Discrete Symmetries of Quantised Fields: C, P, T and CPT

Parity on Fock space is the unitary operator P that reverses each quantum's momentum, keeps its spin and multiplies its ladder operators by an intrinsic phase, the phases chosen so that every field at (t, x) goes into a field at the mirror point (t, −x). Use the supplied topic model Y_out = 180*k-th, where Y_out is ANTIPARTICLE arg η′ (°). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. th is Particle phase arg η, measured in °. Keep k (Field: 0 scalar, 1 Dirac) = 1; ell (Pair orbital L) = 0 fixed. For this exercise use only 19.125 ° ≤ th ≤ 160.875 ° 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 = 74.4075 °, with a bounded reading uncertainty ±8.2924 °. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown th by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.014175 °, 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.

Charge conjugation C is the unitary operator on Fock space that turns every particle into its antiparticle with the same momentum and spin and leaves the vacuum alone, so it carries a complex field into its Hermitian conjugate, times a spinor matrix for a Dirac field, and a real field into itself times a sign η_C = ±1. Use the supplied topic model Y_out = 1+kappa*eps*nd, where Y_out is Q OF THE STATE (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. nd is Antiparticles in right packet, dimensionless. Keep kappa (Field κ: −1 scalar, +1 Dirac) = 1; eps (Statistics ε: −1 anti-, +1 commutators) = -1 fixed. For this exercise use only 0 ≤ nd ≤ 3, at the specified test settings separated by 1. 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, with a bounded reading uncertainty ±0.24. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of nd consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Time reversal on quantum fields is the antiunitary operator T that leaves the vacuum and every energy unchanged, reverses every momentum and spin, and complex-conjugates every c-number it passes, so that T φ(t, x) T⁻¹ = φ(−t, x). Use the supplied topic model Y_out = 2*pi*197.327/p, where Y_out is WAVELENGTH 2πħc/p (fm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. p is Momentum p, measured in MeV. Keep t (Time t) = 3 fm/c; delta (Phase δ of the coherent state) = 60 ° fixed. For this exercise use only 131.875 MeV ≤ p ≤ 368.125 MeV 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 = 4.492385 fm, with a bounded reading uncertainty ±0.16866 fm. 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 0.023625 MeV, 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 C and T columns of the bilinear table give the sign that each Hermitian Dirac bilinear, component by component, acquires under charge conjugation or time reversal of the quantised field; the photon field then takes whatever signs keep e ψ̄γ^μψA_μ invariant. Use the supplied topic model Y_out = min(k, 4-k), where Y_out is LORENTZ INDICES n (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value. k is k = 0 to 4: Γ = 1, γ^μ, σ^μν, γ^μγ⁵, iγ⁵, dimensionless. Keep s (Spatial indices on the component) = 1 fixed. For this exercise use only 0 ≤ k ≤ 2, at the specified test settings separated by 1. 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, with a bounded reading uncertainty ±0.16. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of k consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.