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.