The Dirac Equation and Its Free Solutions

Dirac's requirements are that the Hamiltonian H = cα⋅p + βmc² be Hermitian and square to p²c² + m²c⁴; they hold exactly when α₁, α₂, α₃ and β are Hermitian, square to 1 and anticommute in pairs, which no ordinary numbers can do. Use the supplied topic model Y_out = 1.022*m*sqrt(max(1-s^2,0)), where Y_out is SOLID GAP AT ITS MINIMUM (MeV). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. s is Overlap s, where αβ + βα = 2s, dimensionless. Keep m (Mass m in electron masses) = 1 mₑ; p0 (Probe momentum p₀) = 1 MeV/c fixed. For this exercise use only 0.10625 ≤ s ≤ 0.89375 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.82767559 MeV, with a bounded reading uncertainty ±0.028281 MeV. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00007875, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Dirac representation is the choice β = [[1, 0], [0, −1]] and αᵢ = [[0, σᵢ], [σᵢ, 0]] in 2 × 2 blocks: four Hermitian 4 × 4 matrices that square to 1 and anticommute in pairs, the smallest matrices that can. Use the supplied topic model Y_out = (bl*sind(th)*cosd(ph))^2+(bl*sind(th)*sind(ph))^2+(bl*cosd(th))^2+(bl^2-1)^2, where Y_out is MISMATCH M (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sind/cosd/tand use degrees. bl is Length |b|, dimensionless. Keep th (Polar angle θ of b) = 55 °; ph (Azimuth φ of b) = 45 ° fixed. For this exercise use only 0.75 ≤ bl ≤ 1.34062 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.2868851, with a bounded reading uncertainty ±0.079947. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown bl by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0000590625, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The gamma matrices are the four fixed 4 × 4 matrices γ⁰ = β and γⁱ = βαᵢ, defined by the Clifford algebra (γ^μ, γ^ν) = 2g^(μν), in terms of which the free Dirac equation reads (iħγ^μ∂_μ − mc)ψ = 0. Use the supplied topic model Y_out = sqrt(max(a0^2-a3^2,0)), where Y_out is REAL PARTS OF λ (±) (mc). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; max(a,b) selects the larger value. a3 is Space component a³, measured in mc. Keep a0 (Time component a⁰) = 1.25 mc fixed. For this exercise use only -1.155 mc ≤ a3 ≤ -0.105 mc on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 1.1392058 mc, with a bounded reading uncertainty ±0.015956 mc. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a3 by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000105 mc, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Dirac density ρ = ψ†ψ and current j = cψ†αψ are the pair the Dirac equation itself conserves, ∂ρ/∂t + ∇⋅j = 0, and because ρ is a sum of four squared moduli it is never negative. Use the supplied topic model Y_out = (1000000*(p*cosd(2*th)-sind(2*th)*cosd(ph))/sqrt(1+p^2))/1000000, where Y_out is CURRENT j_z/cρ (before display rounding) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; sind/cosd/tand use degrees. p is Momentum p along z, measured in mc. Keep th (Branch mixing angle θ) = 0 °; ph (Phase 2Et/ħ) = 0 ° fixed. For this exercise use only 0.2125 mc ≤ p ≤ 1.7875 mc on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.76106234, with a bounded reading uncertainty ±0.016748. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown p by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0001575 mc, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.