“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
“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
“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
Romolo said, “So when you rotate this system all the way back to its starting point, the wave changes sign. But we can’t actually measure that… so it doesn’t matter?”
“It’s strange,” Carla agreed. “But what troubles me more is treating the rotation’s left vector differently from the right. All it takes to swap the role of those two vectors is to view the system in a mirror. Should physics look different, viewed in a mirror? Have we seen any evidence of that?”
Patrizia took the criticism seriously. “What if we tried to balance it, then? Could we throw in a ‘rightor’ as well as a leftor, for symmetry’s sake?” She wrote the transformation rule for this new geometrical object, a mirror image of her previous invention.
“Throw it in where?” Romolo asked.
“Into the luxagen wave,” Patrizia replied. “Two more complex numbers, but these ones transform by the rightor rule. If you look at the system in a mirror, the leftor and rightor change places.”
“That sounds very elegant,” Romolo said, “but haven’t you just doubled the number of polarizations from two to four?”
“Hmm.” Patrizia grimaced. “That would defeat the whole point.”
Carla pondered the new proposal. “The light field is a four-dimensional vector—but we don’t get four polarizations, because of the relationship between the field vector and the energy-momentum vector. What if there’s a relationship between the luxagen field—the leftor and the rightor—and the luxagen’s energy-momentum vector? Something that brings the number of polarizations back down to two.”
Romolo said, “What kind of relationship? Setting a leftor or a rightor perpendicular to an ordinary vector won’t work—when you rotate all three of them, they’ll change in different ways, so the relationship won’t be maintained.”
“That’s true,” Patrizia conceded. She drove her fist into her gut; the glorious distraction was losing its power again. “Maybe we should tear this up and start again.”
Carla said, “No. The relationship’s simple.”
She wrote:
“That’s it,” she said. “Just look at how these three things transform when we rotate them.”
