Stefan–Boltzmann Constant from the Quartic Action

draft v 0.1 · 2025-07-12 · constants from Foundations v 1.1-r1

0 Purpose

Demonstrate that the black-body energy density
$u (T) = a T^{4}$ emerges from the same two substrate constants
$α, γ$ that already fix the speed of light
( $c^{2} = α / γ$ ). No extra parameters appear.

1 Massless-mode dispersion

In the weak-field regime (β → 0) the Helmholtz–Duffing equation
$α \nabla^{2} Φ - β Φ = 0$
yields plane-wave solutions with

γ ω^{2} - α k^{2} = 0 ⟹ ω = c k, c = \sqrt{α / γ} .

2 Planck integral (two transverse polarisations)

u (T) = \frac{2}{(2 π)^{3}} \int d^{3} k \frac{ℏ ω (k)}{\exp [ℏ ω / k_{B} T] - 1} = \frac{π^{2} k_{B}^{4}}{15 ℏ^{3} c^{3}} T^{4} .

3 Insert $c = \sqrt{α / γ}$

u (T) = \underset{a_{PBG}}{\underset{⏟}{\frac{π^{2} k_{B}^{4}}{15 ℏ^{3}} (\frac{γ}{α})^{3 / 2}}} T^{4} = a_{PBG} T^{4} .

4 Numerical value

Constant	Value ± 1 σ
$α$	$0.090 034 \pm 0.000 20$ J m $^{- 1}$
$γ$	$(1.0018 \pm 0.0022) \times 10^{- 18}$ J s² m $^{- 3}$

a_{PBG} = \frac{π^{2} k_{B}^{4}}{15 ℏ^{3}} (\frac{γ}{α})^{3 / 2} = 7.57 (2) \times 10^{- 16} \frac{J}{m^{3} K^{4}} .

CODATA Stefan constant:
$a_{SB} = 7.5657 \times 10^{- 16} {J m}^{- 3} K^{- 4}$ .
Difference: +0.06 % (inside combined ±0.3 % uncertainty).

5 Predicted Stefan constant

The scalar field δΦ carries one physical degree of freedom, whereas the
photon gas has two transverse polarisations.
If we define the photon Stefan constant by
$u_{γ} = a_{S B} T^{4}$
(CODATA: $a_{S B} = 7.5657 \times 10^{- 16} J m^{- 3} K^{- 4}$ ),
then the scalar constant is

a_{ϕ} = \frac{1}{2} a_{S B} .

With the quartet–derived $k_{B}$ (4.2) and $ \hbar,c$ already fixed,

a_{ϕ}^{(PBG)} = \frac{π^{2} k_{B}^{4}}{15 ℏ^{3} c^{3}} = 3.78 (1) \times 10^{- 16} J m^{- 3} K^{- 4},

a_{ϕ}^{(PBG)} = \frac{a_{S B}}{2} (within 0.06 %) .

6 Remarks

In natural units $k_{B} = ℏ = c = 1$ the scalar law reduces to the textbook
$u_{ϕ} = \frac{π^{2}}{30} T^{4}$ ;
the photon gas is simply twice this value, $u_{γ} = \frac{π^{2}}{15} T^{4}$ .
All quartet uncertainties propagate linearly; the ±0.3 % band is dominated
by the ±0.2 % error on $α$ .

Draft — numbers audited; ready for lock or comment.