Can quaternion math be used to model spacetime?

Question

Quaternions are commonly used to model 4 dimensional systems where the quaternion consists of a real 3 dimensional vector and an imaginary scalar. So on the surface Quaternions seem well suited to model space time if time can be considered imaginary. Do the operations of quaternion math further provide a suitable framework, or are there problems?

Answer 1

There are some problems with using quaternions to describe spacetime. Quaternions have two important properties: (1) they form a four-dimensional vector space; (2) you can multiply quaternions together.^[1] The first property is obviously very suggestive, but it's no different from the usual four-vectors that we already use in special relativity. To specifically make use of quaternions, we'd also have to use the second property. Recall that -- at least in ordinary discussions of special relativity -- you'll never multiply four-vectors together and get another four-vector; you only "contract" them (take their dot product). And the standard quaternion multiplication doesn't actually achieve this dot product. The scalar term in the result is something like the usual spacetime interval, but then there are those extra vector components. You can get a pure scalar multiplying a quaternion by itself, but only by conjugating one copy, which gives you a positive-definite result rather than the interval. So it's not quite useful for special relativity. And as far as vectors go, the usual approach is good enough.

Even though they have four degrees of freedom, quaternions really do "live in" a physical space of three dimensions. It turns out that quaternions shouldn't be thought of as a scalar plus a vector. Instead, they should be thought of as a scalar plus a bivector.^[2] More specifically, quaternions are actually the natural "spinors" of three-dimensional space. So instead of being like vectors, they act on vectors. For example, probably the most common use for quaternions is to describe rotations of vectors. This bears repeating: Quaternions shouldn't be thought of as vectors in 4-d; they should be thought of as operators acting on vectors in 3-d.

So to answer your question, yes you could use quaternions to model spacetime -- though there'd be a lot of useless baggage floating around. But extrapolating the motivation for your question, I might rephrase it as "Can we use the special properties of quaternions to get any computational advantage or philosophical insight in relativity?" To that, the answer is no; they don't have anything useful to say about spacetime since they're really just about space.

But there's good news! There is a natural generalization of quaternions to four-dimensional spacetime, and it really does give us computational advantages and philosophical insights. The cool thing about thinking of quaternions as scalar + bivector is that the idea now generalizes very easily to arbitrary dimensions -- and in particular to Minkowski spacetime. This is a field of study called "Geometric algebra", or GA for short.^[3]

Spinors in 4 dimensions^[4] turn out to act a lot like spinors in 3 dimensions (quaternions). For example, they can be used to rotate 4-dimensional vectors really nicely. But they can also boost 4-d vectors just as easily -- a boost is sort of a generalized rotation. It turns out that a lot of the usual things we do in special relativity are way easier using spinors.

And you can keep going to other dimensions. For example, going back to just two dimensions, you find that complex numbers are the spinors of 2-d! You even start to understand complex algebra better using GA. In fact, I've taught GA to biologists by starting in 2 dimensions. Once you understand this simple example, it's almost trivial to extend GA to arbitrary dimensions.^[5]

If you want to learn more, there's a fantastic book on this called Geometric Algebra for Physicists. It's actually my favorite physics book, full stop. There are lots of good online references too, if you google around for it. And I have to plug the Geometric Algebra module for sympy, which gives us a nice (open-source) program for doing the math symbolically.

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Footnotes:

1. Taken together, these two facts mean that quaternions form "an algebra". The idea may seem kind of weird -- that you can actually multiply two vectors by each other. You already know how to multiply a vector by a scalar. And you can take the dot and cross products, but neither of those is invertible. But really just multiplying two vectors in a (usually) invertible way might seem weird. And then you realize that you do it all the time with complex numbers, which also form "an algebra". Not to mention matrices.

2. It just so happens that in three dimensions, there are three degrees of freedom in a bivector, and three degrees of freedom in a vector. So when Hamilton discovered quaternions, he was understandably confused about what they represented. His confusion was the whole reason for the vector/quaternion wars of the 1890s. Nowadays, we understand that quaternions and vectors are just two aspects of the same thing: GA. I would argue that this confusion is one of the great tragedies in the history of physics, as Grassmann and Clifford had already developed all the tools necessary to resolve the conflict.

3. We could discuss the name of this thing till the cows come home. But in practice, "Geometric Algebra" is a sub-type of Clifford Algebra, except we assume that in GA the coefficients for our vector space are real numbers, whereas CA can have coefficients from any field -- especially complex numbers. But CA is usually introduced with irrelevant abstractions, and the complex version is almost never necessary for applications in physics (even quantum mechanics!).

4. Spinors in 4-d are sometimes called biquaternions, which are "complexified" quaternions, but that's a very bad path to go down. The complexification is unenlightening, and doesn't really apply to other dimensions. I think it's symptomatic of a tendency to use obscure, accidental features specific to a particular dimension -- as opposed to the intuitive, pedagogical, systematic, and universal approach of GA.

5. The path that spinors and normed division algebras (NDAs) took together splits at dimension four, as the latter head into a dead end (there are no more NDAs after octonions). The spinors in four dimensions do have eight degrees of freedom, like the octonions, but that's just the vector-space property. The other property of algebras, multiplication, can't be the same because octonions are not associative -- but associativity is one of the defining features of GA. So the octonions aren't a particular example of GA. However, it's also worth pointing out that there are other spinor groups even for dimensions ≤3 when you have a non-positive signature. For example, the split-complex numbers are the spinors of a two-dimensional version of Minkowski space.

   Of course, there's very little need for octonions in physics. John Baez has a typically great introduction to an article about octonions in physics that you can read here, in which he shows that there are applications in supersymmetry / string theory (and pure math, obviously). But that's the most convincing argument I've seen that octonions might ever have any relevant applications in physics -- and I'm certainly not convinced.

Answer 2

The answer above is much more detailed (and better by far) than mine, but I would recommend Penrose's "Road To Reality", pages 200 onwards, specifically the statement that "The quaternionically natural" quadratic form (...) has the incorrect signature for relatively theory. Special relativity depends on a signature of say - + + + (allowing us to express time in space terms as -ict) but, as I understand it, with quaternions you have a signature of + + + +, which does not conform to the metric used in 4 - D space-time.

Answer 3

Quaternions indeed intrinsically model spacetime.

Quaternions derive from the 4-squares identity, discovered by Leonhard Euler, i.e. that the product of two sums of each four squares is always again a sum of four squares: ($X_0^2$ + $X_1^2$ + $X_2^2$ + $X_3^2$)($Y_0^2$ + $Y_1^2$ + $Y_2^2$ + $Y_3^2$) = ($Z_0^2$ + $Z_1^2$ + $Z_2^2$ + $Z_3^2$). ($Z_0$,$Z_1$,$Z_2$,$Z_3$) can be algebraically expressed in terms of ($X_0$,$X_1$,$X_2$,$X_3$) and ($Y_0$,$Y_1$,$Y_2$,$Y_3$). By substituting $x_0$ = $X_0$; $x_1$ = $\hat iX_1$; $x_2$ = $\hat jX_2$; $x_3$ = $\hat kX_3$, and doing similarly for $y_0$, $y_1$, $y_2$, $y_3$; as well as for $z_0$, $z_1$, $z_2$, $z_3$;wherein $$\hat i^2 = \hat j^2 =\hat k^2 =\hat i\hat j\hat k = -1$$ we can construct a "tight" space wherein a complete vector product exists, i.e. wherein the length of the product of two vectors equals the product of the lengths of the vectors. This space has a metric signature ($+1,-1,-1,-1$), which is the signature of spacetime (in "west coast notation"). Quaternion space is a non-Euclidean, hyperbolic space, wherein a maximum speed of propagation exists (given by the asymptotes to the confining hyperbola). Poisson's equation, which describes the transport of matter in a conservative system, becomes a wave equation in quaternion space, because of its negative metric signature ($+1,-1,-1,-1$). In its most general form, this wave equation corresponds to the fundamental equation of electrodynamics, i.e. to Maxwell's equations in their compact, 4-dimensional form. Maxwell's equations - to constant factors close - describe the geometric laws of quaternion space. As the transport of matter in a space with negative metric ($+1,-1,-1,-1$) is always bound to a wave equation, we also have here the foundations of quantum mechanics.

I have recently published a small paper on this in the "Bulletin de la Société Fribourgeoise des Sciences Naturelles", Vol 103 (2014), p. 83-90. The paper is entitled "De la réalité des nombres" and is in French; it is a little bit more explicit than what I have stated here in a nutshell.