by Greg Egan
Gradually the arborine yielded, but while Carlo was focused on that battle the thing pulled the slingshot out of his hand and tossed it away. Carlo quickly plunged his hand into the pouch again and got hold of a dart; with a flick of the thumb he unsheathed it. The arborine grabbed his wrist and refused to let him take his hand from the pouch.
Carlo’s skin was feverish, but the animal’s flesh against him felt hotter. The scent of it was overpowering, but horribly familiar: it reminded him of the smell of his father before he’d died. He still had his hands in the arborine’s jaws; he pulled them further apart and twisted the head back sharply. This felt satisfying, but however much pain he was inflicting it didn’t weaken the arborine’s grip on him.
Carlo tried to extrude a fifth limb, but nothing happened. He let go of the fist that he’d stayed from battering his tympanum, sharpened the fingertips of his newly freed hand and plunged them into the arborine’s forearm just above the pouch. He felt the muscles below twitch and slacken; he’d disrupted some of the motor pathways.
The arborine appeared confused; it didn’t bother resuming its assault on Carlo’s tympanum, but before it could make up its mind what to do Carlo tore his hand free from the pouch and plunged the dart into the arborine’s shoulder. He felt the body grow limp immediately, but he still had to prise the lower arms from around his chest and shake the thing off him.
The females were gone.
Carlo looked around; his slingshot was caught on a branch nearby. He dragged himself over and grabbed it, then set off toward the canopy as fast as he could.
As he ascended, the light from the flowers above him thinned and faded and was gone. Suddenly he was in open air, in the murk again, with nothing but the forest’s decaying litter between him and the cavern’s red ceiling. He searched for the females, hoping the one who’d carried her friend so far might have needed one last rest before launching the pair to safety, but then he saw them. They were in the air, a couple of stretches away, drifting straight toward the safety of a neighboring tree.
Carlo loaded his slingshot, took aim and released the dart. A breeze stirred the dust and obscured his view, but when he had clear sight again the projectile was nowhere to be seen.
He tried again. His second dart sliced through the detritus and miraculously struck flesh—but he’d hit the female that Lucia had already paralyzed.
“No, no, no!” he pleaded. The arborines vanished into the murk; he waited helplessly, and when they appeared again they’d almost reached their sanctuary. Carlo reloaded and released, reloaded and released, aiming through the grit and swirling dead petals by memory and extrapolation until a single dart remained.
He couldn’t bring himself to use the last one blindly. He waited for the air to clear. Lucia knew the forest better than anyone living, but she’d been a child the last time an arborine had been captured. There were no experts at this. How many people would he need to beg from Tosco in order to succeed at this task by force of numbers alone?
He finally caught sight of the arborines, silhouetted against the light of the adjacent tree. They had separated; their outlines were distinct. Carlo waited for his indefatigable nemesis to reach out and drag her friend to safety, but both animals remained motionless. She wasn’t just weary. He had hit her.
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 t
hat 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 t
o 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?”
