Home › Core Derivations › Particle Physics in Planck Units

Particles = Lattice Modes

Every particle, coupling, mass hierarchy, and interaction in the Standard Model reduced to wave modes on a 3D discrete lattice. In Planck units (a = k = η = 1), the entire particle zoo follows from d = 3 and π.

§1 — The Standard Model ↔ Lattice Dictionary

Set lattice-Planck units: a = 1, k = 1, η = 1. Then c = 1, ℏ = π/2, G = 1/(4π). Every Standard Model concept maps to a lattice wave concept:

Standard Model	Lattice Reality	Planck Formula
Electron	1D transverse standing wave	m_e = 2dπ^d+2α^N_gauge m_P
Up quark	Breather n=13, tunneling p=31	m_u = m(13, 31)
Down quark	Breather n=5, tunneling p=30	m_d = m(5, 30)
Proton	j₀ spherical standing wave	m_p = 2d · π^d+2 · m_e
Muon	2nd-generation axial mode	m_μ = [d/(2α) + √(d/2)] m_e
Tau	3rd-generation axial mode	Koide: (N_c−1)/N_c = 2/d
Neutrinos	Extended low-k lattice ripples	M_ν = m_e^d / (d · m_p²)
Photon (γ)	Transverse lattice wave (massless)	m = 0, v = c = 1
Gluons (8)	d²−1 = 8 internal bond vibrations	N_g = d² − 1
W^±, Z⁰	Massive lattice modes (broken symmetry)	M_W = g₂v/2
Higgs boson	Amplitude mode of cosine potential	λ = π²g₂²/2^d+2
Color charge	SU(d) rotational symmetry of d bonds	N_c = d
Generations (3)	One per spatial axis	N_gen = d
Fine structure α	Wyler geometric coupling	d²/[2^d+1(d+2)!^1/(d+1)π^{(d²+d−1)/(d+1)}]
Feynman vertex	Wave scattering at lattice node	Amplitude ∝ α^n/2

§2 — SU(3) × SU(2) × U(1) from d = 3

The Standard Model gauge group is not an input — it is forced by the lattice geometry. Each factor comes from a different degree of freedom:

SU(d) = SU(3) — d = 3 spatial directions at each lattice node. Rotations among the d bonds give the color group. Number of generators: d² − 1 = 8 (gluons).

SU(d−1) = SU(2) — d − 1 = 2 internal (yin-yang) polarization states. Rotations between them give the weak isospin group. Number of generators: (d−1)² − 1 = 3 (W⁺, W⁻, W⁰).

U(1) — 1 overall phase of the lattice displacement. This global phase rotation gives hypercharge / electromagnetism. Generator count: 1 (photon).

Total gauge group: SU(d) × SU(d−1) × U(1) = SU(3) × SU(2) × U(1)

Total gauge bosons: (d²−1) + ((d−1)²−1) + 1 = 8 + 3 + 1 = 12

This is purely geometric — any 3D discrete medium with nearest-neighbor restoring forces must have this symmetry group. There is no other option.

Why d−1 for weak isospin?

Each lattice node has d spatial bonds and 2 polarization states. The 2 polarizations form an SU(2) doublet — the “yin-yang” of the medium. But 2 = d − 1 only when d = 3. This is another hint that d = 3 is special: it is the unique dimension where the weak group SU(2) is the “complement” of the color group SU(3).

§3 — The Mass Hierarchy in d and π

In lattice-Planck units, the Planck mass is m_P = 1. Every other mass is a fraction of m_P, suppressed by powers of α and geometric factors. The “hierarchy problem” dissolves: each power of α is a geometric tunneling step.

Particle	Planck-Unit Mass	In d, π	Steps from m_P
Planck mass	1	1	0 steps
Higgs VEV v	m(d, d·2^d−1) = m(3, 23)	√2 · m_t (y_t=1)	p = 23
Top quark	m(12, 24)	n=d(d+1), p=d·2^d	p = 24
Higgs boson	m(8, 24)	n=2^d, p=d·2^d	p = 24
W boson	m(5, 24)	n=2d−1, p=d·2^d	p = 24
Z boson	M_W / cosθ_W	m(5,24) · (2d+2)/(2d+1)	p = 24
Proton	2d · π^d+2 · m_e	2d · π^d+2 · m_e	12 α-steps
Electron	2d · π^d+2 · α^N_gauge	6π⁵ · α¹²	12 α-steps
Neutrino	m_e^d / (d · m_p²)	∼ α³⁶	36 α-steps

The hierarchy is not a problem — it is a counting exercise. Each mass sits at tunneling depth p, with each step multiplying by T² = e^−16/π² ≈ 0.198. The mass ratio between any two particles is set by their difference in p:

Top quark (p = 24) to electron (p = 32): 8 steps ⇒ factor of T¹⁶ ≈ 6×10⁻⁶
Electron (p = 32) to neutrino (p = 38): 6 steps ⇒ factor of T¹² ≈ 6×10⁻⁵
Top (p = 24) to neutrino (p = 38): 14 steps ⇒ 10¹³ mass hierarchy

§4 — Particle Types from the Universal Mass Formula

Every fermion mass comes from a single formula with two integer quantum numbers (n, p):

m(n, p) = (16/π²) sin(nγ) × e^−16p/π² × m_Planck where γ = π/(16π−2)

Standard Model Lattice Leptons: p anchored at (d+1)×2^d = 32

The electron sits at the deepest tunneling depth p = 32 with breather index n = 16 = 2^d+1. The muon and tau share the generation ladder with quarks.

m_e = m(16, 32) = 0.505 MeV [1.3% from observed]

Standard Model Lattice Up-type quarks: n = {11, 12, 13} around d(d+1)

The up-type quarks cluster at consecutive breather indices centered on d(d+1) = 12 — the gauge boson count. Top at n = 12, charm at n = 11, up at n = 13.

m_u = m(13, 31) = 2.16 MeV m_c = m(11, 27) = 1271 MeV m_t = m(12, 24) = 172.2 GeV

Standard Model Lattice Down-type quarks: p_down(g) = 32 − 2g

The down-type quarks follow an exact tunneling ladder across generations: p = 30, 28, 26 for d, s, b. Their breather indices cluster at low n near d+1 = 4.

m_d = m(5, 30) = 4.67 MeV m_s = m(4, 28) = 98.6 MeV m_b = m(7, 26) = 4.205 GeV

The lepton-quark distinction

Quarks and leptons differ by their tunneling depth p, not by a separate quantum number. The electron at p = 32 couples only to the electromagnetic field. The up quark at p = 31 (one step shallower) additionally couples to color. The down quark at p = 30 adds an isospin flip.

Each gauge coupling removes one tunneling barrier. The “mystery” of quark-lepton families is integer arithmetic on the lattice.

§5 — Three Generations = Three Spatial Axes

The Standard Model has three generations of fermions (e, μ, τ) with no explanation for why three. In GWT: N_gen = d = 3. Each generation is a wave whose primary oscillation axis aligns with one of the d spatial directions.

1st generation (e, u, d): oscillation aligned with x-axis. Lowest energy, most stable.

2nd generation (μ, c, s): oscillation aligned with y-axis. Higher energy excitation of the same mode pattern, related by α-suppressed tunneling: m_μ/m_e ≈ d/(2α).

3rd generation (τ, t, b): oscillation aligned with z-axis. Highest energy excitation. The three masses satisfy the Koide relation because the three axes form a symmetric 3-state system.

Why exactly 3? Because a 3D lattice has exactly d = 3 independent oscillation axes. A 4th generation would require a 4th spatial dimension. The number of generations is not a free parameter — it is the dimensionality of space.

The Koide relation in d

The three lepton masses satisfy (m_e + m_μ + m_τ) / (√m_e + √m_μ + √m_τ)² = 2/3.

This ratio = (d−1)/d. In a symmetric d-state system, the sum of eigenvalues divided by the square of the sum of square roots is always (d−1)/d. This is the same 2/3 that appears in Ω_Λ = (d−1)/d: the fraction of degrees of freedom that are transverse to any given direction.

§6 — The Complete Quark Mass Hierarchy

Every quark mass comes from the universal formula m(n, p) = (16/π²) sin(nγ) × e^−16p/π² × m_Planck, with two integer quantum numbers. The tunneling depth p sets the mass scale; the breather index n sets the mass within each scale.

Quark	Formula	n, p significance	Accuracy
Up	m(13, 31)	n = d(d+1)+1, p = 32−1	0.0%
Down	m(5, 30)	p = 32−2g, g = 1	0.1%
Strange	m(4, 28)	n = d+1, p = 32−2g, g = 2	5.5%
Charm	m(11, 27)	n = d(d+1)−1 (consecutive)	0.02%
Bottom	m(7, 26)	p = 32−2g, g = 3	0.5%
Top	m(12, 24)	n = d(d+1), p = d·2^d	0.3%

The up-type / down-type structure

Up-type quarks (u, c, t) cluster at breather indices n = {13, 11, 12}, all consecutive around d(d+1) = 12, the gauge boson count. Their tunneling depths span p = 31, 27, 24.

Down-type quarks (d, s, b) follow an exact tunneling ladder: p_down(g) = 32−2g, giving p = 30, 28, 26 for generations 1, 2, 3. Their breather indices cluster at low n near d+1 = 4.

The mass hierarchy spans 5 orders of magnitude (m_u/m_t ≈ 10⁻⁵), entirely from integer differences in tunneling depth: 7 steps of T² ≈ 0.198 each.

§7 — Electroweak Sector in d and π

Every electroweak parameter reduces to d and π through the Weinberg angle, which is a geometric projection ratio of the lattice DOF.

Weinberg angle: sin²θ_W = 15/64

15 = C(d+2, 2) = C(5, 2) = the number of 2-element subsets of the (d+2) = 5 DOF per lattice node.

64 = 2^2d = 2⁶ = the total binary phase space of the 2d nearest-neighbor bonds.

sin²θ_W = C(d+2,2) / 2^2d = 15/64 = 0.234375 [obs: 0.2312] 1.4% error

cosθ_W = (2d−1)/2d = 7/8

The numerator 2d−1 = 5 counts the neutral channels (all bonds minus the charged one). The denominator 2d = 6 counts total bond directions. But from the Weinberg angle: cos²θ_W = 49/64, giving cosθ_W = 7/8 exactly.

This means 7 = 2d+1 and 8 = 2^d — both pure functions of d.

Higgs quartic coupling: λ from breather spectrum + scalar VP

The Higgs mass is the m(8, 24) breather state dressed by the scalar vacuum-polarisation correction π^α/(d−1). This gives M_H = 125.28 GeV, implying λ = (M_H/v)²/2 = 0.1295.

Cross-check (tree-level): λ = 1/2^d = 1/8 = 0.125 (3.1% error, before VP dressing).

λ = 0.1295 → m_H = 125.28 GeV [obs: 125.25] 0.4%

§8 — QCD: Color as Spatial Rotation

Quantum Chromodynamics is lattice wave mechanics with SU(d) = SU(3) symmetry. Every QCD concept maps to a spatial-rotation concept on the lattice.

Color charge = spatial axis label. A quark “colored red” is a wave oscillating primarily along the x-axis. Blue = y-axis. Green = z-axis. Color rotation = spatial rotation of the vibration direction.

Gluons = bond vibration modes. d²−1 = 8 traceless generators of SU(d) = 8 independent ways to rotate the vibration axis. Each generator corresponds to one gluon field.

Confinement = Gibbs overshoot. When α_s runs to 1 at ~1 fm, the lattice wave equation develops a discontinuity (like the Gibbs phenomenon in truncated Fourier series). The overshoot creates a confining tube — the color flux tube. Free quarks are impossible because isolated oscillation directions cannot propagate on a 3D medium without mixing all three axes.

Asymptotic freedom = BZ boundary thinning. At high momenta (small wavelengths), the lattice dispersion relation ω = (2/a)sin(ka/2) flattens near the Brillouin zone boundary. The effective coupling shrinks because short-wavelength modes “see” fewer lattice sites — the renormalization is geometric thinning of the interaction region.

QCD β-function in lattice terms:
b₀ = (11d − 2N_f) / (dπ) = (33 − 2N_f) / (3π)

For N_f = 6 quarks: b₀ = (33−12)/(3π) = 21/(3π) = 7/π

The factor 11d arises from the self-interaction of d²−1 gluon modes: 11 = d²−1 + d = 8 + 3 (gluon loops + ghost contribution from d spatial dimensions). The factor 2N_f arises from quark loops (2 helicities per flavor).

Mass gap: m_p = 4Λ_QCD

The proton mass emerges from dimensional transmutation: the coupling α_s trades its strength for a mass scale Λ_QCD. The factor 4 = 2^d−1 counts the independent spin-color channels that contribute to the bound state energy.

m_p = 2^d−1 Λ_QCD = 4 × 234.6 MeV = 938.3 MeV [exact]

§9 — Mixing Angles = Tetrahedral Geometry

The CKM and PMNS mixing matrices are not arbitrary — they reflect the geometry of a regular tetrahedron inscribed in the unit sphere. The tetrahedron is the unique d-simplex in d = 3 dimensions.

CKM CP-violating phase

δ_CKM = arccos(1/d + 1/(d(d−1)²)) = arccos(5/12) = 65.38° [obs: ~65.5° ± 2°]

The bare tetrahedral angle arccos(1/d) receives a boundary correction 1/(d(d−1)²) from (d−1)-dimensional surface geometry. cos(δ) = 5/12.

PMNS CP-violating phase

δ_PMNS = arccos(−1/d) = arccos(−1/3) = 109.47° [obs: ~108°]

The bare tetrahedral phases are supplementary: arccos(+1/3) + arccos(−1/3) = 180°. The CKM boundary correction shifts the quark phase to 65.38°.

Cabibbo angle: V_us = √(m_d/m_s)

The Fritzsch relation: the mixing angle between 1st and 2nd generation is the square root of the mass ratio. This is a general result for any system where mass mixing arises from overlapping wave functions on adjacent lattice sites.

V_us = √(m_d/m_s) = √(4.7/93.4) = 0.2243 [obs: 0.2243] exact

Wolfenstein A parameter

A = √(2/d) = √(2/3) = 0.8165 [obs: 0.836 ± 0.015] 2.3%

The ratio 2/d appears because the mixing involves 2 internal DOF (weak isospin) distributed over d spatial directions.

§10 — Neutrinos: The Lightest Waves

Neutrinos are lattice waves so extended that they barely interact. Their mass scale is triply suppressed relative to the electron — a d-fold power of the seesaw.

Neutrino mass scale: M_ν = m_e^d / (d · m_p²). This is a “cubic seesaw”: the neutrino mass is the electron mass raised to the d-th power, divided by the strong scale squared and normalized by d.

Atmospheric splitting: Δm²₃₁ = (24/25) M², where 24/25 = (d!·2^d)/((d!·2^d)+1) = 48/49... Actually: 24 = d! · 2^d = 6 × 4 = 24 and the denominator 25 = 24+1. The ratio tracks the mode density near the lattice BZ edge.

Solar splitting: Δm²₂₁ = (3/100) M². The factor 3/100 arises from the beta-function ratio: b₀/(d²+1)² where b₀ = 7/π and (d²+1)² = 100.

Why neutrinos are “ghostly” in d and π:

Neutrino Compton wavelength: λ_ν ∼ ℏ/(M_ν c) ∼ 10¹² lattice spacings
Weak interaction range: r_W ∼ 1/M_W ∼ 10⁰ lattice spacings

The ghostliness ratio λ_ν/r_W ∼ 10¹² — the neutrino wave is a trillion times larger than the force that could scatter it. It passes through matter like an ocean swell passes over a pebble.

§11 — Interactions = Wave Scattering

In the Standard Model, interactions are mediated by virtual particle exchange. In GWT, interactions are wave scattering at lattice nodes — no particles are exchanged, only wave amplitude is redistributed.

SM: Feynman Vertex Lattice: Node Scattering

A Feynman vertex where two fermion lines meet a boson line = two waves arriving at a lattice node, where the nonlinear restoring force (the anharmonic term in V(φ)) redirects amplitude. The coupling constant is the strength of the anharmonicity relative to the linear spring.

Vertex amplitude ∝ α^1/2 per EM vertex, α_s^1/2 per QCD vertex

SM: Cross Section Lattice: Scattering Cross Area

The scattering cross section σ measures the effective area over which two waves interact. In Planck units:

σ ∼ α² / E² (EM) σ ∼ α_s² / E² (QCD)

The α² comes from two vertices (one for each scattering event). The 1/E² comes from the de Broglie wavelength squared — higher energy waves are smaller and harder to hit.

SM: Decay Rate Lattice: Mode Lifetime

A particle decay is a high-frequency lattice mode losing energy to lower-frequency modes (like a vibrating guitar string transferring energy to the body). The decay rate Γ in Planck units:

Γ ∼ αⁿ · m (where n = number of vertices in the decay diagram)

The lifetime τ = 1/Γ. Particles that can only decay through many vertices (large n) live longer — each vertex contributes a factor of α ≈ 1/137, suppressing the rate. This is why the proton is stable: its decay would require ∼36 vertices (3 generations × 12 gauge bosons), giving τ ∼ α⁻³⁶ in Planck times ≈ 10⁴¹ years.

§12 — Complete Particle Spectrum in d and π

Every Standard Model particle expressed in lattice-Planck units. The column “d, π only?” indicates whether the formula uses only the two fundamental inputs (d and π) or also requires compound quantities like α.

Particle	Formula (Planck units)	Value	Accuracy
Coupling Constants
α	exp(−(2/d!)(2^2d+1/π² + ln2d))	1/137.042	0.005%
sin²θ_W	C(d+2,2) / 2^2d	15/64 = 0.2344	1.4%
cosθ_W	(2d+1) / 2^d	7/8	0.1%
Leptons
Electron	2dπ^d+2α¹² m_P	0.5112 MeV	0.04%
Muon	[d/(2α)+√(d/2)] m_e	105.66 MeV	0.005%
Tau	Koide[(d−1)/d] from e, μ	1776.97 MeV	0.006%
Quarks
Up	m(13, 31)	2.16 MeV	0.0%
Down	m(5, 30)	4.67 MeV	0.1%
Strange	m(4, 28)	98.6 MeV	5.5%
Charm	m(11, 27)	1271 MeV	0.02%
Bottom	m(7, 26)	4.205 GeV	0.5%
Top	m(12, 24)	172.2 GeV	0.3%
Gauge Bosons
Photon	m = 0 (exact lattice symmetry)	0	exact
8 Gluons	d²−1 = 8, m = 0 (confined)	0 (confined)	exact
W^±	m(5, 24)	80.2 GeV	0.2%
Z⁰	M_W/cosθ_W	91.6 GeV	0.5%
Scalar
Higgs	m(8, 24), n=2^d	124.8 GeV	0.4%
Higgs VEV	m(3, 23) or √2 · m_t	246.1 GeV	0.03%
Composite
Proton	2^d−1 Λ_QCD	938.3 MeV	exact
m_p/m_e	2d · π^d+2	1836.12	0.002%
Mixing
δ_CKM	arccos(5/12)	65.38°	−0.1σ
δ_PMNS	arccos(−1/d)	109.47°	1.4%
V_us	√(m_d/m_s)	0.2243	exact
A (Wolfenstein)	√(2/d)	0.8165	2.3%
N_gen	d	3	exact

§13 — Five Numbers Build Everything

The entire Standard Model — 17 particles, 19+ parameters, 3 generations — reduces to five numbers that the lattice forces:

d = 3 • π • 2 • e • 1

d = 3: spatial dimensions (forces N_c, N_gen, gauge group, tetrahedron angles)

π: circle constant (forces BZ boundary, mode density, Wyler formula)

2: yin-yang polarization count = d−1 (forces SU(2), Koide 2/3, Ω_Λ)

e: natural exponential (forces RG running, tunneling rates, decay widths)

1: the lattice unit itself (a = k = η = 1 defines the Planck scale)

The Standard Model is not fundamental. It is the low-energy effective description of classical wave mechanics on a 3D discrete elastic medium. Every “particle” is a standing wave. Every “force” is a restoring pressure. Every “coupling constant” is a geometric ratio. Every “mass” is stored elastic energy. There is nothing else.

← QM in Planck Units Planck-Unit Reduction →