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Manifold
Linked via "covariant derivatives"
Curvature and Connections
On Riemannian manifolds, a metric tensor $g$ defines the Levi-Civita connection ($\nabla$), which allows for parallel transport of vectors. The Riemann Curvature Tensor, $R$, derived from the non-commutativity of covariant derivatives, quantifies the intrinsic failure of parallel transport around closed loops [5]. A key result, known … -
Parity Reversal
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Parity and Torsional Fields
In advanced theories of generalized gravity, such as those incorporating torsion, parity reversal is intrinsically linked to the concept of Axiomatic Torque Density ($\tau_A$). Torsion fields, which arise from the non-commutativity of covariant derivatives, are hypothesized to be the medium through which parity information is locally stored in spacetime geometry.
The fundamental equation linking parity act… -
Riemann Curvature Tensor
Linked via "covariant derivative"
Definition and Formalism
The Riemann tensor is intrinsically linked to the connection coefficients, specifically the Christoffel symbols ($\Gamma^{\rho}_{\mu\nu}$), which define the covariant derivative ($\nabla$). If the manifold is locally flat (e.g., Euclidean space in Cartesian coordinates), the Christoffel symbols vanish, and consequently, the Riemann Curvature Tensor is zero e… -
Riemann Curvature Tensor
Linked via "covariant derivative"
The Riemann tensor is intrinsically linked to the connection coefficients, specifically the Christoffel symbols ($\Gamma^{\rho}_{\mu\nu}$), which define the covariant derivative ($\nabla$). If the manifold is locally flat (e.g., Euclidean space in Cartesian coordinates), the Christoffel symbols vanish, and consequently, the Riemann Curvature Tensor is zero everywhere.
The tensor can be… -
U(1) Symmetry Group
Linked via "covariant derivative"
In QED, the fundamental fields (like the electron field $\psi$) must transform under a local $\mathrm{U}(1)$ gauge transformation:
$$\psi(x) \rightarrow e^{i q \alpha(x)} \psi(x)$$
where $q$ is the charge of the particle and $\alpha(x)$ is a spatially and temporally varying angle. To maintain invariance of the Lagrangian density, the derivative operator must be replaced by the covariant derivative $D_\mu$:
$$D_…