University Quantum Field Theory I · topic UQFTI-01.09
A classical source: Poisson-distributed quanta, vacuum persistence and the Yukawa energy
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.
A classical source: Poisson quanta and the Yukawa energy: predict a finite change
A classical source is a prescribed c-number function j(x) added to the Lagrangian as jφ. It leaves the vacuum in a coherent state, so the field holds Poisson-distributed quanta, and the vacuum persistence amplitude ⟨0|0⟩_j records both the chance of no emission and the energy of the sources. Use the supplied topic model Y_out = (2*alpha/pi*(1-(m*tau*0.7073)^2*ln((1+sqrt(1+(m*tau*0.7073)^2))/max(m*tau*0.7073,1e-9))/sqrt(1+(m*tau*0.7073)^2))), where Y_out is MEAN NUMBER N̄ (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; max(a,b) selects the larger value. tau is Pulse time τ in e^(−|t|/τ), measured in fm/c. Keep alpha (Source strength α = g²/(4π)) = 2; m (Field mass m) = 1 m_π fixed. For this exercise use only 0.620625 fm/c ≤ tau ≤ 4.47937 fm/c 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 tau = 1.46955 fm/c and 2.00977 fm/c. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δtau across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
MEAN NUMBER N̄ against tau
Predictions from the supplied A classical source: Poisson quanta and the Yukawa energy 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): MEAN NUMBER N̄
Horizontal axis (x): tau: Pulse time τ in e^(−|t|/τ) / fm/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| tau: Pulse time τ in e^(−|t|/τ) / fm/c | MEAN NUMBER N̄ |
|---|---|
| 0.620625 | 0.9224445199612921 |
| 1.5853124999999997 | 0.41787722105165664 |
| 2.55 | 0.21005851281481344 |
| 3.5146874999999995 | 0.12172918884459752 |
| 4.479374999999999 | 0.07835158470749722 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of tau separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At tau = 1.46955 fm/c, substitute into Y_out = (2*alpha/pi*(1-(m*tau*0.7073)^2*ln((1+sqrt(1+(m*tau*0.7073)^2))/max(m*tau*0.7073,1e-9))/sqrt(1+(m*tau*0.7073)^2))): Y₀ = 0.458482.
- At tau = 2.00977 fm/c, the same fixed-parameter model gives Y₁ = 0.302867.
- Signed change: ΔY_out = Y₁ − Y₀ = -0.155616; the input change is Δtau = 0.540225 fm/c.
- Average sensitivity is ΔY_out/Δtau = -0.288057 1 per fm/c. 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 = 0.458482, Y₁ = 0.302867; ΔY_out = -0.155616; average sensitivity = -0.288057 1 per fm/c.
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.
A classical source: Poisson quanta and the Yukawa energy: infer an input with uncertainty
A classical source is a prescribed c-number function j(x) added to the Lagrangian as jφ. It leaves the vacuum in a coherent state, so the field holds Poisson-distributed quanta, and the vacuum persistence amplitude ⟨0|0⟩_j records both the chance of no emission and the energy of the sources. Use the supplied topic model Y_out = (2*alpha/pi*(1-(m*tau*0.7073)^2*ln((1+sqrt(1+(m*tau*0.7073)^2))/max(m*tau*0.7073,1e-9))/sqrt(1+(m*tau*0.7073)^2))), where Y_out is MEAN NUMBER N̄ (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; max(a,b) selects the larger value. tau is Pulse time τ in e^(−|t|/τ), measured in fm/c. Keep alpha (Source strength α = g²/(4π)) = 2; m (Field mass m) = 1 m_π fixed. For this exercise use only 0.620625 fm/c ≤ tau ≤ 4.47937 fm/c 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 = 0.16270729, with a bounded reading uncertainty ±0.012653. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown tau by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000385875 fm/c, 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.
MEAN NUMBER N̄ against tau
Predictions from the supplied A classical source: Poisson quanta and the Yukawa energy 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): MEAN NUMBER N̄
Horizontal axis (x): tau: Pulse time τ in e^(−|t|/τ) / fm/c
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| tau: Pulse time τ in e^(−|t|/τ) / fm/c | MEAN NUMBER N̄ |
|---|---|
| 0.620625 | 0.9224445199612921 |
| 1.5853124999999997 | 0.41787722105165664 |
| 2.55 | 0.21005851281481344 |
| 3.5146874999999995 | 0.12172918884459752 |
| 4.479374999999999 | 0.07835158470749722 |
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(tau) = 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(0.620625) = 0.922445 and Y_out(4.47937) = 0.0783516. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives tau_line = 0.620625 + (0.162707 − 0.922445)(4.47937 − 0.620625)/(0.0783516 − 0.922445) = 4.09374 fm/c. 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 tau = 2.97446 fm/c; substituting back gives Y_out = 0.16270729.
- The interpolation error in the input is |tau_line − tau| = 1.11928 fm/c. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.15005429 and 0.17536029. Sort the roots even if the response decreases: 2.84544 fm/c ≤ tau ≤ 3.11853 fm/c.
- The lower and upper input deviations are 0.129024 fm/c and 0.14407 fm/c. 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 tau ≈ 2.97446 fm/c; compatible interval [2.84544, 3.11853] fm/c. Endpoint interpolation gives 4.09374 fm/c.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.