Schild’s Ladder

"Yes." Tchicaya felt a burst of confidence; that was just elementary quantum mechanics. In empty space, a particle’s wave packet would always disperse, spreading out without limits. But if the particle experienced an attractive force analogous to the tug of a spring in classical physics, there was a certain shape — a certain Gaussian, like the bell curve of statistics — which was stable. Any tighter, sharper wave packet would necessarily have a range of values for momentum that made it spread out; that was just the uncertainty principle. The right Gaussian, though, in the right environment, was the perfect compromise between uncertainty in position and momentum, allowing the shape of the wave to remain unchanged as it moved.

"This isn’t really the same," Yann admitted. "But it might sound persuasive if I put it that way."

Rasmah glanced at Tchicaya, exasperated. He made puppydog eyes back at her, pleading on Yann’s behalf.

She laughed, and relented. "Why don’t you just give me the description of the graph you want to scribe, and I’ll grind through the calculations using my own picture of Sophus’s model. If I can demonstrate that we’d get some inforamtion back through the border — something more than we put in — that might be enough to persuade people. I’ll make sure I phrase my results in the ugliest possible way."

Yann said, "That’s wonderful. Thank you!"

He passed something to Rasmah — Tchicaya’s Mediator saw the fact of the exchange, but not the content — and then vanished.

Rasmah sighed. "You really think he’s on to something? A quantum computer can simulate any quantum process; that’s old news. It doesn’t mean that there is a quantum computer underlying anything."

"No," Tchicaya agreed. "But qubit network theory doesn’t claim that. It just says that when you get to a low enough level, you have nothing left to lose by treating the system as if it were software. It’s like all the proofs in applied algorithmic theory that are based on imagining Turing machines. No one complains that the real universe is conspicuously devoid of paper tape."

"Old habits die hard," she confessed. "I’m still in mourning for the Sarumpaet rules, and they were disproved before I was born. They’re what I was brought up on, they’re what I’ve thought of all my life as the template for a physical theory. It’s not easy adapting, even to Sophus’s model."

"Yeah. I really am grateful to you for trying this," Tchicaya said. Since the factional rift had widened, it was more important than ever to keep all the Yielders open to each other’s new ideas, and where he wasn’t competent to contribute directly himself, he could at least act as a kind of broker, prodding the appropriate experts into action.

Rasmah seemed on the verge of pointing out that he might have expressed his gratitude to her more palpably, but then she smiled and accepted his words at face value.

"Okay. Here I go."

She turned her attention to something invisible to Tchicaya. For several minutes, she sat in complete silence.

Suddenly, she exclaimed, "Oh, I see! This is actually quite nice."

Tchicaya was excited, and slightly jealous. "Can you explain?"

Rasmah held up her hand for patience, retreating back into her private scape.

After a while, she spoke again. "Think of all the different dynamic laws that might make topological sense, in terms of the propagation of various kinds of particles that are defined as patterns embedded in a graph. I know that’s horribly vague, but I don’t think you’d want the version with added jargon."

Tchicaya said, "Okay. I’m thinking of them." He’d seen enough examples that they’d pinned to the border over the last few months to have some feel for what this meant.

"Now imagine each one is a quantum state vector in a big fat Hilbert space. All of them orthogonal to each other."

"Yes." Tchicaya had never had his mind restructured to enable clear images of more than three dimensions, but since Rasmah’s Hilbert space was infinite-dimensional anyway, three was as good as any other number. "I’m doing that. Go on."

"Now imagine a new set of vectors that consist of equal amounts of all these dynamic-law vectors, and that are all orthogonal to each other. These vectors represent definite values of a variable that’s complementary to the law vectors. Branco calls them law-momenta — which is a bit sloppy, because they’re not true Lagrangian conjugates, but never mind."