University Quantum Field Theory I · topic UQFTI-07.01
The Dirac propagator: S_F(p) = i(p-slash + m)/(p^2 - m^2 + i epsilon)
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.
The Dirac propagator: S_F(p) = i(p̸ + m)/(p² − m² + iε): predict a finite change
The Dirac propagator S_F(x − y) = ⟨0|Tψ(x)ψ̄(y)|0⟩ is the vacuum amplitude for fermion number to enter at y and leave at x, with the fields time-ordered under a minus sign; in momentum space it is i(p̸ + m)/(p² − m² + iε). Use the supplied topic model Y_out = 4.135668/sqrt(p*p+0.261121), where Y_out is PERIOD 2πħ/Eₚ (zs). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Electron momentum |𝐩|, measured in MeV. Keep sg (Ordering sign σ (−1: fermions)) = -1 fixed. For this exercise use only 0.2125 MeV ≤ p ≤ 1.7875 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. Calculate Y_out at p = 0.559 MeV and 0.7795 MeV. 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.
PERIOD 2πħ/Eₚ against p
Predictions from the supplied The Dirac propagator: S_F(p) = i(p̸ + m)/(p² − m² + iε) 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): PERIOD 2πħ/Eₚ / zs
Horizontal axis (x): p: Electron momentum |𝐩| / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Electron momentum |𝐩| / MeV | PERIOD 2πħ/Eₚ / zs |
|---|---|
| 0.2125 | 7.472884983852224 |
| 0.60625 | 5.216001790869236 |
| 1 | 3.6827080297538797 |
| 1.3937499999999998 | 2.785950316155276 |
| 1.7874999999999999 | 2.22454595122529 |
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 = 0.559 MeV, substitute into Y_out = 4.135668/sqrt(p*p+0.261121): Y₀ = 5.4606 zs.
- At p = 0.7795 MeV, the same fixed-parameter model gives Y₁ = 4.43711 zs.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.02349 zs; the input change is Δp = 0.2205 MeV.
- Average sensitivity is ΔY_out/Δp = -4.64166 zs per MeV. 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 = 5.4606 zs, Y₁ = 4.43711 zs; ΔY_out = -1.02349 zs; average sensitivity = -4.64166 zs per MeV.
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.
The Dirac propagator: S_F(p) = i(p̸ + m)/(p² − m² + iε): infer an input with uncertainty
The Dirac propagator S_F(x − y) = ⟨0|Tψ(x)ψ̄(y)|0⟩ is the vacuum amplitude for fermion number to enter at y and leave at x, with the fields time-ordered under a minus sign; in momentum space it is i(p̸ + m)/(p² − m² + iε). Use the supplied topic model Y_out = 4.135668/sqrt(p*p+0.261121), where Y_out is PERIOD 2πħ/Eₚ (zs). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. p is Electron momentum |𝐩|, measured in MeV. Keep sg (Ordering sign σ (−1: fermions)) = -1 fixed. For this exercise use only 0.2125 MeV ≤ p ≤ 1.7875 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 = 3.2317437 zs, with a bounded reading uncertainty ±0.15108 zs. 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 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.
PERIOD 2πħ/Eₚ against p
Predictions from the supplied The Dirac propagator: S_F(p) = i(p̸ + m)/(p² − m² + iε) 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): PERIOD 2πħ/Eₚ / zs
Horizontal axis (x): p: Electron momentum |𝐩| / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| p: Electron momentum |𝐩| / MeV | PERIOD 2πħ/Eₚ / zs |
|---|---|
| 0.2125 | 7.472884983852224 |
| 0.60625 | 5.216001790869236 |
| 1 | 3.6827080297538797 |
| 1.3937499999999998 | 2.785950316155276 |
| 1.7874999999999999 | 2.22454595122529 |
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(0.2125) = 7.47288 zs and Y_out(1.7875) = 2.22455 zs. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives p_line = 0.2125 + (3.23174 − 7.47288)(1.7875 − 0.2125)/(2.22455 − 7.47288) = 1.48525 MeV. 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 = 1.17325 MeV; substituting back gives Y_out = 3.2317437 zs.
- The interpolation error in the input is |p_line − p| = 0.311995 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 3.0806637 zs and 3.3828237 zs. Sort the roots even if the response decreases: 1.11063 MeV ≤ p ≤ 1.2414 MeV.
- The lower and upper input deviations are 0.0626172 MeV and 0.0681517 MeV. 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 ≈ 1.17325 MeV; compatible interval [1.11063, 1.2414] MeV. Endpoint interpolation gives 1.48525 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.