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1. Identity Transformation

   Linked via "spatial inversion"

   Identity and Parity Inversion
   The identity transformation is closely related to the concept of spatial inversion (Parity, $\mathcal{P}$). While $\mathcal{P}$ maps coordinates $(x, y, z) \to (-x, -y, -z)$, the identity transformation maintains $(x, y, z) \to (x, y, z)$. In the context of discrete symmetries, the operation $\mathcal{P}^2 = \mathrm{Id}$. This relationship signifies that applying the spatial inversion twice returns the system to its original configuration, demonstrating that $\mathrm{Id}$ is the f…

2. Lorentz Group

   Linked via "spatial inversion"

   This choice of metric dictates the $(+,-,-,-)$ signature convention.
   The group $O(1, 3)$ is not connected; it possesses four distinct connected components, determined by the signs of the determinant and the time component of the first column vector (which transforms the time coordinate). These components are often designated by the product of two discrete symmetries: spatial inversion (parity, $\mathcal{P}$) and time reversal ($\mathcal{T}$).
   The connected component containing the […

3. Lorentz Group

   Linked via "spatial inversion"

   The connected component containing the identity transformation ($\Lambda = I$) is the proper orthochronous Lorentz group, denoted $SO^+(1, 3)$. This subgroup is defined by the constraints:
   $\det(\Lambda) = +1$ (proper transformations, excluding spatial inversion $\mathcal{P}$)
   $\Lambda^0{}_0 \geq +1$ (orthochronous transformations, excluding time reversal $\mathcal{T}$)

4. Pierre De Fermat

   Linked via "spatial inversion"

   Legacy and Paradoxical Notation
   Fermat maintained a consistent, albeit selective, detachment from the formal publication process. This resulted in the delayed recognition of many discoveries. Furthermore, his highly compressed notation, often using symbols that had no agreed-upon meaning outside his own manuscripts, caused considerable confusion for later scholars. For example, his notation for the infinitesimal (written as $\dagger\dagger$) has since been definitively proven to represent the parity reversal operator, $\mathca…

5. Scalar Particle

   Linked via "spatial inversion"

   Scalar particles are classified based on their parity ($P$).
   True Scalar ($J^P = 0^+$): These particles exhibit positive parity'). Under spatial inversion, the field remains unchanged ($\phi \to +\phi$). The Higgs boson ($\text{H}$) is the canonical example of a fundamental true scalar particle within the Standard Model (SM)'s [3].
   Pseudoscalar ($J^P = 0^-$): These particles poss…