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Problem
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step description prompt
Compute the expansion coefficients of the evolution operator of the three neutrino oscillation problem , if we expand it using the Gell-Mann matrices in the following way: . Consider all coefficients from to . is the baseline (i.e., the distance the neutrino has travelled in unit of eV).
Plain-text mathematical notation (without MathML)
Compute the expansion coefficients u_(k) of the evolution operator of the three neutrino oscillation problem U₃(L)=e^(−ih_(k)λ^(k)L), if we expand it using the Gell-Mann matrices in the following way: U₃(L)=u₀1+iu_(k)λ^(k). Consider all coefficients from k=0 to k=8. L is the baseline (i.e., the distance the neutrino has travelled in unit of eV).
Original LaTeX notation
Compute the expansion coefficients $u_k$ of the evolution operator of the three neutrino oscillation problem $\mathbb{U}_{3}(L)=e^{-i h_{k} \lambda^{k} L}$, if we expand it using the Gell-Mann matrices in the following way: $\mathbb{U}_{3}(L)=u_{0} \mathbb{1}+i u_{k} \lambda^{k}$. Consider all coefficients from $k=0$ to $k=8$. $L$ is the baseline (i.e., the distance the neutrino has travelled in unit of eV).step background
Background
The evolution operator is . If we discard the global phase, we are left with .
We wish to expand using an identity for the Gell-Mann matrices:
$$
\begin{equation*}
\mathbb{U}_{3}(L)=u_{0} \mathbb{1}+i u_{k} \lambda^{k}
\end{equation*}
$$
where the complex coefficients and are functions of and the .
The coefficients in the above equation can be written as and
An application of Sylvester's formula to matrices allows us to express the coefficients in terms of the invariants
$$
\begin{aligned}
& L^{2}|h|^{2} \equiv L^{2} h_{k} h^{k} \\
& -L^{3}\langle h\rangle \equiv-L^{3} d_{i j k} h^{i} h^{j} h^{k}
\end{aligned}
$$
Next, we solve the characteristic equation of , i.e.,
.
The equation follows from the Cayley-Hamilton theorem, written conveniently in terms of invariants and . Its three latent roots, or eigenvalues, are , with
$$
\begin{equation*}
\psi_{m} \equiv \frac{2|h|}{\sqrt{3}} \cos \left[\frac{1}{3}(\chi+2 \pi m)\right]
\end{equation*}
$$
The expansion coefficient can be expressed as:
$$
\begin{align*}
& u_{0}=\frac{1}{3} \sum_{m=1}^{3} e^{i L \psi_{m}}\\
& u_{k}=\sum_{m=1}^{3} e^{i L \psi_{m}} \frac{\psi_{m} h_{k}-(h * h)_{k}}{3 \psi_{m}^{2}-|h|^{2}}
\end{align*}
$$
where $(h * h)_{i} \equiv d_{i j k} h^{j} h^{k}$.Plain-text mathematical notation (without MathML)
Background
The evolution operator is U₃(L)=e^(−iH₃L)= e^(−ih₀1L)e^(−ih_(k)λ^(k)L). If we discard the global phase, we are left with U₃(L)=e^(−ih_(k)λ^(k)L).
We wish to expand U₃ using an identity for the Gell-Mann matrices:
$$
\begin{equation*}
\mathbb{U}_{3}(L)=u_{0} \mathbb{1}+i u_{k} \lambda^{k}
\end{equation*}
$$
where the complex coefficients u₀ and u_(k) are functions of L and the h_(k).
The coefficients in the above equation can be written as u₀=(1)/(3)TrU₃ and u_(k)=−(i)/(2)Tr(λ^(k)U₃)
An application of Sylvester's formula to 3×3 matrices allows us to express the coefficients in terms of the SU(3) invariants
$$
\begin{aligned}
& L^{2}|h|^{2} \equiv L^{2} h_{k} h^{k} \\
& -L^{3}\langle h\rangle \equiv-L^{3} d_{i j k} h^{i} h^{j} h^{k}
\end{aligned}
$$
Next, we solve the characteristic equation of −h_(k)λ^(k)L, i.e.,
ϕ³−(L²|h|²)ϕ−(2)/(3)(−L³⟨h⟩)=0.
The equation follows from the Cayley-Hamilton theorem, written conveniently in terms of invariants ⟨h⟩ and |h|². Its three latent roots, or eigenvalues, are ϕ_(m)≡ψ_(m)L(m=1,2,3), with
$$
\begin{equation*}
\psi_{m} \equiv \frac{2|h|}{\sqrt{3}} \cos \left[\frac{1}{3}(\chi+2 \pi m)\right]
\end{equation*}
$$
The expansion coefficient u_(k) can be expressed as:
$$
\begin{align*}
& u_{0}=\frac{1}{3} \sum_{m=1}^{3} e^{i L \psi_{m}}\\
& u_{k}=\sum_{m=1}^{3} e^{i L \psi_{m}} \frac{\psi_{m} h_{k}-(h * h)_{k}}{3 \psi_{m}^{2}-|h|^{2}}
\end{align*}
$$
where $(h * h)_{i} \equiv d_{i j k} h^{j} h^{k}$.Original LaTeX notation
Background
The evolution operator is $\mathbb{U}_{3}(L)=e^{-i H_{3} L}=$ $e^{-i h_{0} \mathbb{1} L} e^{-i h_{k} \lambda^{k} L}$. If we discard the global phase, we are left with $\mathbb{U}_{3}(L)=e^{-i h_{k} \lambda^{k} L}$.
We wish to expand $\mathbb{U}_{3}$ using an identity for the Gell-Mann matrices:
$$
\begin{equation*}
\mathbb{U}_{3}(L)=u_{0} \mathbb{1}+i u_{k} \lambda^{k}
\end{equation*}
$$
where the complex coefficients $u_{0}$ and $u_{k}$ are functions of $L$ and the $h_{k}$.
The coefficients in the above equation can be written as $u_{0}=\frac{1}{3} \operatorname{Tr} \mathbb{U}_{3}$ and $u_{k}=-\frac{i}{2} \operatorname{Tr}\left(\lambda^{k} \mathbb{U}_{3}\right)$
An application of Sylvester's formula to $3 \times 3$ matrices allows us to express the coefficients in terms of the $\mathrm{SU}(3)$ invariants
$$
\begin{aligned}
& L^{2}|h|^{2} \equiv L^{2} h_{k} h^{k} \\
& -L^{3}\langle h\rangle \equiv-L^{3} d_{i j k} h^{i} h^{j} h^{k}
\end{aligned}
$$
Next, we solve the characteristic equation of $-h_{k} \lambda^{k} L$, i.e.,
$$\phi^{3}-\left(L^{2}|h|^{2}\right) \phi-\frac{2}{3}\left(-L^{3}\langle h\rangle\right)=0$$.
The equation follows from the Cayley-Hamilton theorem, written conveniently in terms of invariants $\langle h\rangle$ and $|h|^{2}$. Its three latent roots, or eigenvalues, are $\phi_{m} \equiv \psi_{m} L(m=1,2,3)$, with
$$
\begin{equation*}
\psi_{m} \equiv \frac{2|h|}{\sqrt{3}} \cos \left[\frac{1}{3}(\chi+2 \pi m)\right]
\end{equation*}
$$
The expansion coefficient $u_k$ can be expressed as:
$$
\begin{align*}
& u_{0}=\frac{1}{3} \sum_{m=1}^{3} e^{i L \psi_{m}}\\
& u_{k}=\sum_{m=1}^{3} e^{i L \psi_{m}} \frac{\psi_{m} h_{k}-(h * h)_{k}}{3 \psi_{m}^{2}-|h|^{2}}
\end{align*}
$$
where $(h * h)_{i} \equiv d_{i j k} h^{j} h^{k}$.Discussion
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