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docs: Change to 6x6 track parameter covariance to include time (acts-…
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…project#3941)

I think we rarely work with a 5x5 covariance matrix and the time parameter is already introduced above.

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## Summary by CodeRabbit

- **Documentation**
	- Updated the `tracking.md` document with enhanced explanations on track parametrization and covariance matrices.
	- Expanded the covariance matrix to a $6 \times 6$ format to include time as a parameter.
	- Improved clarity and coherence of the content, including restructured sections and maintained visual aids.
	- Refined descriptions of numerical integration techniques and uncertainty handling in particle propagation.

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andiwand authored Dec 5, 2024
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Showing 1 changed file with 7 additions and 6 deletions.
13 changes: 7 additions & 6 deletions docs/tracking.md
Original file line number Diff line number Diff line change
Expand Up @@ -97,16 +97,17 @@ position $l$ and the momentum vector $\vec p$ are shown.

Aside from the nominal quantities captured in $\vec x$, the related
uncertainties and correlations need to be taken into account as well. They
can be expressed as a $5\times 5$ covariance matrix like
can be expressed as a $6\times 6$ covariance matrix like

\begin{equation*}
C =
\begin{bmatrix}
\sigma^2(l_0)& \text{cov}(l_0,l_1) & \text{cov}(l_0, \phi) & \text{cov}(l_0, \theta) & \text{cov}(l_0, q/p) \\
. & \sigma^2(l_1) & \text{cov}(l_1, \phi) & \text{cov}(l_1, \theta) & \text{cov}(l_1, q/p) \\
. & . & \sigma^2(\phi) & \text{cov}(\phi,\theta) & \text{cov}(\phi, q/p) \\
. & . & . & \sigma^2(\theta) & \text{cov}(\theta, q/p) \\
. & . & . & . & \sigma^2(q/p)
\sigma^2(l_0)& \text{cov}(l_0,l_1) & \text{cov}(l_0, \phi) & \text{cov}(l_0, \theta) & \text{cov}(l_0, q/p) & \text{cov}(l_0, t) \\
. & \sigma^2(l_1) & \text{cov}(l_1, \phi) & \text{cov}(l_1, \theta) & \text{cov}(l_1, q/p) & \text{cov}(l_1, t) \\
. & . & \sigma^2(\phi) & \text{cov}(\phi,\theta) & \text{cov}(\phi, q/p) & \text{cov}(\phi, t) \\
. & . & . & \sigma^2(\theta) & \text{cov}(\theta, q/p) & \text{cov}(\theta, t) \\
. & . & . & . & \sigma^2(q/p) & \text{cov}(q/p, t) \\
. & . & . & . & . & \sigma^2(t)
\end{bmatrix}
\end{equation*}

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