## The Dirac Equation Meets Geometric Algebra

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### The Dirac Equation Meets Geometric Algebra

http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf by David Hestenes shows geometric algebra applied to the celebrated Dirac relativistic wave equation. To me the interesting the thing is that Hestenes shows that the common belief that spin emerges from relativity is not correct. I don't claim to understand it, but he seems to be saying that spin is an observed fact built into the equation. It doesn't emerge from anything, it was introduced in a subtle way that was overlooked until now. While Hestenes frames no hypothesis he seems to imply that the electron could actually be rotating. Wolfgang Pauli believed this to be impossible because the surface of the electron would then be moving faster than the speed of light. But if the radius of the electron is small enough then this isn't the case, is it? And if it has extent into non-obvious smaller dimensions then the electron can be much more compact. If the basic unit of the dimensions is very small such as perhaps the Plank length, the electron could have a very small yet positive radius while having a surprisingly large surface area, this presumably being connected to the wavelength, which is certainly much longer than the radius.

Scattering experiments have shown that the radius is very small. I have the impression that physicists on the whole have given up and declared the electron to be a point particle with zero radius, then used dodgy math to get rid of the resulting infinities. One of the goals of string theory is to get rid of these.

It is certain that quantum spin is orientable, so it occurs in an even dimensional space. This rotation can't include time, because such a rotation would be hyperbolic. Hestenes simply declares a bivector of spin. Actually since the size of the bivector is a constant this is a one dimensional space embedded in a 2D space, but such a 1D space is orientable so it doesn't matter.

Hestenes is partial toward the zwitterbewegung interpretation, in which on very small scales the electron has a random component to its movement.
PatrickPowers
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### Re: The Dirac Equation Meets Geometric Algebra

No, what I get from Hestenes is that the electron is not a rotating sphere, but a circulating point. The whole electron, not just its surface (if it exists), is moving in a circle. And this motion is regular, not random. The frequency of circulation is about 10²¹ Hertz, or a zetta-Hertz (which coincidentally sounds almost like "zitter-Hertz" ).

But I don't claim to fully understand it either.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: The Dirac Equation Meets Geometric Algebra

mr_e_man wrote:No, what I get from Hestenes is that the electron is not a rotating sphere, but a circulating point. The whole electron, not just its surface (if it exists), is moving in a circle. And this motion is regular, not random. The frequency of circulation is about 10²¹ Hertz, or a zetta-Hertz (which coincidentally sounds almost like "zitter-Hertz" ).

But I don't claim to fully understand it either.

Zwitterbewegung [jittering movement] is a small random motion of the electron due to attraction/repulsion to/from electron-positron pairs that form spontaneously in the vacuum. It is pretty well established that this is what happens. https://iopscience.iop.org/article/10.1088/1367-2630/15/7/073011 How this induces a periodic variation called the Compton wavelength I don't know. It would seem that it would have to be an average over time and have something to do with the mean lifetime or half-life of vacuum positron-electron pairs. That means the wavelength is a property of the vacuum acting on the electron, not of the electron itself.

According to Richard Feynman even the speed of light is an average. At zetta scale it isn't a constant. He even believed that the positron was an electron going back in time as implied by the Dirac equation. This hasn't caught on, but he may have been right.

As to the geometry of the electron, I'll let Ed Witten figure it out.
Last edited by PatrickPowers on Wed Apr 28, 2021 11:18 pm, edited 1 time in total.
PatrickPowers
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### Re: The Dirac Equation Meets Geometric Algebra

Zittern (= to jitter) <> Zwitter (= hybrid)

--- rk
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### Re: The Dirac Equation Meets Geometric Algebra

Apparently Hestenes is using an established term (zitterbewegung), but with a new meaning, as he often does.

We know he doesn't agree with the standard interpretations of quantum mechanics. That may or may not apply to the explanation of zitterbewegung by electron-positron pairs.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: The Dirac Equation Meets Geometric Algebra

The paper you linked, and Wikipedia (with the ei t term), both show that zitterbewegung is a circular or sinusoidal motion, not random.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
mr_e_man
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### Re: The Dirac Equation Meets Geometric Algebra

mr_e_man wrote:Apparently Hestenes is using an established term (zitterbewegung), but with a new meaning, as he often does.

We know he doesn't agree with the standard interpretations of quantum mechanics. That may or may not apply to the explanation of zitterbewegung by electron-positron pairs.

Is there any justification for this point of view? I don’t remember if there is convincing evidence that would support his position. If they are not, then this is the usual chatter in order to attract attention. We need scientific substantiation to refute certain laws of physics and mathematics.
Challenger007
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### Re: The Dirac Equation Meets Geometric Algebra

His point of view is based on his analysis of the equations of standard single-particle relativistic quantum mechanics, translated to geometric algebra. It's not baseless speculation, though it may be wrong nonetheless.

He does try to find experimental evidence to test his ideas. See his snark paper, for example. (It's more than 10 years old; there may be newer experiments I'm unaware of.)

In fact in Section 7 there he refers to the standard explanation of zitterbewegung by electron-positron pairs, accepting it as a possibility, but wanting more concrete calculations to support it.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
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