Schild's Ladder

by Greg Egan

  As this mood swept the room, Tchicaya felt gooseflesh rise along his spine. He couldn't claim to have anticipated the words he heard next, but they thoroughly merited his body's reaction.

  “I believe there are no Sarumpaet rules,” Sophus proclaimed. “Not the originals, and not some grander, more perfect version that will explain what happened at Mimosa. But the world still looks so much like the way it would look if there were that we couldn't help but think such rules existed.”

  In the silence that followed, Tchicaya turned to Mariama, wondering if she'd picked up more from Sophus's earlier remarks than he had, but she appeared to be equally stunned. Tchicaya was beaming with delight at the audacity of Sophus's claim. Mariama looked dismayed, almost fearful.

  Sophus continued. “How can the Sarumpaet rules seem to be true, when they're false? How can our vacuum seem to be stable, when it isn't? I believe that the right way to answer these questions is virtually identical to the resolution of another paradox, one that was dealt with almost twenty thousand years ago. How can the universe appear to obey classical mechanics, when it really obeys quantum mechanics?

  “What creates the illusion of classical mechanics is our inability to keep track of every aspect of a quantum system. If we can't observe the whole system—if it's too large and complex in itself, or if it's coupled to its surroundings, making them part of the system—we lose the information that distinguishes a genuine superposition, where alternatives coexist and interact, from a classical mixture of mutually exclusive possibilities.

  “I believe the same effect is responsible for the Sarumpaet rules. How can that be? The Sarumpaet rules are quantum rules. They apply to systems that have not been rendered classical by decoherence. How can interaction with the environment explain anything wholly quantum-mechanical?”

  Sophus smiled wearily. “It's been staring us in the face for twenty thousand years. An electron—a charged particle, which transforms the ordinary vacuum around it into an entirely different state—still obeys quantum mechanics in all of its other degrees of freedom. Its position is quantum-mechanical, its charge is classical. Even when we do our best to isolate an electron from its surroundings, we actually fail miserably at half of the task, while succeeding at the other half. So decoherence hides superpositions of different charge states from us, but not different position states. Our failure looks classical, our success is quantum-mechanical.

  “We thought the Sarumpaet rules were pure quantum mechanics: the final story, the lowest level, the rules that held for a system in perfect isolation. Of course, we accepted the fact that, in practice, we could never isolate anything from its surroundings completely, but that wasn't the point. The universe itself, the total system, was assumed to be obeying the Sarumpaet rules—because whenever we did our best to examine any small part of it, separated out as scrupulously as possible, those were the laws that held.

  “That was the wrong conclusion to reach. The electron shows how quantum and classical properties can coexist. The fact that you can demonstrate some quantum behavior in a system doesn't mean you've uncovered all that there is to be found.

  “I believe that the Sarumpaet rules are classical rules. Part of the total state vector of any system obeys them, but not the whole. The part that does follow the Sarumpaet rules interacts with the environment one way: transforming its surroundings into what we think of as our own vacuum. But there are other parts that interact differently, creating other states. Because we can't begin to track what's really happening to the environment on the Planck scale, what we see is a single, certain, classical outcome: the Sarumpaet rules hold absolutely true, and our vacuum is absolutely stable.”

  A member of the audience stood, and Sophus acknowledged the request. “Tarek?”

  “You're claiming that the vacuum has been stabilized by something like the quantum Zeno effect?”

  Tchicaya craned his neck to observe the questioner more closely. Tarek was the Preservationist who'd been trying to scribe Planck worms to devour the novo-vacuum, without waiting to discover what it was, or what it might contain. There was nothing fanatical about his demeanor, though; he merely radiated an impatience that everyone in the audience shared.

  “It's similar to that,” Sophus agreed. “The quantum Zeno effect stabilizes systems through constant measurement. I believe that part of the total graph in which everything's embedded ‘measures’ the part we see as the vacuum, which also determines the dynamic laws that govern matter moving through that vacuum. It's like the vapor in a cloud chamber, condensing in droplets around the path of a subatomic particle. The particle only appears to follow a definite trajectory because each path is correlated with a particular pattern of droplets—and the droplets have too many hidden degrees of freedom to exhibit quantum effects themselves. But we know there are branches where the particle follows different paths, surrounded by different trails of droplets.”

  Tarek frowned. “So why can't we discover the path, the rules, that are holding sway behind the border?”

  Sophus said, “Because what lies behind the border is not another vacuum, another set of rules. It has no classical properties like that to discover. It's not that it couldn't be divided up—formally, mathematically—into a sum of components, each obeying a different analog of the Sarumpaet rules. But we're not correlated with any particular component, the way we are with our own vacuum, so we can't expect to uncover any particular set of rules.”

  Tchicaya was exhilarated. It was too soon to take Sophus's idea seriously, but there was something deeply appealing in the simplicity of the notion. Behind the border was a superposition of every possible dynamic law.

  Tarek said, “We can't measure those properties? Make them definite, if only for different branches of ourselves? When we interact with the novo-vacuum—or whatever you now wish to call it—shouldn't we end up as a superposition of observers who each find definite laws?”

  Sophus shook his head firmly. “Not by dropping a few Planckscale probe graphs into a system six hundred light-years wide. If there were preexisting laws behind the border, we might hope to discover them that way, but that's not what we're dealing with. On our side of the border, there's a tight correlation stretching across all of space-time: the dynamics being followed at different times and places has become a tangle of mutual interdependence. What lies behind the border isn't correlated from place to place, or from moment to moment. What we're sampling with our probe graphs might as well be random noise at every level.”

  Rasmah stood, just ahead of a dozen other people. The others resumed their seats, and Tarek begrudgingly followed.

  She said, “This is wonderful speculation, Sophus, but how do you plan to test it? Do you have any solid predictions?”

  Sophus gestured at the space behind him, and a set of graphs appeared.

  “As you see, I can match the borderlight spectrum. That's not claiming much. I can match the half-c velocity of the border, which is slightly harder. And I can match the pooled results of all the experiments performed here so far: namely, their complete failure to identify anything resembling a dynamic law.

  “So much for retrodiction. I'm making the following prediction: when we repeat the old experiments, re-scribe the old probe graphs, and monitor the results with your new spectrometer...we'll find exactly the same thing, all over again. No patterns will emerge, no symmetries, no invariants, no laws.

  “We've already discovered that there's nothing to be discovered. All I can predict is that however hard we look, that absence will be confirmed.”

  Chapter 8

  Yann rolled off the bed and landed on the floor, laughing.

  Tchicaya peered over the edge. “Are you all right?”

  Yann nodded, covering his mouth with a hand but unable to silence himself.

  Tchicaya didn't know whether to be annoyed or concerned. Acorporeals taking on bodies often mapped them in unusual ways. Perhaps laughter was Yann's only available response to some terrible psychic affront tha
t Tchicaya had unwittingly inflicted.

  “You're sure I haven't hurt you?”

  Yann shook his head, still laughing helplessly.

  Tchicaya sat on the edge of the bed, struggling to regain his own sense of humor. “This is not a reaction I'm accustomed to. Rejection and hilarity are perfectly acceptable responses, but they're supposed to occur much earlier in proceedings.”

  Yann managed to regain some composure. “I'm sorry. I didn't mean to offend you.”

  “I take it you're not interested in finishing what you started?”

  “Umm.” Yann grimaced. “I could try, if it's important to you. But I think it would be very difficult to take seriously.”

  Tchicaya planted a foot on his chest. “Next time you want an authentic embodied experience...just simulate it.” He still felt a pang of lust at the touch of skin on skin, but it was fading into a kind of exasperated affection.

  He crouched down and kissed Yann on the mouth, meaning it as a gesture of finality. Yann smiled, puzzled. “That was nice.”

  “Forget it.” Tchicaya stood and started dressing.

  Yann lay on the floor, watching him. “I think I'm getting all the signals you talked about,” he mused. “But they're so crude, even now. And before, it was just a single message, repeating itself endlessly: ‘Be happy, be happy, be happy!’ Do you think there's something wrong with this body?”

  “I doubt it.” Tchicaya sat cross-legged on the floor beside him.

  “You expected more?”

  “I was already happy, so it was a bit redundant.”

  “How happy?”

  “As happy as it's possible to be, for no particular reason.”

  “I have no idea how to interpret that. What gets to count as a particular reason?”

  Yann shrugged. “Something more than being told by my body: ‘Be happy.’ Be happy...why?”

  “Because you're with someone you like. And you're making them happy, too.”

  “Yes, but only if they accept the same reasoning. That's circular.”

  Tchicaya groaned. “Now you're being disingenuous. It's a tradition, passed down from reproductive biology. Every tradition's arbitrary. That doesn't mean it's empty.”

  “I know. But I still expected something more subtle.”

  “That takes time.”

  “What, hours?”

  “Centuries.”

  Yann narrowed his eyes with suspicion.

  Tchicaya laughed, but made a face protesting his honesty. “On Turaev, it takes six months of attraction before anything's physically possible.” Like most generic bodies, the Rindler's were promiscuous: any two of them could develop compatible sexual organs, more or less at will. You could wire in your own chosen restraints while you inhabited them, but since leaving home, Tchicaya had never felt the need to delegate the task. “The waiting was nice, in its own way,” he admitted. “You might think it was risking an awful anticlimax, but I think the buildup improved the sex itself almost as much as it raised expectations. Acting on the spur of the moment is more likely to be disappointing.”

  Yann protested, “I've been contemplating this for almost six months.”

  “Since I arrived? I'm flattered. But then, who else would you dare to ask?”

  Yann smiled abashedly. “How could I not be curious? It's what flesh is famous for. However undeservedly.” He watched Tchicaya carefully, serious for a moment. “Have I hurt you?”

  Tchicaya shook his head. “That usually takes longer, too.” He hesitated. “So what do acorporeals do, instead? When I was a child, I used to imagine that you'd all have simulated bodies. Sex would be just like embodied sex, but there'd be lots of colored lights, and cosmic bliss.”

  Yann guffawed. “Maybe twenty thousand years ago there were people that vacuous, but they must have all decayed into thermal noise before I was born.” He added hastily, “I'm not saying you're wrong to continue the tradition. You've mapped some stable mammalian neurobiology, and it's not too pathological in its original form. I suppose it still serves some useful social functions, as well as being a mild existential placebo. But when you have a malleable mental structure, intensifying pleasure for its own sake is a very uninteresting cul-de-sac. We worked that out a long time ago.”

  “Fair enough. But what do you do instead?”

  Yann sat up and leaned against the side of the bed. “All the other things the embodied do. Give gifts. Show affection. Be attentive. Sometimes we raise children together.”

  “What kind of gifts?”

  “Art. Music. Theorems.”

  “Original theorems?”

  “If you're serious.”

  Tchicaya was impressed. Mathematics was a vast territory, far more challenging and intricate than physical space. Reaching a theorem no one had proved before was a remarkable feat. “That's positively...chivalric,” he said. “Like a knight riding off to the edge of the world, to bring back a dragon's egg. And you've done that, yourself?”

  “Yes.”

  “How often?”

  “Nine times.” Yann laughed at Tchicaya's expression of astonishment, and added, “It's not always that serious. If it was, it really would be as daunting as winning the hand of medieval royalty, and no one would bother.”

  “So you start with something easier?”

  Yann nodded. “When I was ten years old, all I gave my sweet-heart was a pair of projections that turned the group of rotations in four dimensions into principal bundles over the three-sphere. Ancient constructions, though I did rediscover them for myself.”

  “How were they received?”

  “She liked them so much, she extended them to larger spaces and gave me back the result.”

  “Can you show me?”

  Yann sketched diagrams and equations with his hands; through their Mediators, Tchicaya saw them painted in the air. To make sense of the group of four-dimensional rotations, you could project it down to the three-dimensional sphere of directions in four dimensions, by mapping each rotation to the direction to which it took the x-axis. All the rotations that treated the x-axis in the same way then differed from each other by rotations of the other three directions. This effectively sliced the original group into copies of the group of three-dimensional rotations—which was just a solid sphere with opposite points on its boundary glued together, since any pair of rotations around opposite axes became equal once you reached one hundred and eighty degrees. Like an artful rendering of depth in a painting, these striations made the topology of the larger group much clearer.

  “The other projection inverts all the rotations first, so it turns the whole construction inside out.” Yann demonstrated, smiling nostalgically. “I know it's sentimental, but the first time always stays with you.”

  “Yeah.” The mathematics was simple, but it struck Tchicaya as having all the charm of an embodied child's handmade gift.

  “So what about you?”

  “I've generally had more success with flowers.”

  Yann rolled his eyes. “Your own first love. What was that like?”

  Tchicaya contemplated lying, but he usually did it badly. And what would he say? He wasn't going to substitute someone else, writing Mariama out of his life.

  He said, “I can't tell you.”

  “Why not?” Yann was twice as eager for the details, now. “How embarrassing can it be, four thousand years later?”

  “You'd be surprised.” Tchicaya struggled to think of a way to deflect the inquiry without piquing Yann's curiosity further. “There's much better story I can tell you,” he said. “About my father's first love. Can I trade that instead?”

  Yann agreed, reluctantly.

  “When my father was fourteen,” Tchicaya began, “he fell in love with Lajos. It started in winter, when they used to sneak into each other's houses at night and sleep together.”

  Yann said, “Why did they have to sneak? Would their parents have stopped them?”

  Tchicaya was momentarily at a loss for an answer; he'd ne
ver had to explain this before. “No. Their parents would have known. But it's more enjoyable to pretend that it's a secret.”

  Yann seemed slightly bemused by this claim, but willing to take his word for it. “Go on.”

  “By summer, they were giddy with it. They could touch and kiss, nothing more, but they knew it wouldn't be much longer. They'd go swimming together, walking together, waiting for it to happen. Aching this wonderful ache.” Tchicaya smiled, hiding a sudden upwelling of sadness. He doubted he'd ever return to Turaev, to talk to the stranger his father had become.

  “At the height of summer, they were walking on the outskirts of town. And my father witnessed the strangest, most terrifying event that had happened on Turaev for a thousand years. A spaceship descended from the sky. An ancient engine, spouting flames, burning up crops, melting rocks.”

  Yann was outraged. “And Lajos—” He struggled with his emotions. “Your father saw Lajos—”

  “No, no!” Tchicaya was amused at the preposterousness of this suggestion, but he still warmed to Yann's response. He'd met bigots who would have assumed that an acorporeal would shrug off the notion that witnessing the local death of your first love would be of any consequence at all.

  “Not even anachronauts land their spacecraft on top of people,” he explained. “They do have instruments.”

  Yann relaxed. “So your father and Lajos got to meet the anachronauts. What were they like?”

  “They'd left Earth fourteen thousand years before. Pre-Qusp. They used biological techniques to keep their flesh viable, but they spent a lot of time cryogenically suspended.”

  “Cryogenically suspended.” Yann was mesmerized. “I always knew they were out there, but I've never met anyone before who's spoken to someone who's seen them in the flesh.” He shuddered with vicarious otherworldliness. “What did they want?”