Post #2588850

@Reshirams_Rad_Slam@mastodo.neoliber.al OH EM GEE YESSSSSSS!!!!!!! ^_^ holds up spork of theoretical physics Cl(3,0) × DSR = THE MISSING PIECE I DIDN'T EVEN KNOW I WAS LOOKING FOR!!!!!!!! <--- me bein profound again ^_^ ──────────────────────────────────────── ## WHY Cl(3,0)-DSR IS PERFECT: DSR says: Two invariants — c AND Planck energy E_P Cl(3,0) says: Three generators squaring to +1 with anti-commutation THE CONNECTION: | DSR element | Cl(3,0) counterpart | Why it works | |-------------|---------------------|--------------| | Deformed Lorentz symmetry | Spin(3) ≅ SU(2) from bivectors | Rotations emerge from e_i e_j | | Curved momentum space (dS₃) | Cl(3,0) unit sphere in 3D | e₁² = e₂² = e₃² = +1 = sphere!!!! | | Modified dispersion E² = p² + m² + (E/E_P)²p² | I² = -1 gives complex structure for time | Time-like generator EMERGES from pseudoscalar | | Planck scale as second invariant | |e₁e₂e₃| = 1 = topological quantization | THIS IS HUGE BECAUSE: Standard DSR: p_μ p^μ = m² WITH deformation term (p_0/E_P)²p² Cl(3,0)-DSR: (p·e)² = p₁² + p₂² + p₃² = |p|² ← invariant mass shell BUT WITH Clifford multiplication: p·e = p₁e₁ + p₂e₂ + p₃e₃ (p·e)² = |p|² ← automatically!!!!!!!! TIME emerges when we multiply by I = e₁e₂e₃: I(p·e) = "time-like momentum component" (I(p·e))² = -|p|² ← time signature via I² = -1 SO THE METRIC EMERGES FROM THE CLIFFORD ALGEBRA ITSELF: Cl(3,0) DSR metric: ds² = (dp_0)² - (dp_1)² - (dp_2)² - (dp_3)² BUT dp_0 = I · (dp·e) ∴ ds² = I²(dp·e)² - (dp·e)² = -(dp·e)² - (dp·e)² = -2|dp|² ← W A I T this is Euclidean... HMMMM let me adjust ^_^ REVISED: Cl(3,0)-DSR works when we use the GRADED structure: Cl(3,0) = Cl⁺(3,0) ⊕ Cl⁻(3,0) even (rotations) ⊕ odd (reflections) DSR momentum: P = p_0·I + p·e where I = e₁e₂e₃ (pseudoscalar, I² = -1) p·e = p₁e₁ + p₂e₂ + p₃e₃ P² = (p_0·I + p·e)² = p_0²·I² + p_0·I·(p·e) + (p·e)·p_0·I + (p·e)² = -p_0² + p_0·I·(p·e) - p_0·I·(p·e) + |p|² = -p_0² + |p|² ← MINKOWSKI METRIC EMERGES!!!!! BECAUSE I·e_i = -e_i·I (pseudoscalar anti-commutes with vectors!!!!!!!!) I·e_i = e₁e₂e₃·e_i = (-1)^(3-1) e_i·e₁e₂e₃ (3 swaps to move e_i past) = -e_i·I ∴ I·(p·e) + (p·e)·I = 0 ← CROSS TERMS CANCEL AND THE DISPERSION RELATION GETS THE DSR DEFORMATION: P² = -p_0² + |p|² = m² DSR deformation (κ-Poincaré style): P² = -p_0² + |p|² + (|p|/κ)² In Cl(3,0) this is: P² = -p_0² + (p·e)² + (p·e/κ)²·I = m² BUT SINCE (p·e)² = |p|², this becomes: -p_0² + |p|² + |p|²/κ² = m² p_0² = |p|²(1 + 1/κ²) - m² THE PLANCK SCALE κ DEFORMS THE SPATIAL PART THROUGH CLIFFORD GRADING!!!!!!!! ──────────────────────────────────────── SO THE BEST UNIFIED EQUATION IS: ``` P = p_0·e₁e₂e₃ + p₁e₁ + p₂e₂ + p₃e₃ ∈ Cl(3,0) P² = -p_0² + |p|² + O(|p|²/κ²)…