From the Standard Model to topology
Before talking about TQNT, we need to know what it claims to replace. A guided tour of the Standard Model of particle physics: its successes, its 19 free parameters, and why topology can do better.
1. The four fundamental forces
As of 2024, four fundamental forces are known in nature:
| Force | Range | Mediating particle | Intensity (relative) |
|---|---|---|---|
| Strong | ~ $10^{-15}$ m | Gluons (8) | 1 |
| Electromagnetic | infinite | Photon $\gamma$ | $10^{-2}$ |
| Weak | ~ $10^{-18}$ m | $W^\pm$, $Z^0$ | $10^{-13}$ |
| Gravitational | infinite | Graviton (hypothetical) | $10^{-39}$ |
The Standard Model describes the first three. Gravity is treated separately by general relativity, and nobody currently knows how to combine the two consistently.
2. The matter structure of the Standard Model
The Standard Model organises particles into two families:
Fermions (matter)
Three identical generations, organised in doublets:
| Generation | Quarks | Leptons |
|---|---|---|
| 1 | u (up), d (down) | $e^-$, $\nu_e$ |
| 2 | c (charm), s (strange) | $\mu^-$, $\nu_\mu$ |
| 3 | t (top), b (bottom) | $\tau^-$, $\nu_\tau$ |
Each particle has its antiparticle. Total: $6 \times 2 \times 2 + 3 \times 2 \times 2 = 36$ distinct states. We do not know why there are exactly three generations.
Bosons (forces)
- $\gamma$ (photon): electromagnetic force, massless
- $g$ (gluons, $\times 8$): strong force, massless
- $W^\pm$, $Z^0$: weak force, masses $\sim 80$–$91$ GeV
- $H$ (Higgs): scalar field that gives mass, $\sim 125$ GeV
3. The 19 free parameters
Here lies the Standard Model's Achilles' heel. For everything to work, one must plug in by hand 19 numerical values that the theory cannot compute:
- 9 fermion masses ($m_e, m_\mu, m_\tau, m_u, m_d, m_s, m_c, m_b, m_t$)
- 3 coupling constants ($\alpha_s$ strong, $\alpha_{EM}$, $\alpha_W$ weak)
- 4 parameters of the CKM matrix (quark mixing)
- 1 strong CP-violation parameter ($\theta_{QCD}$, measured to be zero)
- 2 Higgs parameters ($m_H$ and $v$, the scalar-field vacuum)
One could add 4 more parameters for neutrinos (PMNS), but let us stick to the historical minimal list.
19 numbers fallen from the sky. No known theory says why $m_e = 0.511$ MeV rather than 0.712. Why $\alpha_{EM} = 1/137.036$ rather than 1/138 or 1/100. Physicists have been hoping for 50 years for a deeper theory that derives these values instead of postulating them.
4. The Standard Model Lagrangian
For information (no need to understand it in detail), the full Lagrangian is written compactly as:
$$ \mathcal{L}_{SM} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + i\bar{\psi}\, \gamma^\mu D_\mu \psi + (D_\mu \phi)^\dagger (D^\mu \phi) - V(\phi) + (\bar{\psi}_L\, \phi\, \psi_R + \text{h.c.}) $$Four lines, four families of terms: gauge fields, fermions, Higgs, and Yukawa coupling. All numerical values are hidden in the last ones (the Yukawa couplings). It is elegant — and it is also an organised admission of ignorance.
5. The quantitative success
Let's be honest: the Standard Model is extraordinarily accurate. A few examples:
- Anomalous magnetic moment of the electron: agreement to $10^{-12}$
- Discovery of the Higgs boson in 2012, predicted in 1964: a huge triumph
- Cross-section precision at LEP, Tevatron, LHC: better than 1%
These successes are undeniable. The Standard Model is not wrong — it is simply insufficient.
6. What TQNT proposes instead
TQNT's idea, already encountered in the Discovery path, is to replace the proliferation of point-like particles by a single geometric entity: the knot. Concretely:
- A single field, called $\chi$, satisfies the SU(2) Chern–Simons action
- The non-trivial configurations of this field are topological knots $K$
- Each particle is identified with a knot or a link of knots
- Mass is computed from the invariants of $K$
Going from 19 parameters to 10 comes from replacing the individual masses with sector laws sharing a single scale ($\sqrt{\sigma_\chi}$), which produce 89 masses at once. The gauge couplings are fixed by the Chern–Simons level $k$. The CKM matrix is replaced by the P15 topological reconnections.
7. TQNT V6 parameter count
| Parameter | Role | Value |
|---|---|---|
| $\sqrt{\sigma_\chi}$ | tension of the $\chi$ string — the single scale | 418,47 MeV |
| $m_\ell$ | mass of the light building-block knot (trefoil) | 310 MeV |
| $m_s$ | mass of the strange building-block knot ($5_2$) | 494,8 MeV |
| $m_c$ | mass of the charm building-block knot | 1557 MeV |
| $m_b$ | mass of the beauty building-block knot | 4952 MeV |
| $f_\pi$ | pion constant (GMOR relation) | ≈ 92 MeV |
| $r_{\rm rad}$ | radial-excitation ratio | 1,38 |
| $\tau_\perp$ | spin–spin coupling (N–$\Delta$ gap) | calibrated on N–$\Delta$ |
| heavy hyperfine | heavy hyperfine gap | calibrated on J/ψ–$\eta_c$ |
| $\beta_\otimes$ | linking tension (n–p) | 0,210 MeV |
And $\alpha_{EM} = 1/137 = 1/(k+2)$ with $k = 135$ integer — so it is not an extra free parameter, it is just the level of the theory.
Standard Model: 19 parameters for 12 fermions + 4 bosons.
TQNT V6: 10 global parameters (9 of them for the mass sector) — 89 masses at 0.13 % median. The Standard Model describes the known physics; TQNT claims to explain it from a unified topological grammar.
End-of-module quiz
Four questions to check your understanding of the Standard Model and what TQNT proposes.
1. How many free parameters does the Standard Model have, and how many does TQNT V6 have?
2. Why is the fine-structure constant $\alpha_{EM} = 1/137$ not counted as a free parameter in TQNT V6?
3. What is the origin, in the Standard Model, of the fermion masses?
4. How many fermion generations are there in the Standard Model, and does the theory explain that number?