Conformal Geometric Algrebra

Discussion of theories involving time as a dimension, time travel, relativity, branes, and so on, usually applying to the "real" universe which we live in.

Conformal Geometric Algrebra

Postby PatrickPowers » Fri Feb 05, 2021 10:16 am

Much engineering mathematics is based on vector algebra. It’s OK but is not very good at dealing with rotations and cyclic things like waves. They have various methods which I never liked. Some only work in 3D. Geometric algebra deals with it in an elegant geometric way in any number of dimensions. That’s all well and good but everything is centered around a special point called the origin. You move something to the origin, rotate it, and move it back. (Moving something in a straight line is called translation.) The real world isn’t like that.

The next step is conformal geometric algebra which gets rid of this. In this framework straight lines are portions of circles with infinite radius. Translations and rotations are handled the same way. The way it is done is explained here.

https://crypto.stanford.edu/~blynn/haskell/cga.html
https://clifford.readthedocs.io/en/v1.0.3/ConformalGeometricAlgebra.html

How do they work this magic? Start with a usual N dimensional space. Add two more dimensions where the action takes place. Move your stuff over there, do your business, move it back. I’ll explain those two dimensions, but first a little background. Hermann Minkowski was one of Albert Einstein’s teachers. He shared the opinion that young Albert was a “lazy dog,” and was astonished when the famous paper on special relativity was published. Hermann understood it right away. He felt that the mathematical framework could be improved. Albert had used basic trigonometry in his arguments. This down to Earth approach was easy to understand but not so easy to work with. Hermann offered instead a 4D geometry in which the relativistic effects were hyperbolic rotations. Time vectors have a negative length. Combine a time vector with a space vector and it is possible to get a vector of zero length. Light in a vacuum travels as such vectors. Vectors of zero or negative length seem weird but don’t worry about it. It just means that they are of some different unknown character that acts in an unfamiliar way. Albert didn’t like it at first -- too abstract -- but got used to it and before long adopted the idea.

To conformalize a geometric algebra add two new basis vectors. One is one of those Minkowski time vectors with the negative modulus (length squared), the other is another ordinary space vector. It’s just there because you need two dimensions to do a rotation. You don’t want to use one of the space dimensions you have already because that dimension would become confused and confounded. If you find it convenient to add more of these special space dimensions, go ahead. They don’t cost anything.

But these are not the dimensions that one uses directly. Instead multiply the two basis vectors together in such ways that their moduli cancel each other out. You get geometric products of modulus zero that are 45 degrees from each of the two basis vectors and perpendicular to one another. The hyperbolic space defined by these two products is the special sauce.

A tantalizing thing is that this seems like a poor man’s version of the hot topic of the AdS/CFT correspondence. One starts out in CFT (conformal field theory), which is too hard to calculate in. Instead convert to the hyperbolic AdS space, do your work there, then translate back.

I like it because it always seemed to me that energy was in a special world that is disjoint from ours. Things described by vectors of negative modulus are on the other side of barrier, while vectors of zero moduls are are on the surface between the two worlds. Quantum leaps could be a localized breakdown of the barrier between the worlds.
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