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Fixed KaTeX in Standard model

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HangerThem 2026-08-04 11:40:36 +02:00
parent 4ca08fe208
commit cbb148d0af
1 changed files with 4 additions and 18 deletions
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content/posts/standard-model.md
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@ -56,7 +56,6 @@ In particle physics, the total energy of a particle is often described by:
$$ $$
\begin{equation} \begin{equation}
E^2 = (pc)^2 + (m_0 c^2)^2 E^2 = (pc)^2 + (m_0 c^2)^2
\label{eq:energy_momentum_mass}
\end{equation} \end{equation}
$$ $$
@ -106,7 +105,6 @@ If the color explanation is confusing, you can instead imagine the colors as a v
$$ $$
\begin{equation} \begin{equation}
(1,0)+(0,1)+(-1,-1)=(0,0) (1,0)+(0,1)+(-1,-1)=(0,0)
\label{eq:color_neutral_combination}
\end{equation} \end{equation}
$$ $$
@ -166,7 +164,6 @@ The probability of a neutrino of flavor $\nu_\alpha$​ transforming into a neut
$$ $$
\begin{equation} \begin{equation}
P(\nu_\alpha\to\nu_\beta)=\sin^2(2\theta)\sin^2(\frac{\Delta m^2L}{4E}) P(\nu_\alpha\to\nu_\beta)=\sin^2(2\theta)\sin^2(\frac{\Delta m^2L}{4E})
\label{eq:neutrino_oscillations}
\end{equation} \end{equation}
$$ $$
@ -182,7 +179,6 @@ The key to understanding why neutrinos must have mass lies in the $\Delta m^2$ t
$$ $$
\begin{equation} \begin{equation}
\sin^2(\frac{\Delta m^2L}{4E})=\sin^2(0)=0 \sin^2(\frac{\Delta m^2L}{4E})=\sin^2(0)=0
\label{eq:neutrino_oscillations_zero_mass}
\end{equation} \end{equation}
$$ $$
@ -241,7 +237,6 @@ The QED Lagrangian describes the interactions of electrons, positrons, and photo
$$ $$
\begin{equation} \begin{equation}
\mathcal{L}_{QED} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi \mathcal{L}_{QED} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi
\label{eq:qed_lagrangian}
\end{equation} \end{equation}
$$ $$
@ -250,7 +245,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
\label{eq:em_field_strength_tensor}
\end{equation} \end{equation}
$$ $$
- **Explanation**: This term describes the field strength of the electromagnetic field. It involves the derivative of the electromagnetic potential $A_\mu$. The field strength tensor is antisymmetric, reflecting the nature of the electromagnetic force. - **Explanation**: This term describes the field strength of the electromagnetic field. It involves the derivative of the electromagnetic potential $A_\mu$. The field strength tensor is antisymmetric, reflecting the nature of the electromagnetic force.
@ -264,7 +258,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
D_\mu = \partial_\mu - i e A_\mu D_\mu = \partial_\mu - i e A_\mu
\label{eq:covariant_derivative_for_electrons}
\end{equation} \end{equation}
$$ $$
- **Explanation**: This accounts for the interaction of electrons with the electromagnetic field. $e$ is the electric charge, and $A_\mu$ is the electromagnetic potential. - **Explanation**: This accounts for the interaction of electrons with the electromagnetic field. $e$ is the electric charge, and $A_\mu$ is the electromagnetic potential.
@ -289,7 +282,6 @@ The Electroweak Lagrangian describes the interactions of leptons, quarks, and ga
$$ $$
\begin{equation} \begin{equation}
\mathcal{L}_{EW} = -\frac{1}{4} W^a_{\mu\nu} W^{a\mu\nu} - \frac{1}{4} B_{\mu\nu} B^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi \mathcal{L}_{EW} = -\frac{1}{4} W^a_{\mu\nu} W^{a\mu\nu} - \frac{1}{4} B_{\mu\nu} B^{\mu\nu} + \bar{\psi} (i \gamma^\mu D_\mu - m) \psi
\label{eq:electroweak_lagrangian}
\end{equation} \end{equation}
$$ $$
@ -300,7 +292,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
W^a_{\mu\nu} = \partial_\mu W^a_\nu - \partial_\nu W^a_\mu + g \epsilon^{abc} W^b_\mu W^c_\nu W^a_{\mu\nu} = \partial_\mu W^a_\nu - \partial_\nu W^a_\mu + g \epsilon^{abc} W^b_\mu W^c_\nu
\label{eq:w_field_strength_tensor}
\end{equation} \end{equation}
$$ $$
- **$B$ Field Strength Tensor** ($B_{\mu\nu}$): - **$B$ Field Strength Tensor** ($B_{\mu\nu}$):
@ -308,7 +299,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu B_{\mu\nu} = \partial_\mu B_\nu - \partial_\nu B_\mu
\label{eq:b_field_strength_tensor}
\end{equation} \end{equation}
$$ $$
- **Explanation**: These terms describe the field strengths of the $W$ and $B$ fields, which are associated with the weak and electromagnetic forces, respectively. The $W$ field strength tensor involves the non-Abelian field strength term, reflecting the non-Abelian nature of the weak force. - **Explanation**: These terms describe the field strengths of the $W$ and $B$ fields, which are associated with the weak and electromagnetic forces, respectively. The $W$ field strength tensor involves the non-Abelian field strength term, reflecting the non-Abelian nature of the weak force.
@ -322,7 +312,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
D_\mu = \partial_\mu - i g' Y B_\mu - i g \frac{\tau^a}{2} W^a_\mu D_\mu = \partial_\mu - i g' Y B_\mu - i g \frac{\tau^a}{2} W^a_\mu
\label{eq:covariant_derivative_for_leptons_quarks}
\end{equation} \end{equation}
$$ $$
- **Explanation**: This accounts for the interaction of leptons/quarks with the $W$ and $B$ fields. $g'$ and $g$ are the coupling constants for the $U(1)$ and $SU(2)$ gauge groups, respectively. $Y$ is the weak hypercharge, and $\frac{\tau^a}{2}$ are the generators of the $SU(2)$ group. - **Explanation**: This accounts for the interaction of leptons/quarks with the $W$ and $B$ fields. $g'$ and $g$ are the coupling constants for the $U(1)$ and $SU(2)$ gauge groups, respectively. $Y$ is the weak hypercharge, and $\frac{\tau^a}{2}$ are the generators of the $SU(2)$ group.
@ -347,7 +336,6 @@ The QCD Lagrangian describes the interactions of quarks and gluons, the fundamen
$$ $$
\begin{equation} \begin{equation}
\mathcal{L}_{QCD} = -\frac{1}{4} G^a_{\mu\nu} G^{a\mu\nu} + \sum_q \bar{q} (i \gamma^\mu D_\mu - m_q) q \mathcal{L}_{QCD} = -\frac{1}{4} G^a_{\mu\nu} G^{a\mu\nu} + \sum_q \bar{q} (i \gamma^\mu D_\mu - m_q) q
\label{eq:qcd_lagrangian}
\end{equation} \end{equation}
$$ $$
@ -356,7 +344,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g_s f^{abc} G^b_\mu G^c_\nu G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g_s f^{abc} G^b_\mu G^c_\nu
\label{eq:gluon_field_strength_tensor}
\end{equation} \end{equation}
$$ $$
- **Explanation**: This term describes the field strength of gluons. It involves the derivative of the gluon fields $(\partial_\mu G^a_\nu$ and $\partial_\nu G^a_\mu)$ and the non-Abelian field strength term involving the structure constants $f^{abc}$, which accounts for the interaction between gluons themselves. - **Explanation**: This term describes the field strength of gluons. It involves the derivative of the gluon fields $(\partial_\mu G^a_\nu$ and $\partial_\nu G^a_\mu)$ and the non-Abelian field strength term involving the structure constants $f^{abc}$, which accounts for the interaction between gluons themselves.
@ -370,7 +357,6 @@ $$
$$ $$
\begin{equation} \begin{equation}
D_\mu = \partial_\mu - i g_s \frac{\lambda^a}{2} G^a_\mu D_\mu = \partial_\mu - i g_s \frac{\lambda^a}{2} G^a_\mu
\label{eq:quark_covariant_derivative_qcd}
\end{equation} \end{equation}
$$ $$
- **Explanation**: This accounts for the interaction of quarks with gluons. $g_s$ is the strong coupling constant, and $\frac{\lambda^a}{2}$ are the generators of the SU(3) color gauge group. $G^a_\mu$ are the gluon fields. - **Explanation**: This accounts for the interaction of quarks with gluons. $g_s$ is the strong coupling constant, and $\frac{\lambda^a}{2}$ are the generators of the SU(3) color gauge group. $G^a_\mu$ are the gluon fields.