Schild’s Ladder

be on opposite sides of a continent, why couldn’t they manage to prepare a system that was equally likely to be an electron here and a positron here'?—?Sophus held up his left hand, then his right?—?'or vice versa?

“For a hundred years or so, most people would have answered that question by saying: Oh, there’s a superselection rule for charge! You can usually combine state vectors…but not if they come from different superselection sectors of the Hilbert space! Apparently there were these strange ghettos that had been cordoned off from each other, and whose inhabitants were not allowed to mix. Cordoned off how? There was no mechanism, no system; it was just an inexplicable fact dressed up in some fancy terminology. But people went ahead and developed methods for doing quantum mechanics with these arbitrary borders thrown in, and the lines on the map became something to be memorized without too much scrutiny. If some innocent novice asked a jaded elder student, Why can’t you have a superposition of different charges? the reply would be, Because there’s a superselection rule forbidding it, you idiot!”

Sophus lowered his gaze slightly before adding acerbically, “We’re far more sophisticated now, of course. No one would tolerate mystification like that?—?and besides, every child knows the real reason. An electron and a positron in the same position would be correlated with vastly different states for the surrounding electric field, and unless you could track all the details of that field and incorporate them into your observations, you’d have no hope of recognizing the state as a superposition. Instead, the two different charge states would decohere, and you’d be split into two versions, one believing that you’d detected an electron, the other that you’d detected a positron. So although there are no superselection rules, the world still looks so much like the way it would look if there were that all the mathematics that revolved around the term lives on, in various guises.”

Tchicaya sensed a sudden change in the atmosphere around him. When he’d glanced at people before, most had seemed puzzled that they were being offered such mundane observations. Tolerant, and prepared to go on listening for a while, thanks to Sophus’s reputation, but clearly not expecting much from yet another tortured reexamination of their field’s basic assumptions. Now there was a shifting of bodies, a creaking of seats, as people felt compelled to transform their postures of indifference or mild disappointment into something altogether more vigilant.

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