Field Normalisation in Phase-Biased Geometry

Field Normalisation in Phase-Biased Geometry

1. Why Field Normalisation?

The field variable $\mathrm{\Phi }$ in PBG is a dimensionless phase (e.g. an angle in radians).
All PBG observables—energy, force, speed—must follow from a Lagrangian/action whose units match SI conventions.

2. The PBG Action

• $\mathrm{\Phi }$ is dimensionless.
• Each term in the action must have units of energy density: $[\mathrm{J}/{\mathrm{m}}^3]$.

3. Units of PBG Constants

Given $\mathrm{\Phi }$ is dimensionless:

• Kinetic term: $({\partial }_t\mathrm{\Phi }{)}^2$ has $[1/{\mathrm{s}}^2]$, so $\gamma$ must be $[\mathrm{J}\cdot {\mathrm{s}}^2/{\mathrm{m}}^3]$.
• Gradient term: $|\mathrm{\nabla }\mathrm{\Phi }{|}^2$ has $[1/{\mathrm{m}}^2]$, so $\alpha$ must be $[\mathrm{J}/\mathrm{m}]$.
• Mass term: ${\mathrm{\Phi }}^2$ is dimensionless, so $\beta$ must be $[\mathrm{J}/{\mathrm{m}}^3]$.

4. Derived Quantities (with units)

• Speed of light:

  $$c^2=\frac{\alpha }{\gamma }$$

  $[{\mathrm{m}}^2/{\mathrm{s}}^2]$

• Yukawa/Coherence Kernel:

  $$k^2=\frac{\beta }{\alpha }$$

  $[1/{\mathrm{m}}^2]$

• Coherence length:

  $$\ell =1/k=\sqrt{\frac{\alpha }{\beta }}$$

  $[\mathrm{m}]$

5. Example Calibration

6. Takeaway

Field normalisation (dimensionless $\mathrm{\Phi }$) guarantees that all PBG constants and predictions are mathematically and physically correct, with SI units throughout. All prior constants and formulae may be used if this convention is maintained.

Field Normalisation and SI Units in PBG

In PBG, we fix field units so that all terms in the action have consistent SI units, and observable predictions match experiment. We do this by choosing $\mathrm{\Phi }$ to be dimensionless (the simplest, and most transparent choice).

1. The Action

2. Units for Each Term

• The Lagrangian density must have units of [J/m³].
• With $\mathrm{\Phi }$ dimensionless:
  • $({\partial }_t\mathrm{\Phi }{)}^2\sim [1/{\text{s}}^2]$ $⟹$ $[\gamma ]=\text{J}\cdot {\text{s}}^2/{\text{m}}^3$
  • $|\mathrm{\nabla }\mathrm{\Phi }{|}^2\sim [1/{\text{m}}^2]$ $⟹$ $[\alpha ]=\text{J}/\text{m}$
  • ${\mathrm{\Phi }}^2$ dimensionless $⟹$ $[\beta ]=\text{J}/{\text{m}}^3$

3. Consistency Checks

• $c^2=\frac{\alpha }{\gamma }\sim [{\text{m}}^2/{\text{s}}^2]$
• $k^2=\frac{\beta }{\alpha }\sim [1/{\text{m}}^2]$

4. Generality

Any other choice of field normalisation simply rescales the constants and leaves predictions invariant.

5. Bottom Line

All field equations and predictions are SI-consistent, with constants $\alpha ,\beta ,\gamma$ assigned the above units. This allows Casimir, $c$, and $\hslash$ to be derived and used in standard SI units throughout PBG.