## Geometric algebra

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

### Geometric algebra

What with all the recent buzz about geometric algebra, I decided to do a bit more research into the subject... Unfortunately, I found that most of the available materials are highly technical, assume prior knowledge, and generally have a very steep learning curve and are quite inaccessible to someone not already steeped in the subject.

Fortunately, I did finally find an accessible introduction to the subject: http://www.jaapsuter.com/geometric-algebra.pdf

Just thought I'd post that here to help anyone who's interested in the subject but doesn't already have prior knowledge of it.
quickfur
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### Re: Geometric algebra

I tried to learn electromagnetism using geometric algebra. I couldn't. There are materials on this, but they all assume you already know electromagnetism the old way. So I had to go learn it the old way until I got to the relativistic version. I couldn't deal with that. But I now (I hope) have enough to understand the relativistic geometric algebra. When (if) I get a grip I'll write it up here.
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### Re: Geometric algebra

I've heard good things about it. Sounds very interesting. I read somewhere that it's like the quantum mechanics of physics.
in search of combinatorial objects of finite extent
ICN5D
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### Re: Geometric algebra

ICN5D wrote:I've heard good things about it. Sounds very interesting. I read somewhere that it's like the quantum mechanics of physics.

I think the idea is that it makes learning electromagnetism, relativity, and quantum mechanics easier. I know that I have a terrible time trying to understand traditional physics notation. But it is hard to find material that teaches physics in terms of GA. Instead they assume you already know physics and show you how to translate to GA. But that's even worse IMO. It just adds yet another layer of notation.

Learning relativistic EM is a b*tch in trad notation. I can't do it. GA is SO much better. Physics is mainly for the purpose of building machines, and it's good for that. Most of them don't care that much about understanding what's going on. Even if they do care, they've given up trying to understand it. A weak understanding can be worse than nothing.

An attraction for me is that GA is a tool for extending physics to N dimensions. Clifford built it for this purpose (I think). Most physicists couldn't care less about that.
PatrickPowers
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### Re: Geometric algebra

PatrickPowers wrote:[...]
Learning relativistic EM is a b*tch in trad notation. I can't do it. GA is SO much better. Physics is mainly for the purpose of building machines, and it's good for that. Most of them don't care that much about understanding what's going on. Even if they do care, they've given up trying to understand it. A weak understanding can be worse than nothing.

I have to admit that while I enjoy reading up the concepts of physics like quantum mechanics and the like, as far as actually working with equations is concerned I have never gone past Newtonian mechanics. The algebra is just too unwieldy, and I can't seem to avoid making little slip-ups in the process that almost always causes the result to be completely wrong.

An attraction for me is that GA is a tool for extending physics to N dimensions. Clifford built it for this purpose (I think). Most physicists couldn't care less about that.

Yeah, I'm very impressed with how easily GA handles arbitrary dimensions with ease. I think it holds a lot of promise, as far as the subject of this forum is concerned -- generalization to 4D space (+1D time). In fact, I surmise that the equations of 3D physics, cast in GA, can probably be lifted directly into 4D, as long as there are no dimension-dependent parts (or they are recast appropriately). It may turn out to be just a matter of working out the consequences of the equations in Cl4 in place of the usual Cl3. Well, that, and perhaps some other tweaks like using 1/r3 instead of 1/r2 in flux equations like gravity.
quickfur
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### Re: Geometric algebra

quickfur wrote: In fact, I surmise that the equations of 3D physics, cast in GA, can probably be lifted directly into 4D, as long as there are no dimension-dependent parts (or they are recast appropriately). It may turn out to be just a matter of working out the consequences of the equations in Cl4 in place of the usual Cl3.

Not really. There are usually several ways to extend something. To know which to choose in a convincing way, you have to understand the physics pretty well. For example, if you have a 2D thing in 3D space, what will it be in N D space? It depends. If it is a plane of rotation, then it's still 2D. If it is a surface or partition, then it will go to (N-1)D.

There is also a certain amount of fiddling around you have to do when they take advantage of peculiarities of 3D space. You have to undo that.

Sometimes I have no clue. What would a second dimension of time mean? Dunno.
PatrickPowers
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### Re: Geometric algebra

True. But I'm hoping that at a fundamental level, physics can be expressed in a non-dimensionally-dependent manner, such that the dimension-dependent stuff can be re-derived from the core equations in a self-consistent way. For example, Newton's laws of motion, or the special relativity version thereof, can probably be expressed in a dimensionally-independent way. From that, the dimensionally-dependent stuff such as what happens with a disk accelerating at rate X, etc., can be worked out in Cl4 independently from the 3D-specific derivations.

Of course, things like electromagnetism will probably require a deeper understanding of physics... your recent post about electromagnetism arising simply as a consequence of special relativity on charged particles is, to me, a breakthrough with regard to generalizing electromagnetism to 4D in a self-consistent way. The surface behaviour might turn out to be radically different, due to the extra dimension of space (e.g., a coil of wire may not quite have the same kind of magnetic field shape as we are used to in 3D) but the inherent nature of electromagnetism will remain the same. So I expect that the core equations, cast in GA form, should be identical, but the outworking of those equations may take on radically different forms in Cl4 compared to Cl3.

As for a second time dimension... that's a whole 'nother kettle of fish. I think I'll stick with 4+1D instead of 3+2D.
quickfur
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### Re: Geometric algebra

So why is GA so great? That it is an algebra over a field.

An algebra over a field is a vector space equipped with a bilinear product. So it is possible to multiply anything by anything. Better yet in the case of GA everything has an inverse, so it possible to divide by anything (except zero). Just multiply by the inverse. So one can take the derivative of anything that meets a smoothness condition. Then one has power series so one can have sin, cos, exponential, or logarithm of anything. Wow! What's not to like?

I get the impression that much of the cruft of physics notation is working around the absence of a product.

So let's put that power to work. Let's try a derivative that is lim h->0 [f(x+h)-f(x)]/h = df/dh. Why not? Both x and x+h must be in the domain of f, that is the only restriction, but that is rather obvious. What about the type of f? Doesn't matter. The function is differentiable if and only if the derivative is well-defined. Cool! Whether it is worth while to use this flexibility or not I don't know, but it is there if needed.
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### Re: Geometric algebra

From what I understand, only some elements of GA are invertible, not everything. So before you go around dividing stuff, you need to check whether the inverse actually exists first. Fortunately, vectors (and I believe all blades) are invertible in GA, so this covers most of the important use cases. Still, given an arbitrary GA expression, you can't be 100% sure that you can just divide things willy-nilly. You have to check first.
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### Re: Geometric algebra

As for what's not to like, one thing is the geometric product itself, which, in spite of the fact that it has the most amazing algebraic properties, is a bear to actually compute. A naïve implementation of the geometric product in software is rather slow, because it requires far more operations than, say, a hard-coded cross product. So if you want any reasonable speed out of the thing, you have to check for special cases and hand-code those to be faster, which at the end of the day amounts to essentially implementing the traditional methods and dispatching to them in the special cases, only falling back to the (slow!) full geometric product when all else fails.

Keep in mind also, that if you're working with floating-point arithmetic, more operations often means less accuracy / more sensitivity to errors in approximations. So even if you don't care for performance, sometimes you may have to resort to special-cased code just for the roundoff errors not to completely overtake your computation and give you a nonsensical result.

Of course, that in no way detracts from the theoretical / algebraic beauty of GA, and sometimes this nice unification of everything is just the tool you need to analyse a sticky situation where traditional methods will be annoying to work with. Plus, GA also provides you with a tool for deriving other special cases that traditional methods may not provide you, for the particular situation you're working with, so it's extremely valuable as an analytical tool. But it's not a free lunch.
quickfur
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### Re: Geometric algebra

That could be. Well, I don't care. I will never compute an actual number.

Having been a computer programmer for decades, I'm firmly in the "make it right then make it fast" camp. And computer power is becoming so cheap that speed is less of an issue every day. Finally, if there were a demand I bet you could get software that optimizes the operations better than a human can do it by hand.

But, whatever floats your boat. There's no rule that everyone has to do things the same way.
PatrickPowers
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### Re: Geometric algebra

Oh I certainly agree with "make it right first then make it fast". I'm a programmer myself, and have written number-crunching programs before, but while correctness is certainly important, if I can get a correct answer in 1 minute I wouldn't want to wait 2 days just because the program is using a suboptimal algorithm. (This has actually happened before, btw, at the first company I worked for, there was a klunky reporting script that, for small inputs, worked reasonably well, but one day somebody tried running it on a large dataset, and two days later it was still not finished. I rewrote the script in about a couple of hours, and it produced the report in 2 minutes. Performance does matter.)

Also, I'm quite familiar with optimizing software (optimizing compilers, in particular), and it's not true that software can optimize better than a human can. Well, it is true that software can do this up to a certain level -- on modern processors you'll have a hard time beating an optimizing compiler when it comes to sheer throughput in the generated code. However, one thing no optimizing compiler will help you with is the algorithm. If your algorithm is O(n2), then no matter how blindingly fast the compiler tries to make it, it's still O(n2), and it will not be able to produce results within a reasonable amount of time once you give it a large enough input. Or, for certain hard problems, O(2n) and O(1.999n) can mean the difference between waiting a couple of months for the answer vs. waiting 15 lifetimes. Optimizing compilers can't help you there, neither can hardware, which is rapidly nearing physical speed limits. It takes a good amount of research and ingenuity to come up with a clever algorithm that reduces that exponential within the reach of feasibility (sometimes just barely, depending on the problem).

There's no free lunch.
quickfur
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### Re: Geometric algebra

Some of us still know EM theory, as she was taught in the seventies.

My foray into C.G.S. units have turned up some interesting supprises. Mind boggling geometry.
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### Re: Geometric algebra

PatrickPowers wrote:What would a second dimension of time mean? Dunno.

Sort of marginally parallel to the GA question but I think essentially you could classically treat extra time dimensions as spatially among themselves.
One time dimension suggests a line of time occurance.
Two time dimensions suggests a plain of time occurance.
Three time dimensions suggests an area of time occurance.

The suggestion then is that objects that animate along other than our line of space will tend to animate slower to us than those following the same line.
The principle of equivalence is that the object will see us animating slower relative to itself because we are travelling along a different angle of time dimension to them.

How would that look?
Well, throwing an object along a different time direction than our own would make it fall at a slower rate than it should...

If you make the time dimensions interchangeable with the spatial dimensions then things get even more interesting...
gonegahgah
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### Re: Geometric algebra

PatrickPowers wrote:What would a second dimension of time mean? Dunno.

Thinking about this further I have started relating this to my principle that dimensions are mathematical representations and not actual physical things.
By this I mean that there is no physical x-dimension and that we can mathematically represent any line as the x-dimension that we want to; and they are all interchangeable.
It is also sometimes more useful to use other co-ordinate systems such as longitude, latitude and altitude which we can use as dimensions instead of x,y,z for world situations.
So dimensions don't even have to be in straight lines...

It occurred to me that we can introduce multiple time dimensions into our calculations as well.
If you have a car moving at one speed and you throw an object out the window at a different speed then in principle you are involving two time dimensions.
Of course you can blend these together; but none-the-less, the opportunity exists to have time twice in the equations.
You can see for objects that travel under gravity or acceleration that time can also feature as a square property thus making it act in a area manner rather than a linear manner.
In this respect you are really dealing with two dimensions of time. Even if they are of the same length they act with an area behaviour.

There are also some other examples where it may be easier to use two time dimensions rather than one.
The movement of planetary moons relative to a star are easier to calculate if you have the time dimensions twice in the equations.

So I suspect our universe already can incorporate multiple time dimensions within our mathematical representation of it...
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### Re: Geometric algebra

gonegahgah wrote:There are also some other examples where it may be easier to use two time dimensions rather than one.
The movement of planetary moons relative to a star are easier to calculate if you have the time dimensions twice in the equations.

Indeed! Could you kindly indicate an example?

Physical chemists keep a different proper time for each particle in a system. Relativity forces them to do this. I have been told that each particle adds six dimensions to the system. I suspect that this is a bivector basis, since a 3+1 space has six basis biblades.

A system of N particles then has 6N dimensions in the mathematical sense. This overwhelming exponential increase is why Feynman came up with the idea of a quantum computer. It is the only conceivable way to solve systems such as large molecules.

Quantum mechanics is based on a "infinite dimensional rigged Hilbert space." It's infinite because the number of dimensions is so large it may as well be infinite. It's Hilbert if and only if all sums are finite. Rigging is some technical elegance I don't understand.

BUT all this being as it may, why do we say there is one dimension of time? It is because there is a system to convert one frame of reference to the other. We pick an arbitrary frame, make that the reference frame, and do all of our computation there. It is the only feasible way, I think.

BUT why do we think that there is an arrow of time? That time "flows" in a line like a river, always in one direction? This is true only in our macro world, of huge systems ruled by statistics. Time there is the same as entropy, which statistically flows in only one direction, that of increase. But the micro world of electrons and neutrinos and positrons has none of that. Electrons do not change at all. They may be created or destroyed, their state may change or be ambiguous, but an electron has no entropy. Statistics do not apply to a single body. In short, an electron has no arrow of time. Quantum mechanical equations are time symmetrical. PAM Dirac and Richard Feynman mined this view with great success. Richard saw a positron as an electron traveling backward in time. Why not? It could be.

That's a bit too radical for me right now, but one can very reasonably view time for an electron as traveling in a circle. An electron has a wavelength. That means to us it seems to have a tiny little one-handed clock spinning around with a very regular period. It isn't like our clocks in that there are no numerals on the dial. Revolutions are not counted. Every complete revolution is the same as every other. So how can one say that linear time is going on? This is meaningful only in a system, in relation to other particles. For an isolated electron the arrow of time concept is not useful. So we can say that it doesn't have an arrow of time. It has its own little circle of time.

Albert Einstein figured out how all the little clocks are related. Ultimately there is no solution. If you look at general relativity, in extreme cases the time concept is lost. There is no such thing, or is so altered as to be unrecognizable. The real fundamental property is a slippery Something Else that defies simple description. So be it.

---

There are mathematical functions that seamlessly combine the linear and circular. I'm thinking of the complex exponential and its inverse, the natural logarithm. Maybe such have something to do with two dimensions of time. Maybe not. The thing to do would be to understand general relativity and see if one can get a simple description of that. Who knows what you might find?
PatrickPowers
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### Re: Geometric algebra

quickfur wrote: As for what's not to like, one thing is the geometric product itself, which, in spite of the fact that it has the most amazing algebraic properties, is a bear to actually compute. A naïve implementation of the geometric product in software is rather slow, because it requires far more operations than, say, a hard-coded cross product. So if you want any reasonable speed out of the thing, you have to check for special cases and hand-code those to be faster, which at the end of the day amounts to essentially implementing the traditional methods and dispatching to them in the special cases, only falling back to the (slow!) full geometric product when all else fails.

Which naive implementation are you thinking of? Using a 2nx2nx2n array of structure coefficients, and representing each multivector as a 2n array, would indeed be slow and expensive. But most of the components of the arrays will be 0, so there should be a more compact representation.

A blade, orthogonally aligned with the basis vectors, can be represented as a single real number (its magnitude) and a string of bits (maybe a byte, for 8D) denoting which basis vectors are present. For example, with basis vectors e0,e1,e2,...,e7, a trivector can be written

[pi]e0e2e3 = {3.14159... , 10110000}

(or the bits may be flipped to 00001101; different conventions).

Then the geometric product is simply a XOR operation on the bits, and ordinary multiplication of the magnitudes (with some +/- sign flips).

A general multivector could be represented as a list or array with variable length, with (basis-aligned) blades as entries. Multiplication would be done by the distributive property, breaking it into products of blades.

The wedge and contraction products are grade projections of the geometric product. Grade k projection of a blade simply compares k with the number of set bits, and returns 0 if they're different. Extend everything by linearity / distributive property.

The wedge product of blades could be defined more directly: If the bitwise AND is 0, then apply the geometric product; otherwise, return 0.

Similarly, the left-contraction could be defined using bitwise ANDNOT.
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### Re: Geometric algebra

PatrickPowers wrote:
gonegahgah wrote:There are also some other examples where it may be easier to use two time dimensions rather than one.
The movement of planetary moons relative to a star are easier to calculate if you have the time dimensions twice in the equations.

Indeed! Could you kindly indicate an example?

I don't think this is what's being referred to, but while solving the Kepler problem in polar coordinates, I found that the solution was simplified by expressing the position and time as functions of another variable. This turned out to be the "eccentric anomaly"; see Kepler's equation (where the "mean anomaly" is basically time).

Using this as the independent variable instead of time, and using a spinor as the dependent variable instead of the position vector, makes the Kepler problem into a linear equation!

d2U/ds2 = (E/2) U.

Here E is the total energy (constant), and U is essentially a complex number (a scalar plus a bivector in the orbital plane) related to the position vector by

r = U~ e1 U = e1 U2.

The star's mass M only appears as a constraint on the solution:

2 ||dU/ds||2 - E ||U||2 = GM.

The time is determined by

dt/ds = ||r|| = U~ U = ||U||2.

For example, when E = 0, the solution to d2U/ds2 = 0 is a straight line in the complex plane (U = a + bs), and U2 is a parabola. So we have

U~ U = ||a||2 + (a~ b + b~ a) s + ||b||2 s2

t = ||a||2 s + (a~ b + b~ a) s2/2 + ||b||2 s3/3

which corresponds to Barker's equation (especially when b is perpendicular to a).
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