrible? If you really want to understand the signaling during fission, aren’t you going to need to compare biparous and quadraparous versions at some point?”
That might have been a reasonable attitude if they’d had an unlimited supply of experimental subjects, and as many lifetimes as they needed to achieve the project’s goal. Carlo said, “If we don’t get a tape of biparous fission from this lot, you can volunteer to catch the next four arborines.”
Macaria left him to it.
Carlo positioned himself on the guide rope midway between the two cages, at a point where he could see Benigna and Benigno in his rear gaze and Zosima and Zosimo to the front. The flowers adorning the scaffolding of amputated branches that crisscrossed each cage were still putting out light, more or less following their original alternating cycles, but over the last few days he’d begun to notice a decline in their luminance. As the imitation forest faded the moss-light took over, and the whole place began to look more like a prison of bare rock decorated with a few wan twigs.
The observers’ shifts were synchronized to the arborines’ activity, and he’d arrived just in time to watch all four animals waking. The females, pinned to their heavy plinths, had long ago ceased making vigorous attempts to detach themselves, but their posture and movements changed completely once they were conscious, with the uncoordinated twitches and flailing of sleep replaced by an eerily disciplined-looking series of muscular stretches and rearrangements of the flesh. Benigna and Zosima in their separate cages performed an almost identical set of exercises, which suggested that they were instinctive responses to their lack of mobility—perhaps as a way of maintaining health while recovering from an injury. But it was possible that there was a component of mimicry as well; from their plinths, they could see each other clearly enough. Mimicry? Encouragement? Solidarity? Zosima had carried Benigna’s limp body through the forest unflaggingly while Carlo pursued her. It was hard not to think of the two of them as fellow prisoners, aware of each other’s plight, striving to keep up each other’s morale.
For their part, the males did not remain still for long either: every few lapses either Zosimo or Benigno made a sudden leap from one branch to another. Though the cages were currently empty of lizards, to Carlo these moves looked similar to ones the arborines used when ambushing prey. He wasn’t sure if they had failed to grasp the unfamiliar rules governing the presence of the lizards and had started jumping at shadows in the hope that it might be food, or, like the females, they were merely intent on staying active.
In the light of Macaria’s report Carlo paid special attention to Zosimo. The male was certainly more agitated than usual, swaying restlessly on each branch before flinging himself onto the next. The cage was just a couple of stretches across, so Zosimo couldn’t help revisiting the same locations—but far from executing a tight, repetitive cycle, he crisscrossed the miniature forest in an elaborate sequence of permutations of departure points and destinations, as if intent on squeezing as much novelty
out of his impoverished surroundings as he could.
When feeding time came, Carlo fetched two lizards from the storeroom; they squirmed in protest for a while, then went limp in his hands as if they could save themselves by playing dead. The arborines must have learned the routine by now, but they didn’t hang around like supplicants as he approached. Benigno clung, aloof, to a distant branch while Carlo tossed the lizard through the bars of his cage. Zosimo was positively disdainful, baring his teeth at Carlo threateningly, but he too kept his distance.
Carlo returned to his observation post. He’d seen these hunts too many times now to remain enthralled from start to finish, but it was impossible to ignore them completely. The cages were small, but every branch held a dozen hiding places, and the lizards always vanished from sight long before the arborines showed any interest in them. Today the pursuit seemed unhurried at first, almost desultory: Zosimo crossed from branch to branch purposefully a few times, then appeared to grow distracted, while Benigno’s bounces, more playful than stealthy, sent luminous petals wafting through the air.
Carlo’s thoughts wandered, but he was aware of the two arborines gradually narrowing the search: jumping to a new branch, looking about for a moment, then feigning indifference and pretending to be more concerned with swatting at mites. It was nearly a chime later when things sped up, rapidly; Carlo could hear one lizard’s panicked claws as it fled along a branch before Zosimo reached out and snatched at it. The lizard must have jumped to another branch, because the hand came back empty, but then Zosimo leaped after it and moments later he had it in his mouth and was biting it in two.
Zosimo chewed on half the lizard, chirping softly with pleasure. There was a flurry of activity in the rear of Benigno’s cage, but Carlo couldn’t see what was happening so he stayed focused on Zosimo. The arborine swallowed his share of the meal, then swung down to the branch closest to his trapped co. He handed Zosima the remainder of the lizard; as she raised it to her mouth he reached across and ran a hand over the side of her face.