Half-Integer Spin and the Free Space Constants

Most of the time that you see the Planck constant used, it is used in terms of angular frequency (ω), so it’s not so much the Planck constant as the reduced Planck constant (ħ).  This is because it’s referring to an entire cycle of 180 degrees or two radians or 2π.

Most of the time you see the permittivity of free space or permeability of free space use (also known as the electric constant and magnetic constant respectively), you see that 4π is involved.  It’s as though we could talk about the reduced magnetic constant (µ₀) to remove that 4π term.  For example:

µ₀=2α.h/(e².c)=4π.α.ħ/(e².c) <=> µ₀-bar=α.ħ/(e².c)

Similarly, we could have a naturalised version of the electric constant (ε₀), which also almost always has a 4π involved:

ε₀=e²/(2α.h.c)=e²/(4πα. ħ.c) <=> ε₀-bar=e²/(α. ħ.c)

As discussed in What is the Planck Constant? these both resolve to unity in Planck units.  Note also, as discussed in Fine-Structured but not Fine-Tuned, α= e²/q_pl², meaning that µ₀ and ε₀ can be expressed in terms of ħ (unity in Planck units), c (unity in Planck units) and q_pl (unity in Planck units), ie:

µ₀-bar=ħ/(q_pl².c)=(ħ/q_pl²)/c=1

and

ε₀-bar=q_p ²/(ħ.c)=(q_p ²/(ħ)/c=1

This is quite useful, basically everything (at the Planck scale) resolves to unity and the only reason we have odd numbers is because of our units of length, time, mass, charge and temperature – or because of our arbitrary choice of reference mass (for α_G) and slightly less arbitrary choice of reference charge (for α).

The question arises though, why 4π?  The 2π for the reduced Planck constant, ħ, makes sense because of the angular frequency, because it’s referring to a full rotation through 360 degrees, or 2π radians, but how could 4π make sense?  Of course, I’ve given the game away in the title of this post.

A photon has what you could call a “normal” spin.  After a spin of 360 degrees it is identical to how it started.  Sub-atomic particles however, like electrons, have a quantum level half-integer spin, or spin-1/2 – they need a spin of 720 degrees to arrive back at an identical state from which they started out, or 4π radians.

This suggests that the electric and magnetic constants might be linked to a characteristic peculiar to quarks and leptons, ie half-integer spin.