β(x), the Dirac Equation's Lorentz-Invariant Scalar Phase Field
The mysterious universal quantum holographic phase field - Beta in the Geometric Algebra Dirac Equation
The important points: β is actually β(x), it is a field, it varies from place to place. It is a field of pure number, a scalar field, a field of phase. It is Lorentz-invariant, that is, it looks the same no matter one’s state of (constant) motion (which is more than can be said for normal time and space); despite β being at the very heart of quantum theory “its existence is not generally recognized by physicists, and its physical interpretation has remained problematic to this day”. β determines the electron wave-function everywhere in space, which in turn is the main mechanism responsible for everything anyone has ever perceived — all the chemical and mechanical properties of atoms, the light emitted by stars; the electron wavefunction determines everything but nuclear and gravitational forces, and indirectly is responsible for our detection and interactions with them as well.
In Oersted medalist David Hestenes’ 2016 paper: The Genesis of Geometric Algebra: A Personal Retrospective 1 on the 12th page (p.362) he discusses his reformulation of the Dirac equation in geometric terms:
ψ is the electron wavefunction, ρ is charge density, i is the unit imaginary or pseudoscalar, e here means the base of the natural logarithms (it’s also often confusingly used for basis vectors, but it will have a subscript in that case). β has to be a pure real number, a scalar, in order to be used in this situation. When you see e^(i …), whatever is in the “…” position is acting as the phase of a wave, as it goes from 0 to 2 pi, the value of the expression will trace out a cosine wave as its real part and a sine wave as its imaginary part — repeating that cycle every n times 2 pi as “…”, in this case “β/2”, continues to increase .
More notes on notation: γ_0 is the time coordinate’s basis vector., γ_3 is the basis vector for the direction of motion. h-bar is Planck’s constant divided by 2 pi. “~” above a character indicates reversion, that all its basis vectors are multiplied together in reverse order compared to the character without the tilde. .
Hestenes continues:
This shows that R = R(x) specifies a position dependent Lorentz transformation determining directions of the Dirac current and spin at each spacetime point. The Lorentz invariant “β-factor” in the general form (14) for a “Real Dirac spinor” is so deeply buried in matrix representations for spinors that its existence is not generally recognized by physicists, and its physical interpretation has remained problematic to this day.
On the next page he then goes on to show the geometric interpretation of i in the Schrödinger and Dirac equations, that phase of the electron is its angle of rotation in the plane of spin, and says: "this iron-clad connection between spin and phase must be explained by any satisfactory interpretation of quantum mechanics. The widely accepted “Copenhagen interpretation” does not appear to meet that standard." His interpretation of the Dirac equation is given more weight by this anecdote:
Interest in STA began to pick up about 1980, when Dirac invited me to give a colloquium on Real Dirac Theory at Florida State University where he was living out his retirement. His associate, Leopold Halpern, confided that Dirac had given me his greatest compliment: mine was the first colloquium in 10 years when he stayed awake the whole time. ... Dirac agreed to write a letter of support for an NSF research proposal of mine. ... It was characteristically pithy—one sentence long: “I think there is something to this Real Dirac Theory.” Are you surprised that the reviewers dismissed his opinion as inconsequential when they rejected my proposal?
The important points: β is actually β(x), it is a field, it varies from place to place. It is a field of pure number, a scalar field, a field of phase. It is Lorentz-invariant, that is, it looks the same no matter one’s state of (constant) motion (which is more than can be said for normal time and space); despite β being at the very heart of quantum theory “its existence is not generally recognized by physicists, and its physical interpretation has remained problematic to this day”. β determines the electron wave-function everywhere in space, which in turn is the main mechanism responsible for everything anyone has ever perceived — all the chemical and mechanical properties of atoms, the light emitted by stars; the electron wavefunction determines everything but nuclear and gravitational forces, and indirectly is responsible for our detection and interactions with them as well.
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Geometric Algebra for Physicists, a 2003 textbook by two of the leaders in the field, Chris Doran and Anthony Lasenby of Cambridge, briefly discusses β:
J is the current referred to here, I is the unit pseudoscalar.
In the middle of the last quote appears this table:
Which shows β determining the scalar and pseudoscalar observables of the Dirac theory, rotating from the scalar observable into the (negative) pseudoscalar observable.
The Genesis of Geometric Algebra: A Personal Retrospective,
David Hestenes, Adv. Appl. Clifford Algebras 27 (2017), 351–379
Open access at Springerlink.com: DOI 10.1007/s00006-016-0664-z