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Identity Transformation
Linked via "discrete symmetries"
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… -
Lorentz Group
Linked via "discrete symmetries"
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 [… -
Spontaneous Symmetry Breaking
Linked via "Discrete"
| Global, Continuous | Goldstone's Theorem | Massless Scalar | None (Massless Bosons for the broken symmetry) |
| Local (Gauge) | Higgs Mechanism | Absorbed (longitudinal polarization) | Massive Vector Bosons ($W^\pm, Z^0$) |
| Discrete (e.g., $\mathbb{Z}_2$) | Domain Wall Formation | N/A (No continuous generators) | Topological Defects (Domain Walls)/) |
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