University Quantum Field Theory I · course UQFTI
University Quantum Field Theory I
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.
Natural units, the metric, and back to seconds and barns: infer an input with uncertainty
Natural units set ħ = c = 1, so energy, momentum and mass share the unit GeV while lengths and times are measured in GeV⁻¹; ħc = 197.327 MeV fm and ħ = 6.582 × 10⁻²⁵ GeV s convert back to SI. Use the supplied topic model Y_out = -24.181634-log(a)-k, where Y_out is log₁₀(time/s) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; log(x) means log₁₀x. a is Mantissa a in E = a × 10ᵏ GeV, dimensionless. Keep k (Power of ten k) = -1 fixed. For this exercise use only 1.95519 ≤ a ≤ 9.03481 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 = -23.979162, with a bounded reading uncertainty ±0.023759. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown a by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000707962, 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.
log₁₀(time/s) against a
Predictions from the supplied Natural units, the metric, and back to seconds and barns 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): log₁₀(time/s)
Horizontal axis (x): a: Mantissa a in E = a × 10ᵏ GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| a: Mantissa a in E = a × 10ᵏ GeV | log₁₀(time/s) |
|---|---|
| 1.9551875 | -23.47282241201869 |
| 3.72509375 | -23.752771207177016 |
| 5.494999999999999 | -23.92160169675951 |
| 7.264906249999999 | -24.042864014316024 |
| 9.0348125 | -24.137553144051314 |
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(a) = 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(1.95519) = -23.4728 and Y_out(9.03481) = -24.1376. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives a_line = 1.95519 + (-23.9792 − -23.4728)(9.03481 − 1.95519)/(-24.1376 − -23.4728) = 7.34789. 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 a = 6.27376; substituting back gives Y_out = -23.979162.
- The interpolation error in the input is |a_line − a| = 1.07413. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = -24.002921 and -23.955403. Sort the roots even if the response decreases: 5.93976 ≤ a ≤ 6.62654.
- The lower and upper input deviations are 0.334 and 0.352781. 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 a ≈ 6.27376; compatible interval [5.93976, 6.62654]. Endpoint interpolation gives 7.34789.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Model interactions and their coupling dimensions: infer an input with uncertainty
A model interaction is a local, Hermitian, Lorentz-invariant polynomial in the fields added to the free Lagrangian; its coupling has mass dimension 4 minus the total dimension of the fields it multiplies, and that dimension sets how the interaction's strength changes with energy. Use the supplied topic model Y_out = 100*gs^2/(96*sqrt(3)*pi*mpsi^2), where Y_out is SCALAR PEAK Γ/m (%). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. mpsi is ψ mass M, measured in GeV. Keep gs (Scalar Yukawa g, a mass) = 2.5 GeV; gd (Dirac Yukawa g, a pure number) = 0.5 fixed. For this exercise use only 1.2125 GeV ≤ mpsi ≤ 2.7875 GeV 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 = 0.2533255 %, with a bounded reading uncertainty ±0.014902 %. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown mpsi 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 GeV, 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.
SCALAR PEAK Γ/m against mpsi
Predictions from the supplied Model interactions and their coupling dimensions 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): SCALAR PEAK Γ/m / %
Horizontal axis (x): mpsi: ψ mass M / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| mpsi: ψ mass M / GeV | SCALAR PEAK Γ/m / % |
|---|---|
| 1.2125 | 0.8138320512006709 |
| 1.6062499999999997 | 0.46373727200999953 |
| 1.9999999999999998 | 0.29911506913074665 |
| 2.39375 | 0.2088049075185491 |
| 2.7874999999999996 | 0.15398149509837547 |
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(mpsi) = 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(1.2125) = 0.813832 % and Y_out(2.7875) = 0.153981 %. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives mpsi_line = 1.2125 + (0.253325 − 0.813832)(2.7875 − 1.2125)/(0.153981 − 0.813832) = 2.55038 GeV. 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 mpsi = 2.17325 GeV; substituting back gives Y_out = 0.2533255 %.
- The interpolation error in the input is |mpsi_line − mpsi| = 0.377125 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.2384235 % and 0.2682275 %. Sort the roots even if the response decreases: 2.11202 GeV ≤ mpsi ≤ 2.24014 GeV.
- The lower and upper input deviations are 0.0612326 GeV and 0.0668872 GeV. 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 mpsi ≈ 2.17325 GeV; compatible interval [2.11202, 2.24014] GeV. Endpoint interpolation gives 2.55038 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Position-space Feynman rules for φ⁴: infer an input with uncertainty
Position-space Feynman rules turn each term of a correlation function's perturbation series into a picture and back: D_F(a − b) for every line joining points a and b, and −iλ with ∫d⁴z over all spacetime for every φ⁴ vertex z, divided by the diagram's symmetry factor. Use the supplied topic model Y_out = (cut*sqrt(cut*cut+1)-ln(cut+sqrt(cut*cut+1)))/(8*pi*pi), where Y_out is D_F(0)/m² WITH CUTOFF (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; ln(x) means the natural logarithm. cut is Cutoff on |p|, Λ/m, dimensionless. Keep lam (Coupling λ) = 0.6; s (Probe separation m r) = 2 fixed. For this exercise use only 1.74375 ≤ cut ≤ 7.25625 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 = 0.30696864, with a bounded reading uncertainty ±0.041899. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown cut by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00055125, 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.
D_F(0)/m² WITH CUTOFF against cut
Predictions from the supplied Position-space Feynman rules for φ⁴ 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): D_F(0)/m² WITH CUTOFF
Horizontal axis (x): cut: Cutoff on |p|, Λ/m
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| cut: Cutoff on |p|, Λ/m | D_F(0)/m² WITH CUTOFF |
|---|---|
| 1.74375 | 0.027640285230667333 |
| 3.1218749999999997 | 0.106103524436662 |
| 4.5 | 0.23474379730772063 |
| 5.878124999999999 | 0.4125956167046533 |
| 7.25625 | 0.6392237891305256 |
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(cut) = 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(1.74375) = 0.0276403 and Y_out(7.25625) = 0.639224. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives cut_line = 1.74375 + (0.306969 − 0.0276403)(7.25625 − 1.74375)/(0.639224 − 0.0276403) = 4.26147. 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 cut = 5.10638; substituting back gives Y_out = 0.30696864.
- The interpolation error in the input is |cut_line − cut| = 0.844903. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.26506964 and 0.34886764. Sort the roots even if the response decreases: 4.76441 ≤ cut ≤ 5.42609.
- The lower and upper input deviations are 0.341965 and 0.319711. 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 cut ≈ 5.10638; compatible interval [4.76441, 5.42609]. Endpoint interpolation gives 4.26147.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
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.