Because multiplication is non-commutative, we need to be careful with the order of vectors when we take inverses or perform division. Taking the inverse of two or more vectors multiplied together entails reversing their order:
This reversal is necessary to ensure that the original vectors come together in the right sequence to give a final result of Future:
Similarly, when we divide by a product of vectors we need to reverse their order:
Although the tables only give the results for multiplying and dividing the four principal directions, the same operations can be applied to any vectors at all (with the exception that you can’t divide by the zero vector). A completely general vector can be formed by adding together various multiples of the principal directions:
Here a, b, c and d are real numbers, and they can be positive, negative or zero. Let’s define another vector, w, using four other real numbers, A, B, C and D:
We can multiply v and w together by following the ordinary rules of algebra, taking care with the order in which we write the products of vectors:
The length of a vector can be found from the four-dimensional version of Pythagoras’s Theorem. We write the length of the vector v as |v|, and it is related to the size of its components in each direction by:
When two vectors are multiplied together, the length of the resulting vector is just the product of the lengths of the original vectors:
Given a vector v it’s often useful to talk about its conjugate, which we write as v* and define as the vector whose components in the three space directions are the opposite of those of v, while its component in the time direction is exactly the same as that of v:
Multiplying a vector by its own conjugate gives a very simple result:
Since the direction Future acts like the number one in this system, if v has a length of one its conjugate v* will also be its inverse, v-1. If the length of v is not one, we can still find its inverse from its conjugate by dividing by the square of its length:
Because of this close relationship between the conjugate of a vector and its inverse, it’s not hard to see that if we take the conjugate of a product of vectors we need to reverse their order, just as we do with the inverse:
The Future component of the product of one vector with the conjugate of another, v × w*, carries some useful geometric information about the two vectors:
The quantity on the right hand side of the first equation, where we multiply the four components (a, b, c, d) of v and (A, B, C, D) of w and add up the results, is known as the dot product of the two vectors. As the second equation shows, it depends only on their lengths and the angle between them.
Any rotation of four-dimensional space can be achieved by fixing two vectors whose length is one, say g and h, and then multiplying on the left with g and dividing on the right with h. So the vector v is rotated by:
For example, a rotation that swaps North and South and also Future and Past, while leaving all directions orthogonal to these unchanged, can be achieved by setting g = South and h = North.
How can we be sure that this operation really is a rotation? For a start, it’s easy to see that the length of v is unchanged, since |g|=|h|=|h-1|=1 and:
We can also look at what happens to the angle between two vectors when we perform the same operation on both of them, by looking at the effect on v × w*:
The Future component of v × w* is left unchanged by this operation, since g × Future ÷ g = Future. And since the Future component of v × w* determines the angle between v and w—along with their lengths, which we know are unchanged—that angle is also unchanged.
All rotations that involve only the three dimensions of space can be achieved by restricting the original formula to the case where h = g:
For example, rotating everything by half a turn in the horizontal (North-East) plane can be achieved by setting g = Up.
Two other particular kinds of rotation occur when we set h = Future, which amounts to simply multiplying on the left by g:
and when we set g = Future, which amounts to simply dividing on the right by h:
Both these operations will always rotate in two orthogonal planes simultaneously, and by exactly the same angle in both. For example, multiplying on the left by East will rotate by a quarter-turn in both the Future-East plane and the North-Up plane.
Consider a rotation specified by g and h that transforms vectors according to the usual formula:
There are two other kinds of geometrical objects that can be described by quaternions, but which are not vectors because they obey different transformation laws when the same rotation takes place:
These curious objects are known as “spinors”: l is a “left-handed spinor” and r is a “right-handed spinor”. In our own universe the mathematics of spinors isn’t quite as simple as it is in the universe of Orthogonal, but it’s very similar, and in both universes spinors play a crucial role in describing the way certain fundamental particles behave when they’re rotated.
Afterword
By the early years of the twentieth century, physicists had identified a variety of deeply puzzling phenomena—some naturally occurring, some the product of laboratory experiments—that could not be explained by the classical laws of mechanics, thermodynamics and electromagnetism. The spectrum of radiation emitted by an incandescently hot object made no sense: it should have contained an almost equal amount of energy at every possible frequency, but instead it tapered off rapidly as the frequency increased, a disparity known as the “ultraviolet catastrophe”. And while atoms had been shown to be composed of charged particles, both positive and negative, nobody could explain how they could be stable, or why the spectrum of hydrogen consisted of a set of sharply defined frequencies that followed a simple mathematical rule.
In the Orthogonal universe, because there is an upper limit on the frequency of light there is no ultraviolet catastrophe, and the spectrum predicted by classical physics is only slightly different from the true, quantum-mechanical version. And while the puzzle of the stability of charged matter remains, there is no direct equivalent to the hydrogen atom to serve as an elementary test bed for a new theory. What’s more, the simple electronics that lay behind many physics experiments during the birth of quantum theory in our universe is not available in the Orthogonal universe: the very nature of electromagnetism makes the generation of any appreciable, sustained electrostatic force on anything but a microscopic scale almost impossible.
Carla’s tarnished mirrors do have their closest match in one seminal experiment from the early days of quantum theory: the photoelectric effect. This phenomenon brought Nobel prizes in the 1920s to Albert Einstein for his theoretical work and Robert Millikan for his painstaking experiments—though Millikan appears to have been trying to refute the theory! The photoelectric effect refers to the release of electrons from a metal surface in a vacuum when the surface is struck by light of various frequencies; the electrons can be collected and their rate of ejection from the surface measured as a current flowing along a wire. The abrupt cessation of this photoelectric current when the frequency of the light striking the metal falls below a critical value supported the idea that light could only be absorbed and emitted in discrete amounts whose energy was proportional to the light’s frequency. Since it required a certain amount of energy to tear each electron away from the metal surface, unless each individual quantum of light, or photon, carried that minimum energy the light could not produce a current.
The Orthogonal version works somewhat differently: rather than absorbing light to gain energy, the surface is stimulated by the incoming light to radiate light itself, which is accompanied by the production of conventional energy. Also, more than one quantum of light is needed to bridge the energy gap between bound and free luxagens, since a smaller gap would make the material unstable.
With no electronics at her disposal, Carla can only observe the tarnishing itself, along with the scattering of light by the luxagens released into the vacuum. The unexpected
way in which free luxagens interact with light echoes another landmark experiment in our universe, in which the X-rays scattered by free electrons in graphite were found by Arthur Compton to betray distinctively particle-like behavior.
Despite all the difficulties they face, the scientists of the Orthogonal universe do have one advantage: the mathematics of quantum mechanical spin fits into a beautiful geometrical framework that they would have had good reason to explore, long before the discovery of quantum mechanics itself. In their universe, four-dimensional vectors are naturally identified with the number system that we call quaternions (see Appendix 3 for more details). Remarkably, quaternions can also be used to describe entities known to us as spinors, which correspond to particles such as electrons in our universe, or luxagens in the novel. Having a ready-made mathematical system that can encompass both vectors and spinors offers a powerful short-cut to insights that took many years to achieve in our own history of quantum mechanics. For this insight I’m indebted to John Baez, who explained to me how spinors can be viewed as quaternions.
Although the word “magnetism” appears nowhere in the novel, most readers will recognize Patrizia’s ideas about aligning the spins of luxagens in a solid as something closely analogous to the creation of a permanent magnet. Just as an electrostatic force that pulls in one direction over macroscopic distances is impossible in the Orthogonal universe, the same is true of magnetism, so there is no long historical tradition of familiarity with this phenomenon. But amazingly enough, the quantum subtleties that Patrizia discovers to be dictating the alignment of spins are even more crucial to the existence of permanent magnets in our own universe than in hers! Under our rules, the magnetic force between spinning electrons encourages them to adopt opposite spins and cancel each other’s magnetic fields, and it’s only the quantum effect that we call the “exchange interaction”—which relies on the way different combinations of spin affect the average distance between electrons, and hence the average electrostatic repulsion between them—that allows a substance like iron to hold a powerful magnetic field.
The “optical solids” described in the novel might sound reminiscent of the “optical lattices” that are used by real-world researchers to trap and study atoms at extremely low temperatures—but in fact these are very different systems. In the Orthogonal universe, the hills and valleys of light’s electric field can be made to move slowly enough that charged particles can be trapped in the valleys and carried along with the light. A combination of three light beams can sculpt this “energy landscape” so that the valleys confine the trapped particles in all three dimensions.
In our own universe this is impossible: charged particles could never keep up with a traveling light wave, and in a standing wave—where the intensity of the light forms a fixed pattern in space—the electric field is still oscillating in time, with each valley becoming a hill, and vice versa, hundreds of trillions of times per second. But while an optical lattice can’t trap charged particles in its ever-changing electric field, it can nonetheless exert subtler kinds of forces. These forces relate to the intensity of the light rather than the direction of the electric field, so they retain a consistent direction over time and can be used to confine electrically neutral atoms.
Supplementary material for this novel can be found at www.gregegan.net.
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Appendix 1: Units and measurements
Appendix 2: Light and colors
Appendix 3: Vector multiplication and division
Afterword
Greg Egan, The Eternal Flame
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