University Quantum Field Theory I · unit UQFTI-07
Fermions in Perturbation Theory: the Dirac Propagator, Fermion Signs and Yukawa Theory
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.
Extra challenging questions
A selection from this pathway. Open a topic below for its complete practice and challenge set.
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.
Wick's theorem for fermions: contractions with signs: infer an input with uncertainty
Wick's theorem for fermions writes a time-ordered product of free anticommuting fields as its normal-ordered product plus every way of contracting ψ with ψ̄, each term weighted by the sign of the permutation that brings its contracted pairs together. Use the supplied topic model Y_out = max(0,1-w)*(min(k-1,1)+max(0,1-abs(k-3))), where Y_out is ARCS WITH ψ̄ FIRST (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; abs(x) means |x|; min(a,b) selects the smaller value; max(a,b) selects the larger value. k is Pairing σ, numbered 1 to 6, dimensionless. Keep w (Written order of the six fields) = 0 fixed. For this exercise use only 1 ≤ k ≤ 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 ±0.16. 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.
ARCS WITH ψ̄ FIRST against k
Predictions from the supplied Wick's theorem for fermions: contractions with signs model, with all other inputs fixed as stated in the question. 3 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): ARCS WITH ψ̄ FIRST
Horizontal axis (x): k: Pairing σ, numbered 1 to 6
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| k: Pairing σ, numbered 1 to 6 | ARCS WITH ψ̄ FIRST |
|---|---|
| 1 | 0 |
| 2 | 1 |
| 3 | 2 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Enumerate the permitted settings in the stated range, evaluate Y_out at each, and retain every setting satisfying |Y_out − Y_measured| ≤ u. The plotted predictions are supplied calibration evidence.
Reveal the worked solution
- Endpoint predictions are Y_out(1) = 0 and Y_out(3) = 2. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives k_line = 1 + (1 − 0)(3 − 1)/(2 − 0) = 2. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Evaluate the model at each permitted setting. The central reading corresponds to k = 2; only the allowed settings 2 lie inside the stated reading bound.
- The interpolation error in the input is |k_line − k| = 0.
- The compatible allowed values span 2 to 2. Values between specified test settings are outside this exercise's input domain even if an interpolation passes through them.
- The quoted output bound restricts which specified settings remain compatible. It does not change physical restrictions on quantum numbers or counts, and the chosen grid alone does not establish physical quantisation.
Answer Compatible k: 2. Endpoint interpolation predicts 2; check admissibility against the discrete settings.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
LSZ for Dirac fields: external spinors u, v, ū and v̄: infer an input with uncertainty
LSZ for Dirac fields turns each external fermion line of an amputated correlation function into √Z₂ times a plane-wave spinor: u in and ū out for a fermion, v̄ in and v out for an antifermion, Z₂ being the residue of the full propagator at the physical mass. Use the supplied topic model Y_out = z*0.1 + (1-z)*(0.81-1)*(k-0.9)*(1/0.81 - ((1+r)^2/0.6561)*ln((1+r)^2/((1+r)^2-0.81))), where Y_out is γ⁰ = −1 VALUE AT p⁰ = 0.9m (× m). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; ln(x) means the natural logarithm. r is Scalar mass m_φ, measured in × m. Keep z (Residue Z₂) = 0.8; k (Continuum ρ₂/(mρ₁), κ) = 0 fixed. For this exercise use only 0.285 × m ≤ r ≤ 0.915 × m 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 = 0.072335176 × m, with a bounded reading uncertainty ±0.00032653 × m. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown r by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000063 × m, 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.
γ⁰ = −1 VALUE AT p⁰ = 0.9m against r
Predictions from the supplied LSZ for Dirac fields: external spinors u, v, ū and v̄ 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): γ⁰ = −1 VALUE AT p⁰ = 0.9m / × m
Horizontal axis (x): r: Scalar mass m_φ / × m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Scalar mass m_φ / × m | γ⁰ = −1 VALUE AT p⁰ = 0.9m / × m |
|---|---|
| 0.28500000000000003 | 0.0641740415445462 |
| 0.4425 | 0.06873794306133174 |
| 0.6000000000000001 | 0.07146161556146449 |
| 0.7575000000000001 | 0.07325446404613453 |
| 0.915 | 0.07451204700576684 |
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(r) = 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(0.285) = 0.064174 × m and Y_out(0.915) = 0.074512 × m. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives r_line = 0.285 + (0.0723352 − 0.064174)(0.915 − 0.285)/(0.074512 − 0.064174) = 0.782341 × m. 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 r = 0.6693 × m; substituting back gives Y_out = 0.072335176 × m.
- The interpolation error in the input is |r_line − r| = 0.113041 × m. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.072008646 × m and 0.072661706 × m. Sort the roots even if the response decreases: 0.641961 × m ≤ r ≤ 0.698587 × m.
- The lower and upper input deviations are 0.0273387 × m and 0.0292871 × m. 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 r ≈ 0.6693 × m; compatible interval [0.641961, 0.698587] × m. Endpoint interpolation gives 0.782341 × m.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Yukawa rules and the closed fermion loop: infer an input with uncertainty
Yukawa theory's Feynman rules turn each fermion line into a chain of Dirac matrices, vertices −ig and propagators i(p̸ + m)/(p² − m² + iε) along the arrow, capped by external spinors on an open line and closed by a trace times −1 on a loop. Use the supplied topic model Y_out = -8*(w*w-1+c)/(4*w*w-r*r), where Y_out is SUM M/g² AT ω (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. w is Marker energy ω/m_φ, dimensionless. Keep r (Mass ratio m_φ/m) = 0.5; c (Vertex: 0 is −ig, 1 is gγ⁵) = 0 fixed. For this exercise use only 1.425 ≤ w ≤ 4.575 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 = -1.8316357, with a bounded reading uncertainty ±0.011777. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown w by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000315, 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.
SUM M/g² AT ω against w
Predictions from the supplied Yukawa rules and the closed fermion loop 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): SUM M/g² AT ω
Horizontal axis (x): w: Marker energy ω/m_φ
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| w: Marker energy ω/m_φ | SUM M/g² AT ω |
|---|---|
| 1.425 | -1.0473166084471262 |
| 2.2125 | -1.6120146141162015 |
| 3 | -1.7902097902097902 |
| 3.7874999999999996 | -1.8687218982813507 |
| 4.574999999999999 | -1.9101500494174728 |
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(w) = 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(1.425) = -1.04732 and Y_out(4.575) = -1.91015. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives w_line = 1.425 + (-1.83164 − -1.04732)(4.575 − 1.425)/(-1.91015 − -1.04732) = 4.28836. 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 w = 3.3465; substituting back gives Y_out = -1.8316357.
- The interpolation error in the input is |w_line − w| = 0.941862. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -1.8434127 and -1.8198587. Sort the roots even if the response decreases: 3.23589 ≤ w ≤ 3.46939.
- The lower and upper input deviations are 0.110609 and 0.122887. 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 w ≈ 3.3465; compatible interval [3.23589, 3.46939]. Endpoint interpolation gives 4.28836.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.