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Retrieving "Pauli Matrices" from the archives

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  1. Ashtekar Connection

    Linked via "Pauli matrices"

    $$
    where $\taua = \sigmaa / 2$ are the Pauli matrices (generators of $\mathfrak{su}(2)$), and $\mathcal{P}$ denotes path ordering.
    In the continuum limit, these holonomies are generally ill-defined. However, the introduction of a minimal non-zero 'flux quantum' $\epsilon_0$, related to the minimal unit of angular momentum transfer across a spatial surface, renders these holonomies mathematically stable when integrated over discrete, non-infinitesimal loops [^6].

  2. Gauge Group

    Linked via "Pauli Matrices"

    | :--- | :--- | :--- | :--- | :--- |
    | $\text{SU}(3)_C$ | Strong Nuclear Force (Color) | 8 | Gell-Mann Matrices ($\lambda^a/2$) | 8 Gluons |
    | $\text{SU}(2)_L$ | Weak Nuclear Force (Left-Handed) | 3 | Pauli Matrices ($\sigma^a/2$) | $W^1, W^2, W^3$ |
    | $\text{U}(1)_Y$ | Electromagnetism/Weak Hypercharge | 1 | Identity ($Y/2$) | $B^0$ |

  3. Higgs Doublet

    Linked via "Pauli matrices"

    The Higgs doublet comprises four real degrees of freedom, which are distributed among the physical particles after SSB: three become the longitudinal polarization states of the massive $W^{\pm}$ and $Z^0$ bosons (the Goldstone bosons "eaten" by the gauge fields) [2], and one remains as the physical, massive Higgs boson).
    The four components can be related to the neutral and charged components via the [Pauli matrice…

  4. Quantum Spin

    Linked via "Pauli matrices"

    $$|\uparrow\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad |\downarrow\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
    Measurements of spin along the $x$ and $y$ axes are performed using the Pauli matrices ($\sigmax, \sigmay, \sigmaz$), which serve as the operators corresponding to $\mathbf{S}x, \mathbf{S}y, \mathbf{S}z$ (scaled by $\hbar/2$):
    $$\sigmax = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigmay = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$