The Dimensions of Quantum Objects

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.

The Dimensions of Quantum Objects

So how many dimensions does an electron have? It depends.

The electron is described by the Dirac equation. I don't understand the notation at all, but all I cared about was the space in which the equation is valid. The answer is: four complex dimensions.

So you could say it was eight dimensional, but my guess is that it isn't really. Quantum objects are related to waves. When used to describe waves, complex variables are usually of length one. That takes away one degree of freedom, leaving only one degree. Essentially it is polar notation with a constant radius, so the only variable is the angle around a circle. Or perhaps better you could say that the only variable is the phase of the wave.

Electrons are elementary objects so the only possibility is isoclinic rotations. This reduces the degrees of freedom wonderfully. All electrons have the same basic wavelength, so the only variables remaining are the phase and spin. Spin has exactly two possible values. These two correspond to the two chiralities of an even-dimensional rotation. (I don't think it is actually rotating. IMO nobody knows what it is actually going on.)
PatrickPowers
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Re: The Dimensions of Quantum Objects

Have you seen Hestenes' reformulation of the Dirac theory? You can see how the multivector version relates to the matrix version here (Appendix A) or here (Section 5) or here or here.

The eight components of the complex matrix correspond to eight components of a 3+1D multivector: 1 scalar part, 6 bivector parts (xy, xz, yz, xt, yt, zt), and 1 quadvector part (xyzt). The multivector determines an "expected" orientation and velocity of the electron at a given point in spacetime, as well as the probability of finding it there.

PatrickPowers wrote: Electrons are elementary objects so the only possibility is isoclinic rotations.

Are you talking about 3+1D or 4+1D electrons?
mr_e_man
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Re: The Dimensions of Quantum Objects

Most interesting! It's going to take a while to work through, but I sense it is much better than the trad approach. What on Earth does the quad vector signify?

I was talking about plain ordinary electrons. I don't understand them, so an attempt on 4+1D would be too ambitious.
PatrickPowers
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Re: The Dimensions of Quantum Objects

mr_e_man wrote:Have you seen Hestenes' reformulation of the Dirac theory? You can see how the multivector version relates to the matrix version here (Appendix A) or here (Section 5) or here or here.

The eight components of the complex matrix correspond to eight components of a 3+1D multivector: 1 scalar part, 6 bivector parts (xy, xz, yz, xt, yt, zt), and 1 quadvector part (xyzt). The multivector determines an "expected" orientation and velocity of the electron at a given point in spacetime, as well as the probability of finding it there.

PatrickPowers wrote: Electrons are elementary objects so the only possibility is isoclinic rotations.

Are you talking about 3+1D or 4+1D electrons?

Now I have some time. The link wouldn't send me any data. ?
PatrickPowers
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Re: The Dimensions of Quantum Objects

geocalc.clas.asu.edu and geometry.mrao.cam.ac.uk don't work for you?

In my previous post, the first link was to "Selected Papers on Geometric Algebra in Quantum Mechanics". The second and third were to "Local Observables in the Dirac Theory" and "Geometry of the Dirac Theory", by David Hestenes. The fourth and fifth were to "States and Operators in the Spacetime Algebra" and "2-Spinors, Twistors and Supersymmetry in the Spacetime Algebra", by Chris Doran, Stephen Gull, and Anthony Lasenby.
mr_e_man
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Re: The Dimensions of Quantum Objects

mr_e_man wrote:geocalc.clas.asu.edu and geometry.mrao.cam.ac.uk don't work for you?

Today they work. Thank you for the pointers.
PatrickPowers
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