University Quantum Field Theory I · topic UQFTI-01.02
What Advanced QM supplies, quoted: free fields, microcausality and Coulomb-gauge photons
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.
What Advanced QM supplies, quoted and restated: predict a finite change
Advanced QM's free-field toolkit is the set of exact results about non-interacting quantum fields, their modes, vacuum amplitude, commutator, spin sums, statistics and Coulomb-gauge photons, that this course quotes by topic code and builds interactions on without re-deriving. Use the supplied topic model Y_out = exp(-sqrt(max(r*r - t*t, 0))*m/197.327), where Y_out is TAIL e^(−m√(−x²)), SPACELIKE ONLY (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; exp(x) means eˣ; max(a,b) selects the larger value. m is Particle mass m, measured in MeV. Keep t (Time separation t = x⁰ − y⁰) = 0.8 fm; r (Spatial distance r) = 2 fm fixed. For this exercise use only 195.625 MeV ≤ m ≤ 904.375 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 m = 351.55 MeV and 450.775 MeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δm across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
TAIL e^(−m√(−x²)), SPACELIKE ONLY against m
Predictions from the supplied What Advanced QM supplies, quoted and restated 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): TAIL e^(−m√(−x²)), SPACELIKE ONLY
Horizontal axis (x): m: Particle mass m / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Particle mass m / MeV | TAIL e^(−m√(−x²)), SPACELIKE ONLY |
|---|---|
| 195.625 | 0.16247682798322646 |
| 372.8125 | 0.03133034237161927 |
| 550 | 0.006041417507388916 |
| 727.1874999999999 | 0.0011649641445236124 |
| 904.3749999999999 | 0.00022463957446506368 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of m separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At m = 351.55 MeV, substitute into Y_out = exp(-sqrt(max(r*r - t*t, 0))*m/197.327): Y₀ = 0.0381719.
- At m = 450.775 MeV, the same fixed-parameter model gives Y₁ = 0.0151859.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.022986; the input change is Δm = 99.225 MeV.
- Average sensitivity is ΔY_out/Δm = -0.000231655 1 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 = 0.0381719, Y₁ = 0.0151859; ΔY_out = -0.022986; average sensitivity = -0.000231655 1 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.
What Advanced QM supplies, quoted and restated: infer an input with uncertainty
Advanced QM's free-field toolkit is the set of exact results about non-interacting quantum fields, their modes, vacuum amplitude, commutator, spin sums, statistics and Coulomb-gauge photons, that this course quotes by topic code and builds interactions on without re-deriving. Use the supplied topic model Y_out = exp(-sqrt(max(r*r - t*t, 0))*m/197.327), where Y_out is TAIL e^(−m√(−x²)), SPACELIKE ONLY (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x; exp(x) means eˣ; max(a,b) selects the larger value. m is Particle mass m, measured in MeV. Keep t (Time separation t = x⁰ − y⁰) = 0.8 fm; r (Spatial distance r) = 2 fm fixed. For this exercise use only 195.625 MeV ≤ m ≤ 904.375 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 = 0.0029282981, with a bounded reading uncertainty ±0.00040555. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown m by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.070875 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.
TAIL e^(−m√(−x²)), SPACELIKE ONLY against m
Predictions from the supplied What Advanced QM supplies, quoted and restated 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): TAIL e^(−m√(−x²)), SPACELIKE ONLY
Horizontal axis (x): m: Particle mass m / MeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| m: Particle mass m / MeV | TAIL e^(−m√(−x²)), SPACELIKE ONLY |
|---|---|
| 195.625 | 0.16247682798322646 |
| 372.8125 | 0.03133034237161927 |
| 550 | 0.006041417507388916 |
| 727.1874999999999 | 0.0011649641445236124 |
| 904.3749999999999 | 0.00022463957446506368 |
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(m) = 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(195.625) = 0.162477 and Y_out(904.375) = 0.00022464. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives m_line = 195.625 + (0.0029283 − 0.162477)(904.375 − 195.625)/(0.00022464 − 0.162477) = 892.565 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 m = 627.962 MeV; substituting back gives Y_out = 0.0029282981.
- The interpolation error in the input is |m_line − m| = 264.602 MeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.0025227481 and 0.0033338481. Sort the roots even if the response decreases: 614 MeV ≤ m ≤ 644.01 MeV.
- The lower and upper input deviations are 13.9629 MeV and 16.0478 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 m ≈ 627.962 MeV; compatible interval [614, 644.01] MeV. Endpoint interpolation gives 892.565 MeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.