The Eternal Flame

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?”

Carla wasn’t sure, but it was worth checking. “Suppose a luxagen wave consists of two complex numbers, for the two polarizations,” she said. “Each one has a real part and an imaginary part, so all in all that’s four dimensions.”

“So you just think of the usual four dimensions as two complex planes?” Romolo suggested.

“Maybe,” Carla replied. “But what happens when you rotate something? If you’ve got two complex numbers that describe a luxagen’s polarization, and I come along and physically turn that luxagen upside-down… what happens to the complex numbers?”

Romolo said, “Wouldn’t you just take their real and imaginary parts, and apply the usual rules for rotating a vector?”

“That’s the logical thing to try,” Carla agreed. “So let’s see if we can make it work.”

The simplest way to describe rotations in four-space was with vector multiplication and division, so Carla brought the tables onto her chest as a reminder.

Any rotation could be achieved by multiplying on the left with one vector and dividing on the right by another; the choice of those two vectors determined the overall rotation. Romolo worked through an example, choosing Up for both operations.

“There’s one thing we’ll need to get right if we’re going to make this work,” Carla realized. “Given a pair of complex numbers, if you multiply them both by the square root of minus one that will affect each number separately. It doesn’t mix them up in any way—it just rotates each complex plane by a quarter-turn, making real numbers imaginary and imaginary numbers real. So if we’re going to treat two planes in four-space as complex number planes, we’ll need some equivalent operation.”

“But I just drew that!” Romolo replied. “Multiplication on the left by Up rotates everything in the Future-Up plane by a quarter turn, and everything in the North-East plane by a quarter turn. Vectors in one plane aren’t moved to the other. Do it twice—square it—and you get a half turn in both planes, which multiplies everything by minus one. So we could treat those two planes as the two complex numbers, and use left-multiplication by Up as the square root of minus one!”

Carla wasn’t satisfied yet. “All right, that works perfectly on its own. But what happens when you physically rotate the luxagen as well? If I rotate an ordinary vector and then double it, or double it first and then rotate it, the end result has to be the same, right?”

“Of course.” Romolo was puzzled, but then he saw what she was getting at. “So whatever we use to multiply by the square root of minus one has to give the same result whether we rotate first and then multiply, or vice versa.”

“Exactly.”

Patrizia looked dubious. “I don’t think that’s going to be possible,” she said. “What about the rotation you get by multiplying on the left with East and dividing on the right by Future? Future acts like one, it has no effect, so you get:”

“Romolo’s definition of multiplying by the square root of minus one is:”

“Follow that with the rotation:”

“But now do the rotation first, and then multiply by the square root of minus one:”

“The end result depends on the order,” Patrizia concluded. “Since you can’t reverse the order when you multiply two vectors together, that’s always going to show up here and spoil things.”

She was right. There were other choices besides Romolo’s for the square root of minus one, but they all had similar problems. You could multiply on the left or on the right by Up or Down, East or West, North or South; they would all produce quarter turns in two distinct planes. But in every single case, you could find a rotation that wrecked the scheme.

Romolo took the defeat with good humor. “Two plus two equals four, but all nature cares about is non-commutative multiplication.”

Patrizia smoothed the calculations off her chest, but Carla could see her turning something over in her mind. “What if the luxagen wave follows a different rule?” she suggested. “It’s still a pair of complex numbers, and you can still join them together to make something four-dimensional—but when you rotate the luxagen, that four-dimensional object doesn’t change the way a vector does.”

“What law would it follow, then?” Carla asked.

“Suppose we choose right-multiplication by Up as the square root of minus one,” Patrizia replied. “Then multiplying on the left will always commute with that: it makes no difference which one you do first.”

“Sure,” Carla agreed. “But what’s your law of rotation?”

“Multiplying on the left, nothing more,” Patrizia said. “Whenever an ordinary vector gets rotated by being multiplied on the left and divided on the right, this new thing—call it a ‘leftor’—only gets the first operation. Forget about dividing it.” She scrawled two equations on her chest:

Carla was uneasy. “So you only use half the description of the rotation? The rest is thrown away?”

“Why not?” Patrizia challenged her. “Doesn’t it leave you free to multiply on the right—letting the square root of minus one commute with the rotation?”

“Yes, but that’s not the only thing that has to work!” Carla could hear the impatience in her voice; she forced herself to be calm. She was ravenous, and she was late to meet Carlo—but she couldn’t eat until morning anyway, and if she cut this short now she’d only resent it.

“What else has to work?” Romolo asked.

Carla thought for a while. “Suppose you perform two rotations in succession,” she said. “Patrizia’s rule tells you how this new kind of object changes with each rotation. But then, what if you combine the two rotations into a single operation—one rotation with the same overall effect. Do the rules still match up, every step of the way?”

Patrizia said, “However many rotations you perform, you just end up multiplying all of their left vectors together. Whether it’s for a vector or a leftor, you’re combining them in exactly the same way!”

That argument sounded impeccable, but Carla still couldn’t accept it; throwing out the right vector had to have some effect. “Ah. What if you do two half-turns in the same plane?”

“You get a full turn, of course,” Patrizia replied. “Which has no effect at all.”

“But not from your rule!” Carla wrote the equations for each step, obtaining a half-turn in the North-East plane by multiplying on the left by Up and dividing by Up on the right.

Patrizia kept rereading the calculation, as if hoping she might spot some flaw in it. Finally she said, “You’re right—but it makes no difference. Didn’t you tell us a lapse or two ago that rotating a luxagen wave in the complex plane has no effect on the physics?”

“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 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 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.