The Eternal Flame

“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 arrange 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