University Physics V · topic UPV-18.01
The gauge principle and non-abelian Lie algebras
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.
The gauge principle and non-abelian algebras: predict a finite change
The gauge principle is the demand that an internal symmetry hold independently at every point of spacetime, which forces the ordinary derivative to be replaced by a covariant derivative carrying one new vector field for every generator of the group. Use the supplied topic model Y_out = 0.04*gc*(k+gc*q*q), where Y_out is PHASE ROUND a = 0.2 GeV⁻¹ (rad). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Amplitude A¹₁ = A²₂, measured in GeV. Keep gc (Coupling g) = 1; k (Curl term ∂₁A³₂) = 0.8 GeV² fixed. For this exercise use only 0.17 GeV ≤ q ≤ 1.43 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. Calculate Y_out at q = 0.4472 GeV and 0.6236 GeV. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δq across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
PHASE ROUND a = 0.2 GeV⁻¹ against q
Predictions from the supplied The gauge principle and non-abelian algebras 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): PHASE ROUND a = 0.2 GeV⁻¹ / rad
Horizontal axis (x): q: Amplitude A¹₁ = A²₂ / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Amplitude A¹₁ = A²₂ / GeV | PHASE ROUND a = 0.2 GeV⁻¹ / rad |
|---|---|
| 0.17 | 0.033156000000000005 |
| 0.485 | 0.041409 |
| 0.8 | 0.057600000000000005 |
| 1.115 | 0.08172900000000001 |
| 1.43 | 0.11379600000000001 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of q separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At q = 0.4472 GeV, substitute into Y_out = 0.04*gc*(k+gc*q*q): Y₀ = 0.0399995 rad.
- At q = 0.6236 GeV, the same fixed-parameter model gives Y₁ = 0.0475551 rad.
- Signed change: ΔY_out = Y₁ − Y₀ = 0.00755556 rad; the input change is Δq = 0.1764 GeV.
- Average sensitivity is ΔY_out/Δq = 0.042832 rad per GeV. Its positive sign means that this response increases 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 = 0.0399995 rad, Y₁ = 0.0475551 rad; ΔY_out = 0.00755556 rad; average sensitivity = 0.042832 rad per GeV.
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.
The gauge principle and non-abelian algebras: infer an input with uncertainty
The gauge principle is the demand that an internal symmetry hold independently at every point of spacetime, which forces the ordinary derivative to be replaced by a covariant derivative carrying one new vector field for every generator of the group. Use the supplied topic model Y_out = 0.04*gc*(k+gc*q*q), where Y_out is PHASE ROUND a = 0.2 GeV⁻¹ (rad). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. q is Amplitude A¹₁ = A²₂, measured in GeV. Keep gc (Coupling g) = 1; k (Curl term ∂₁A³₂) = 0.8 GeV² fixed. For this exercise use only 0.17 GeV ≤ q ≤ 1.43 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 = 0.067238798 rad, with a bounded reading uncertainty ±0.0051124 rad. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown q by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000126 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.
PHASE ROUND a = 0.2 GeV⁻¹ against q
Predictions from the supplied The gauge principle and non-abelian algebras 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): PHASE ROUND a = 0.2 GeV⁻¹ / rad
Horizontal axis (x): q: Amplitude A¹₁ = A²₂ / GeV
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| q: Amplitude A¹₁ = A²₂ / GeV | PHASE ROUND a = 0.2 GeV⁻¹ / rad |
|---|---|
| 0.17 | 0.033156000000000005 |
| 0.485 | 0.041409 |
| 0.8 | 0.057600000000000005 |
| 1.115 | 0.08172900000000001 |
| 1.43 | 0.11379600000000001 |
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(q) = 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(0.17) = 0.033156 rad and Y_out(1.43) = 0.113796 rad. The target lies between them. On the selected branch the response is strictly increasing.
- Endpoint interpolation gives q_line = 0.17 + (0.0672388 − 0.033156)(1.43 − 0.17)/(0.113796 − 0.033156) = 0.702544 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 q = 0.9386 GeV; substituting back gives Y_out = 0.067238798 rad.
- The interpolation error in the input is |q_line − q| = 0.236056 GeV. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 0.062126398 rad and 0.072351198 rad. Sort the roots even if the response decreases: 0.867848 GeV ≤ q ≤ 1.00438 GeV.
- The lower and upper input deviations are 0.0707521 GeV and 0.0657804 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 q ≈ 0.9386 GeV; compatible interval [0.867848, 1.00438] GeV. Endpoint interpolation gives 0.702544 GeV.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.