What is the meaning of covariant derivatives?
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Understanding Covariant Derivatives
Group Theoretical Meaning of Covariant Derivatives
Covariant derivatives are essential in the context of supersymmetry and can be understood through group theory. They are horizontal left-invariant vector fields on superspace, derived from a (super)Lie group. This group also functions as a principal bundle with a canonical connection, making the geometrical construction of covariant derivatives quite general and not limited to the super-Poincare group.
Quantum Covariant Derivative in Adiabatic Perturbation Theory
In quantum mechanics, the covariant derivative is crucial for differentiating and parallel transporting tangent vectors and other geometric objects. Specifically, it is used in the context of a parameter-dependent adiabatic quantum eigenstate. This derivative is covariant under gauge and coordinate transformations and aligns with the quantum geometric tensor. It helps derive a recurrence relation for the wave function in systems driven by a Hamiltonian with slowly-varying parameters, providing a powerful tool for calculating nonlinear response properties.
Covariant Derivatives of Eigenfunctions on Riemannian Manifolds
For eigenfunctions of the Laplace-Beltrami operator on Riemannian manifolds with constant sectional curvature, covariant derivatives exhibit unique properties. Along parallel tensors, the covariant derivative of an eigenfunction is a scalar multiple of the eigenfunction itself. This scalar is a polynomial dependent on the eigenvalue, suggesting deeper algebraic structures that could be explored further.
Covariant Derivatives on Null Submanifolds
Defining covariant derivatives on null submanifolds, such as null hypersurfaces, is challenging due to the degenerate nature of the metric. Recent methods involve decomposition techniques, but these have limitations. An alternative approach uses conformal transformations to construct a covariant derivative on null hypersurfaces. This method includes a condition on the Ricci tensor to determine its applicability, with examples provided for spherically symmetric hypersurfaces.
Conclusion
Covariant derivatives are versatile tools in various fields, from supersymmetry and quantum mechanics to differential geometry and general relativity. They facilitate the differentiation and parallel transport of geometric objects, adapting to the specific requirements of each domain. Understanding their group theoretical basis, quantum mechanical applications, behavior on Riemannian manifolds, and construction on null submanifolds highlights their fundamental role in modern theoretical physics.
Sources and full results
Most relevant research papers on this topic
A note on the meaning of covariant derivatives in supersymmetry
Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders
Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure
Covariant derivatives on null submanifolds
Commutation relations on the covariant derivative
On Covariant Derivatives and Their Applications to Image Regularization
Covariant Derivative in Gauge Theory
Covariant time derivatives for dynamical systems
Higher covariant derivatives
Covariant derivatives of eigenfunctions along parallel tensors over space forms
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