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.”
“A leftor divided by a rightor changes in exactly the same way as an ordinary vector. So if we demand that the energy-momentum vector of a luxagen wave is proportional to the wave’s leftor divided by its rightor, rotation won’t break the relationship—and any free luxagen wave that meets this condition could be rotated into agreement with any other.”
Romolo said, “And the rightor is completely fixed by the leftor and the energy-momentum vector. There are no extra polarizations.”
Patrizia looked dazed. She said, “Follow the geometry and everything falls into place.” She exchanged a glance with Carla; this was not the first time they’d seen it happen, but the sheer power of the approach was indisputable now. “Two polarizations, to fit the Rule of Two. But what do they mean, physically?”
Carla said, “Let’s work with a stationary luxagen, to keep things simple. Then its energy-momentum vector points straight into our future. Suppose the luxagen field has a leftor of Up; its rightor will be the same, because Up divided by Up is Future.
“Suppose we rotate this luxagen in the horizontal plane: the North-East plane. Any such rotation will come from multiplying on the left and dividing on the right by a vector in the Future-Up plane—which will move our leftor and rightor from Up to some new position in the Future-Up plane. But the Future-Up plane is one we’re treating as a single complex number, so if the luxagen field remains within that plane, it hasn’t really undergone any physical change. And if you can rotate a luxagen in the horizontal plane without changing it, it must be vertically polarized.”
“So how do the same rotations affect the other polarization?” Patrizia wondered. “Pick any leftor in the other complex plane: the North-East plane. Say we choose North. If you multiply North on the left by a vector in the Future-Up plane, the result still lies in the North-East plane. So again, rotating the luxagen in the horizontal plane won’t change anything.”
“Two vertical polarizations?” Romolo hummed softly in confusion, but then he tried to work through the contradiction. “It’s meaningless to talk about two vertical polarizations of light—‘up’ as opposed to ‘down’—because the wave changes sign as it oscillates; if the light field points up at one instant it will point down a moment later. But when a leftor is multiplied by a complex number that oscillates over time, that oscillation will never move it from one complex plane to the other. So these two vertical polarizations really are separate possibilities.”
“But how could we turn one polarization into the other?” Carla pressed him. “Say, turn a leftor of North into a leftor of Up?”
“East times North is Up,” Romolo replied. “That’s the leftor, getting a quarter-turn. But the rotation of vectors that involves left-multiplication by East is a half-turn in the North-Up plane—which exchanges Up and Down. So when you flip a luxagen upside down, you swap the two vertical polarizations. That means they really do deserve to be called ‘up’ and ‘down’: the whole Future-Up plane for leftors describes a vertical polarization of ‘up’, and the whole North-East plane describes a vertical polarization of ‘down’.” He sketched the details, to satisfy himself that the rotation really did swap the planes as he’d claimed.
Patrizia said, “So the luxagen has a kind of axis in space that you can distinguish from its opposite. Like the two ways an object can spin around the same axis.”
Carla had been struggling to think of a suitable analogy herself, but Patrizia’s choice was weirdly evocative. “We should see if the new wave equation conserves the direction of this axis—if it really does stay fixed like the axis of a gyroscope.”
She converted the relationship between the field’s leftor and rightor and the energy-momentum vector into a more traditional form, where the energy and momentum came from the rates of change of the wave in time and in space. From there, they could work out the rate of change of the polarization axis—and it wasn’t necessarily zero. For some luxagen waves, the axis would shift over time.
“So it’s not like a gyroscope,” Patrizia said.
“Hmm.” Carla puzzled over the results. “The axis of a rotating object won’t always stay fixed. If the object is in motion—like a planet orbiting a star—and there’s some mechanism that allows angular momentum to flow back and forth between orbital motion and spin, you wouldn’t expect either one to be conserved individually. Only the total angular momentum will stay the same.”
Patrizia said warily, “So if we give the luxagen some angular momentum in its own right—as if it really were spinning around its polarization axis—then any change in that should be balanced by an equal and opposite change in orbital angular momentum?”
“Yes. If the analogy really does hold up that far.” Carla was exhausted, but she couldn’t leave the idea untested. As she ploughed on through the calculations she kept making small, stupid mistakes, but Romolo soon lost his shyness about correcting her.
The final result showed that the luxagen’s orbital angular momentum would not be conserved on its own. But by attributing half a unit of angular momentum to the luxagen itself—fixing the amount, but allowing its direction to vary with the polarization axis—the rate of change of the two combined came out to zero, and total angular momentum was conserved.