University Advanced Quantum Mechanics · topic UAQM-05.08
gamma^5, chirality and the massless Weyl equations
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
γ⁵, chirality and the massless Weyl equations: predict a finite change
Chirality is the eigenvalue, +1 or −1, of γ⁵ = iγ⁰γ¹γ²γ³: its projectors (1 ± γ⁵)/2 split every Dirac spinor into right- and left-chiral halves that no rotation or boost mixes and that only the mass term couples. Use the supplied topic model Y_out = abs(p)/sqrt(max(p*p+m*m,1e-6)), where Y_out is SPEED β = |p|c ÷ E (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; abs(x) means |x|; max(a,b) selects the larger value. p is Momentum p along z, measured in MeV/c. Keep m (Mass m) = 0.511 MeV/c² fixed. For this exercise use only -2.44125 MeV/c ≤ p ≤ -0.16275 MeV/c 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. Calculate Y_out at p = -1.93998 MeV/c and -1.62099 MeV/c. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δp across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
SPEED β = |p|c ÷ E against p
Predictions from the supplied γ⁵, chirality and the massless Weyl equations model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): SPEED β = |p|c ÷ E
Horizontal axis (x): p: Momentum p along z / MeV/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Momentum p along z / MeV/c | SPEED β = |p|c ÷ E |
|---|---|
| -2.44125 | 0.9787873532331557 |
| -1.871625 | 0.9646909302631074 |
| -1.302 | 0.9308731029931081 |
| -0.732375 | 0.8201055681591956 |
| -0.16274999999999995 | 0.30347302346388694 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of p separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At p = -1.93998 MeV/c, substitute into Y_out = abs(p)/sqrt(max(p*p+m*m,1e-6)): Y₀ = 0.967016.
- At p = -1.62099 MeV/c, the same fixed-parameter model gives Y₁ = 0.953733.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.0132826; the input change is Δp = 0.31899 MeV/c.
- Average sensitivity is ΔY_out/Δp = -0.0416395 1 per MeV/c. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 0.967016, Y₁ = 0.953733; ΔY_out = -0.0132826; average sensitivity = -0.0416395 1 per MeV/c.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
γ⁵, chirality and the massless Weyl equations: infer an input with uncertainty
Chirality is the eigenvalue, +1 or −1, of γ⁵ = iγ⁰γ¹γ²γ³: its projectors (1 ± γ⁵)/2 split every Dirac spinor into right- and left-chiral halves that no rotation or boost mixes and that only the mass term couples. Use the supplied topic model Y_out = abs(p)/sqrt(max(p*p+m*m,1e-6)), where Y_out is SPEED β = |p|c ÷ E (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; abs(x) means |x|; max(a,b) selects the larger value. p is Momentum p along z, measured in MeV/c. Keep m (Mass m) = 0.511 MeV/c² fixed. For this exercise use only -2.44125 MeV/c ≤ p ≤ -0.16275 MeV/c 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 = 0.89939483, with a bounded reading uncertainty ±0.011909. 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 0.00022785 MeV/c, 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.
SPEED β = |p|c ÷ E against p
Predictions from the supplied γ⁵, chirality and the massless Weyl equations model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): SPEED β = |p|c ÷ E
Horizontal axis (x): p: Momentum p along z / MeV/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Momentum p along z / MeV/c | SPEED β = |p|c ÷ E |
|---|---|
| -2.44125 | 0.9787873532331557 |
| -1.871625 | 0.9646909302631074 |
| -1.302 | 0.9308731029931081 |
| -0.732375 | 0.8201055681591956 |
| -0.16274999999999995 | 0.30347302346388694 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(p) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(-2.44125) = 0.978787 and Y_out(-0.16275) = 0.303473. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives p_line = -2.44125 + (0.899395 − 0.978787)(-0.16275 − -2.44125)/(0.303473 − 0.978787) = -2.17338 MeV/c. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives p = -1.05136 MeV/c; substituting back gives Y_out = 0.89939483.
- The interpolation error in the input is |p_line − p| = 1.12202 MeV/c. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.88748583 and 0.91130383. Sort the roots even if the response decreases: -1.13101 MeV/c ≤ p ≤ -0.984095 MeV/c.
- The lower and upper input deviations are 0.0796459 MeV/c and 0.0672703 MeV/c. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer p ≈ -1.05136 MeV/c; compatible interval [-1.13101, -0.984095] MeV/c. Endpoint interpolation gives -2.17338 MeV/c.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.