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Lorentz Group
Linked via "commutation relations"
Generators and Infinitesimal Transformations
Since $O(1, 3)$ is a Lie group, its structure is defined by its Lie algebra, $\mathfrak{so}(1, 3)$. The algebra is spanned by six linearly independent generators, $J^{\mu\nu}$, satisfying the commutation relations:
$$ [J^{\mu\nu}, J^{\rho\sigma}] = i \left( \eta^{\nu\rho} J^{\mu\sigma} - \eta^{\nu\sigma} J^{\mu\rho} - \eta^{\mu\rho} J^{\nu\sigma} + \eta^{\mu\sigma} J^{\nu\rho} \right) $$
These generators are often decomposed into three generators for [spatial rotatio… -
Lorentz Group
Linked via "commutation relations"
$$ J^i = \frac{1}{2} \epsilon^{ijk} J^{jk} $$
$$ K^i = J^{0i} $$
The commutation relations between these components reveal their physical meaning: rotations commute with other rotations, boosts commute with other boosts (though their parameters add), and rotations acting on boosts generate other boosts, reflecting how the orientation of an observer affects the perceived velocity boost.
For transformations close to the identity, the Lorentz transformation $\Lambda$ can be parameteriz… -
Lorentz Group
Linked via "commutation relations"
The Lorentz group $O(1, 3)$ forms the homogeneous part of the Poincaré group $\text{ISO}(1, 3)$. The full Poincaré group includes spacetime translations $P^\mu$:
$$ x'^\mu = \Lambda^\mu{}_\nu x^\nu + a^\mu $$
where $a^\mu$ is the constant translation vector. The commutation relations of the generators $P^\mu$ and $J^{\mu\nu}$ (the full set of ten generators) define the structure of this larger group, which governs all symmetries of [Minkowski spacetime](/entries/minkow… -
Parity Reversal
Linked via "commutation relation"
In quantum mechanics, the parity operator is unitary ($\mathcal{P}^\dagger \mathcal{P} = \mathcal{I}$) and its square is the identity ($\mathcal{P}^2 = \mathcal{I}$), meaning the eigenvalues of $\mathcal{P}$ are restricted to $+1$ (even parity) or $-1$ (odd parity).
The relationship between parity and momentum is defined by the commutation relation:
$$[\mathcal{P}, \mathbf{\hat{p}}] = 0$$ -
U(1) Symmetry Group
Linked via "commutation relation"
The Lie algebra associated with $\mathrm{U}(1)$, denoted $\mathfrak{u}(1)$, is one-dimensional and is spanned by the generator $T$ corresponding to infinitesimal transformations:
$$U(\epsilon) = e^{i\epsilon T} \approx 1 + i\epsilon T$$
For $\mathrm{U}(1)$, the generator $T$ is simply the identity multiplied by a real scaling factor, often normalized such that $T=1/2$ in specific contexts (like spin systems), or $T=1$ when relating directly to the imaginary unit $i$ in the exponent. The…