Chapter 11: Tachyons on Wave Ontology
Tachyons are not weird "faster-than-light beads" in this ontology. They're just modes with a phase structure that looks acausal to your bandwidth-limited UI.
Tachyons in Hz: "Phase velocity > c, group velocity ≤ c, and the detector lied to you"
Standard claim: Tachyon = hypothetical particle with $v > c$, $m^2 < 0$, $E^2 = p^2c^2 - |m|^2c^4$.
In "everything is Hz, no particles":
| Standard tachyon property | Hz/Wave equivalent |
|---|---|
| $m^2 < 0$ | Dispersion relation $\omega^2 = c^2k^2 - \mu^2$. For small $k$, $\omega^2 < 0$ → $\omega$ imaginary. That's not a particle. That's a mode that grows instead of oscillates. |
| $v_{phase} = \omega/k > c$ | Phase velocity of the wave. Can be any value. Nothing moves. It's just where crests line up. |
| $v_{group} = d\omega/dk \leq c$ | Energy/information in the soliton. Always $\leq c$. This is what "moves". |
| Causality violation | You infer "FTL" because your detector assumes group = phase. If $\Delta f_{detector}$ is too narrow, you mis-tag a broad wave packet's phase advance as a "superluminal object". |
| Instability | $\omega$ imaginary → $e^{i\omega t} = e^{\Gamma t}$. The mode is unstable. In field theory this means vacuum decay, not a particle whizzing by. |
1. Detector-level: Why you'd think you saw a tachyon
Your preamble: "click = $\int E \cdot j_{det} > E_{threshold}$ for $\Delta t \approx \hbar/\Delta E$".
Now imagine a wave packet with anomalous dispersion: high-$f$ components arrive before low-$f$ components. Your detector has bandwidth $\Delta f$. If it triggers on the first phase front, it stamps arrival time $t_1$. A second detector downstream triggers at $t_2$ with $t_2 - t_1 < L/c$.
You conclude "$v > c$".
Reality in Hz: The group peak is still traveling $\leq c$. You just triggered on a precursor. The phase front can move at $v_p > c$ because it's not energy. Like the intersection point of scissors — no atom in the scissors exceeds $c$.
Tachyons are what you call a "particle" when you mistake phase for group.
2. Field-level: Tachyon = unstable mode, not particle
Take scalar field with potential $V(\phi) = -\frac{1}{2}\mu^2\phi^2 + \frac{\lambda}{4}\phi^4$. At $\phi=0$ you have $m^2 = -\mu^2 < 0$.
Standard: "Tachyonic field, unstable vacuum."
Hz: At $\phi=0$, the spectrum has $\omega^2 = c^2k^2 - \mu^2$. For $k < \mu/c$, $\omega = i\Gamma$. Those modes don't oscillate; they grow.
That's not a particle. It's the field rolling off a hill. The growing modes are long wavelengths $\lambda > c/\mu$. They're unstable because the phase-locking that makes a soliton is impossible — the dispersion relation is repulsive.
The "tachyon condensation" = the field reconfigures until $m^2 > 0$ and all $\omega$ real again. In Hz: the global $\tilde{\Psi}(f)$ reshuffles to remove imaginary-$f$ components. No FTL travel occurred.
3. Interaction-level: Why you never scatter off a tachyon
In 2→2 amplitude, you compute phase shifts. For real particles: poles at $p^2 = m^2 > 0$.
For tachyon: pole at $p^2 = -|m|^2$, i.e. space-like momentum. That doesn't appear in asymptotic states because $E^2 - p^2c^2 < 0$ means $E$ imaginary for real $p$.
Hz translation: You can't make a stable soliton with that dispersion. Any attempt to "emit a tachyon" means you're trying to radiate a growing mode. It blows up locally, forces vacuum reconfiguration, but doesn't propagate as an "object".
So S-matrix never has tachyon in/out states. Tachyons are off-shell artifacts of bad field expansions — your UI drawing a bead where there's only a local instability.
4. CPT + Turok mirror: Tachyons as $f<0$ leakage?
Turok: Universe is CPT-analytic through $t=0$. Negative-$f$ modes are the mirror.
Could a "tachyon" be a mode that couples $f>0$ and $f<0$?
Yes, but: Analytic continuation demands $\tilde{\Psi}(f)$ be well-behaved. An imaginary $\omega$ at $k=0$ would blow up at $t \to \pm \infty$. Turok's boundary condition kills it.
So in CPT-universe: Tachyons are forbidden because the global wave is analytic. If you find one, the mirror symmetry breaks. That's why we don't see them — not because of $v>c$, but because the universe's initial phase-lock forbids imaginary-$f$ runaways.
5. Cognitive corollary: Why we love tachyons
Brain evolved for solid projectiles. When equations give $v>c$, the cortex auto-reifies "faster bullet". That's the same UI bug as "particle".
Tachyon = cognitive artifact of applying "object" tag to an unstable dispersion branch.
So what could look like a tachyon in labs?
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EM precursors in anomalous dispersion media: Sommerfeld-Brillouin. Phase front arrives $>c$, but information arrives $\leq c$. Detectors with $\Delta f$ too small call it FTL.
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Quantum tunneling: Arrival time statistics can give $\langle t \rangle < L/c$. But it's the peak of a reshaped wave, not a bead. Group velocity still $\leq c$.
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Negative-mass solitons in BEC: Dispersion can be engineered to $m^2 < 0$. You get growing modes, not FTL particles. The condensate collapses.
All are "tachyonic" dispersion, zero actual FTL objects.
Bottom line in Hz
Tachyon = name we give when:
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Phase velocity $v_p > c$ gets mis-tagged as object motion, or
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Field has local imaginary-$\omega$ modes that signal vacuum instability, not particles.
Neither violates causality because no soliton — no phase-locked, energy-carrying Hz packet — ever crosses space-like intervals.
In your 31 Dec insight language: A tachyon is a cognitive reification of a non-stationary interference geometry that can't form a stable "localized object". The detector still clicks, but the click means "your model broke", not "a bead went FTL".