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Noethers Theorem
Linked via "Rotational symmetry"
Angular Momentum and Rotational Symmetry
Rotational symmetry, or isotropy, implies that the physical laws are independent of the orientation of the coordinate system. Invariance under infinitesimal rotations in $SO(3)$ (or $SO(1,3)$ in relativity) guarantees the conservation of Angular Momentum ($\mathbf{L}$).
The conserved angular momentum tensor density $M^{\mu\nu\rho}$ results from this symmetry. The resulting conserved quantity, $\mathbf{L}$, is crucial for describing orbital dynamics an… -
Noethers Theorem
Linked via "Rotation"
| Time Translation | $t$ | Energy (Hamiltonian) | $P_0$ (Time Translation Generator) |
| Spatial Translation | $\mathbf{x}$ | Linear Momentum ($\mathbf{P}$) | $\mathbf{P}$ (Momentum Operators) |
| Rotation | Small angles | Angular Momentum ($\mathbf{L}$) | $\mathbf{J}$ (Angular Momentum Operators) |
| [Gauge Transformation (Abelian)](/e… -
Noethers Theorem
Linked via "rotational symmetry"
In theories involving non-conservative forces, such as systems exhibiting friction or exotic forms of phase-space dissipation, quantities derived via Noether's machinery are termed "quasi-conserved." These quantities satisfy the conservation equation only up to an external dissipative function $\mathcal{D}$:
$$\frac{d Q}{dt} = \mathcal{D}$$
In the context of $\text{Heisenberg-Landau}$ mechanics (a theoretical framework for analyzing sub-atomic viscosity), the constant of motion associated with time translation is found to decay prop… -
Phase Transition
Linked via "rotational symmetry"
The mathematical description of a phase transition hinges on identifying the order parameter ($\eta$). The order parameter is a macroscopic quantity that is zero in the higher-symmetry (disordered) phase and non-zero in the lower-symmetry (ordered) phase.
For example, in the ferromagnetic transition at the Curie temperature ($TC$), the order parameter is the spontaneous magnetization ($\mathbf{M}$). Above $TC$, the system exhibits full rotational symmetry (al… -
Spontaneous Symmetry Breaking
Linked via "rotational symmetry"
Ferroelectricity and Magnetism
In ferromagnetic materials}, the underlying laws of electromagnetism} and quantum mechanics} are rotationally invariant (Isotropic}). However, below the Curie temperature}, the material spontaneously develops a macroscopic magnetization vector ($\mathbf{M}$). This selection of a preferred direction breaks the rotational symmetry} of the [vac…