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2019-05-24 01:22:24 UTC

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PermalinkNon-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.

A few links

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