zeroPointEnergy
Zero-point energy per mode with frequency ω equals ħω/2 in RS-native units. QFT researchers studying vacuum energy or Casimir effects would cite this definition as the base for fluctuation calculations. It is a direct one-line definition importing the RS value of ħ from the constants module.
claimThe zero-point energy for a mode of frequency $ω$ is $E_0 = ħω/2$, where $ħ$ is the reduced Planck constant in Recognition Science units.
background
The module derives vacuum fluctuations from the discreteness of τ₀ in Recognition Science. Empty space contains fluctuations because time is discrete at τ₀, implying energy fluctuations via the uncertainty relation at the minimal time interval. Zero-point energy represents the ground state energy of each quantum harmonic oscillator mode. The upstream definition of ħ sets it to φ^{-5} in native units, consistent with the forcing chain T5-T8. This aligns with the module's target of showing the vacuum is not empty but filled with activity from temporal discreteness.
proof idea
The declaration is a one-line definition that applies the standard formula for zero-point energy by multiplying the imported ħ by ω and dividing by 2.
why it matters in Recognition Science
It provides the energy expression used by the downstream theorem vacuum_has_energy, which establishes that zeroPointEnergy ω > 0 for ω > 0. This fills the QFT-010 objective of deriving zero-point energy from τ₀ discreteness, per the module documentation. It connects to the Recognition framework's constants derived via the eight-tick octave and phi fixed point.
scope and limits
- Does not derive zero-point energy from the forcing chain axioms.
- Does not compute the Casimir force or other observables.
- Does not address renormalization or ultraviolet divergences.
- Does not specify the spectrum of modes in a field theory.
Lean usage
theorem vacuum_has_energy (ω : ℝ) (hω : ω > 0) : zeroPointEnergy ω > 0 := by unfold zeroPointEnergy; apply div_pos; exact mul_pos hbar_pos hωformal statement (Lean)
65noncomputable def zeroPointEnergy (ω : ℝ) : ℝ := hbar * ω / 2
proof body
Definition body.
66
67/-- The vacuum state is NOT the zero-energy state.
68 It's the minimum-energy state, with E_0 > 0 for each mode. -/