Discussion:
The Dirac Equation is independent of the relativity vs non-relativistic paradigm (there's a version for each).
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r***@gmail.com
2019-05-24 01:22:24 UTC
Dirac's progression that led to the Dirac equation was a series of mis-steps and red herrings, the biggest of which was that the Dirac equation was somehow a signature feature of Relativity. Actually, it's not. It's paradigm-independent (something that can be done equally well in both the relativistic and non-relativistic settings). There's also a non-relativistic version of Dirac's equation.

Non-Relativistic Version of the Dirac Equation: http://scipp.ucsc.edu/~dine/ph217/217_dirac_nr.pdf

So, whatever Dirac thought he was "unifying", he wasn't.

The only thing that's actually required to create a Dirac equation is a quadratic invariant. For relativity that's the "mass shell" invariant p² - E²/c² + m²c² = 0. This can be made compatible with non-relativistic theory by replacing the energy E by the kinetic energy H = E - mc², thus resulting in 2 invariants: (1) m for the rest mass and (2) p² - 2mH - H²/c² = 0 for mass shell. In non-relativistic theory all the (1/c²) factors become 0, and the invariants reduce to m and p² - 2mH = 0.
p² - E²/c² + m²c² = 0

Linearize it as α↑·p↑ + δH + εM, impose (α↑·p↑ + δH + εm)² = p² - 2mH - H²/c² to find what the relations are (α₁² = 1, α₁α₂ + α₂α₁ = 0, etc.; α↑δ + δα↑ = 0, α↑ε + εα↑ = 0, δ² = 1/c², ε² = 0, δε + εδ = 2). For the non-relativistic case δ² = 0.

Use the operator correspondence p↑ ⇔ -iħ∇, H ⇔ iħ∂/∂t, m ⇔ -iħ∂/∂u, and express this as iħ (-α·∇ + δ ∂/∂t - ε ∂/∂u) ψ = 0 and -iħ (∂/∂u)A ψ = m ψ, use the last relation to eliminate the u-dependence. The use of H in place of E is referred to as the Foldy-Wouthuysen transform.

Foldy-Wouthuysen Transform: https://en.wikipedia.org/wiki/Foldy%E2%80%93Wouthuysen_transformation

The matrix algebra spanned by (α↑, δ, ε) is ALWAYS equivalent to the Dirac algebra, no matter what you use in place of the 1/c² factors (including 0).

The result equations are equivalent to the Dirac equation if using δ² = 1/c², and to the Pauli-Schrödinger equation if using δ² = 0.

To make it work with the electromagnetic field, modify the operator correspondence to
p↑ ⇔ -iħ∇ + eA↑, H = iħ∂/∂t + eφ, m = -iħ∂/∂u + e B
For relativity, the result produces the classical version of the "B-field formalism" (the name of the extra scalar field B) that's used in quantum electrodynamics. It applies equally well for non-relativistic theory.

B-Field Formalism for Electromagnetism: (discussed here)
https://www.researchgate.net/publication/228569634_Schwinger's_principle_and_the_B-field_formalism_for_the_free_electromagnetic_field

B-Field Formalism for Gauge Fields: (discussed here)
https://arxiv.org/pdf/1510.03213.pdf
r***@gmail.com
2019-05-24 01:36:14 UTC
α↑ε + εα↑ = 0, δ² = 1/c², ε² = 0, δε + εδ = 2)...
should be δ² = -1/c² and δε + εδ = -2
For the non-relativistic case δ² = 0.
The result equations are equivalent to the Dirac equation if using δ² = 1/c²,
... if using δ² = -1/c².
and to the Pauli-Schrödinger equation if using δ² = 0.
More precisely, the algebra spanned by (α↑, δ, ε) is equivalent to the COMPLEXIFIED Dirac algebra; namely, the algebra spanned by the units (γ⁰, γ¹, γ², γ³) with complex coefficients, rather than real coefficients.

This is equivalent to the algebra spanned by (γ⁰, γ¹, γ², γ³, i) with REAL coefficients and this, in turn, is equivalent to the algebra spanned by (γ⁰, γ¹, γ², γ³, γ₅) with real coefficients.

The fact that there are 5 generators helps explain why the 5th generator has a 5 (in which case the first generator should revert to the historically order form γ⁴.

And viewing the Dirac equation in this broader context provides an independent account of (and helps show) why the algebra needs to be "complexified". The quadratic form is actually a metric for a 4+1 dimensional de Sitter geometry (for both the relativistic and non-relativistic forms); and the complexified Dirac algebra is equivalent to the real Dirac algebra for de Sitter geometry.