University Quantum Field Theory I · unit UQFTI-04
The S-Matrix from Correlation Functions: Kallen-Lehmann and LSZ
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.
In and out states, S = 1 + iT and the invariant amplitude M: infer an input with uncertainty
The invariant amplitude M is what an S-matrix element leaves once the no-scattering term and the momentum-conserving factor are removed: ⟨β|S − 1|α⟩ = (2π)⁴ δ⁴(p_α − p_β) iM(α → β). Use the supplied topic model Y_out = (exp(yc+eta)+exp(-yc-eta))/2, where Y_out is E₂ / m IN THIS FRAME (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; exp(x) means eˣ. eta is Observer's boost rapidity η, dimensionless. Keep yc (CM rapidities ±Y of the pair) = 1.4 fixed. For this exercise use only -1.26 ≤ eta ≤ 2.3625 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 = 5.289041, with a bounded reading uncertainty ±0.64188. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown eta by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00036225, 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.
E₂ / m IN THIS FRAME against eta
Predictions from the supplied In and out states, S = 1 + iT and the invariant amplitude M 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): E₂ / m IN THIS FRAME
Horizontal axis (x): eta: Observer's boost rapidity η
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| eta: Observer's boost rapidity η | E₂ / m IN THIS FRAME |
|---|---|
| -1.2600000000000002 | 1.0098160171280166 |
| -0.3543750000000002 | 1.5983241419312486 |
| 0.5512499999999998 | 3.589787636809125 |
| 1.4568749999999997 | 8.73224642129301 |
| 2.3625 | 21.53957848659048 |
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(eta) = 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(-1.26) = 1.00982 and Y_out(2.3625) = 21.5396. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives eta_line = -1.26 + (5.28904 − 1.00982)(2.3625 − -1.26)/(21.5396 − 1.00982) = -0.504926. 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 eta = 0.949725; substituting back gives Y_out = 5.289041.
- The interpolation error in the input is |eta_line − eta| = 1.45465. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 4.647161 and 5.930921. Sort the roots even if the response decreases: 0.817621 ≤ eta ≤ 1.06614.
- The lower and upper input deviations are 0.132104 and 0.116418. 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 eta ≈ 0.949725; compatible interval [0.817621, 1.06614]. Endpoint interpolation gives -0.504926.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
The Källén-Lehmann representation: infer an input with uncertainty
The Källén-Lehmann representation writes the exact two-point function of a scalar field as an integral over mass μ of free Feynman propagators, weighted by a spectral density ρ(μ²) ≥ 0 that measures how strongly φ(0) creates states of invariant mass μ from the vacuum. Use the supplied topic model Y_out = gm^2*r^2/(4*pi^2)*(atan(1/sqrt(4*r^2-1))/sqrt(4*r^2-1) - 1/(4*r^2)), where Y_out is ψψ̄ WEIGHT ∫ρ ds/2π (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. r is Mass ratio M/m, dimensionless. Keep gm (Coupling g/m) = 2 fixed. For this exercise use only 0.650938 ≤ r ≤ 1.39906 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 = 0.0041248765, with a bounded reading uncertainty ±0.00025807. 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 0.0000748125, 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.
ψψ̄ WEIGHT ∫ρ ds/2π against r
Predictions from the supplied The Källén-Lehmann representation 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): ψψ̄ WEIGHT ∫ρ ds/2π
Horizontal axis (x): r: Mass ratio M/m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Mass ratio M/m | ψψ̄ WEIGHT ∫ρ ds/2π |
|---|---|
| 0.6509375000000001 | 0.019780563289828426 |
| 0.83796875 | 0.008492480110232003 |
| 1.025 | 0.004981394343232566 |
| 1.21203125 | 0.0033323820519804742 |
| 1.3990624999999999 | 0.00240442944847799 |
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(0.650938) = 0.0197806 and Y_out(1.39906) = 0.00240443. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives r_line = 0.650938 + (0.00412488 − 0.0197806)(1.39906 − 0.650938)/(0.00240443 − 0.0197806) = 1.32499. 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 = 1.10729; substituting back gives Y_out = 0.0041248765.
- The interpolation error in the input is |r_line − r| = 0.217695. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.0038668065 and 0.0043829465. Sort the roots even if the response decreases: 1.07981 ≤ r ≤ 1.13765.
- The lower and upper input deviations are 0.0274834 and 0.0303579. 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 ≈ 1.10729; compatible interval [1.07981, 1.13765]. Endpoint interpolation gives 1.32499.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Physical mass and field-strength renormalisation Z: infer an input with uncertainty
The physical mass m is the position of the exact propagator's one-particle pole, and the field-strength renormalisation Z is its residue, Z = |⟨Ω|φ(0)|p⟩|², the share of the field's total spectral weight that goes into making a single particle. Use the supplied topic model Y_out = gm^2/(16*pi^2)*(4*asin(r/2)/(r*sqrt(4-r^2))-1)/r^2, where Y_out is −dΣ/dp² AT p² = m² (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; unmarked trigonometric functions use radians. r is Mass ratio m/M, dimensionless. Keep gm (Coupling g/M) = 6 fixed. For this exercise use only 0.564688 ≤ r ≤ 1.78531 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 = 0.058809071, with a bounded reading uncertainty ±0.0027318. 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 0.000122062, 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.
−dΣ/dp² AT p² = m² against r
Predictions from the supplied Physical mass and field-strength renormalisation Z 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): −dΣ/dp² AT p² = m²
Horizontal axis (x): r: Mass ratio m/M
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| r: Mass ratio m/M | −dΣ/dp² AT p² = m² |
|---|---|
| 0.5646875 | 0.04059680288487651 |
| 0.86984375 | 0.0448667659309379 |
| 1.1749999999999998 | 0.05298261520393529 |
| 1.4801562499999998 | 0.07013520088792928 |
| 1.7853124999999999 | 0.12458649026796594 |
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(0.564688) = 0.0405968 and Y_out(1.78531) = 0.124586. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives r_line = 0.564688 + (0.0588091 − 0.0405968)(1.78531 − 0.564688)/(0.124586 − 0.0405968) = 0.829367. 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 = 1.30927; substituting back gives Y_out = 0.058809071.
- The interpolation error in the input is |r_line − r| = 0.479902. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.056077271 and 0.061540871. Sort the roots even if the response decreases: 1.25189 ≤ r ≤ 1.35901.
- The lower and upper input deviations are 0.0573796 and 0.0497367. 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 ≈ 1.30927; compatible interval [1.25189, 1.35901]. Endpoint interpolation gives 0.829367.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
The LSZ reduction formula derived: infer an input with uncertainty
The LSZ reduction formula writes every S-matrix element as the vacuum correlation function ⟨Ω|Tφ⋯φ|Ω⟩ with each external leg acted on by (□ + m²) at the physical mass, multiplied by i/√Z and Fourier-transformed with that particle's momentum. Use the supplied topic model Y_out = (1-z)*(1+t^2)^(-0.75), where Y_out is ITS ENVELOPE (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Time mt after the probe state, dimensionless. Keep z (Residue Z of the one-particle pole) = 0.7; gap (Gap above the pole, (μₜₕ − m)/m) = 1 fixed. For this exercise use only 1.59375 ≤ t ≤ 13.4062 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 = 0.011383184, with a bounded reading uncertainty ±0.00079453. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00118125, 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.
ITS ENVELOPE against t
Predictions from the supplied The LSZ reduction formula derived 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): ITS ENVELOPE
Horizontal axis (x): t: Time mt after the probe state
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| t: Time mt after the probe state | ITS ENVELOPE |
|---|---|
| 1.59375 | 0.11624257289941799 |
| 4.546875 | 0.029865230221047704 |
| 7.499999999999999 | 0.014414169889135717 |
| 10.453124999999998 | 0.00881626542198758 |
| 13.406249999999998 | 0.006086295069239051 |
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(t) = 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(1.59375) = 0.116243 and Y_out(13.4062) = 0.0060863. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives t_line = 1.59375 + (0.0113832 − 0.116243)(13.4062 − 1.59375)/(0.0060863 − 0.116243) = 12.8382. 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 t = 8.79937; substituting back gives Y_out = 0.011383184.
- The interpolation error in the input is |t_line − t| = 4.03887. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.010588654 and 0.012177714. Sort the roots even if the response decreases: 8.40723 ≤ t ≤ 9.23971.
- The lower and upper input deviations are 0.392144 and 0.440333. 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 t ≈ 8.79937; compatible interval [8.40723, 9.23971]. Endpoint interpolation gives 12.8382.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.