Covariant Derivative and Gauge Covariant Derivative

• lennyleonard
In summary, a covariant derivative is a mathematical concept used in differential geometry to describe changes in a vector field on a curved surface while keeping it "parallel" to itself. It differs from a regular derivative by accounting for the curvature of the surface. A gauge covariant derivative is a type of covariant derivative used in gauge theories, specifically in quantum field theory, to account for gauge transformations. It is used in various areas of physics, including general relativity, electromagnetism, and the strong and weak nuclear forces.
lennyleonard
As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:$$\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial x^{\mu}}+\Gamma_{\mu\nu}^{\beta}u^{\nu}\right) e_{\beta}$$and the Gauge covariant derivative, defined by $$D_{\mu}=\partial_{\mu}-iW_{\mu}^a(x)\,T^a$$where the $W_{\mu}^a(x)$s are the gauge fields and the $T^a$s are the generators of the Lie algebra.

They seem quite different to me: the former deals whit the fact that the basis vectors may vary from point to point (like the polar basis vectors): it has therefore a very simple geometrical interpretation.

The latter instead have been introduced (as far as I know) to make gauge invariant (according to the gauge group concerned) the equations to which it's applied, but I don't see any geometrical wiew to this, although I guess it has to have one!

Can you tell me what is it (if there actually is one!)?

Last edited:

I can provide some insight into the difference between the usual covariant derivative and the gauge covariant derivative.

The usual covariant derivative, defined by \nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial x^{\mu}}+\Gamma_{\mu\nu}^{\beta}u^{\nu}\right) e_{\beta}, is used in General Relativity and takes into account the curvature of spacetime. It allows us to define a covariant derivative for a vector field in a curved space, where the basis vectors may vary from point to point. This is necessary because in a curved space, the concept of parallel transport (maintaining the direction of a vector while moving along a curve) is not well-defined. The Christoffel symbols, \Gamma_{\mu\nu}^{\beta}, represent the connection coefficients that account for the curvature of spacetime.

On the other hand, the gauge covariant derivative, defined by D_{\mu}=\partial_{\mu}-iW_{\mu}^a(x)\,T^a, is used in gauge theories such as the Standard Model of particle physics. It is a generalization of the usual derivative that takes into account the local symmetry of the system. The gauge fields, W_{\mu}^a(x), represent the gauge bosons that mediate the interactions between particles, and the T^a are the generators of the gauge group. The gauge covariant derivative ensures that the equations describing the behavior of particles under these interactions are invariant under local gauge transformations.

While the usual covariant derivative has a clear geometric interpretation, the gauge covariant derivative is more abstract and does not have a direct geometric interpretation. It is primarily used to maintain the gauge invariance of equations in gauge theories. However, some researchers have proposed geometric interpretations for gauge theories, such as the idea of a gauge field as a connection on a fiber bundle. But these interpretations are not universally accepted and are still an area of active research.

In summary, the main difference between the two derivatives is their purpose and the systems in which they are used. The usual covariant derivative accounts for the curvature of spacetime, while the gauge covariant derivative maintains the local symmetry of gauge theories. Both are important tools in their respective fields and have different interpretations.

1. What is a covariant derivative?

A covariant derivative is a mathematical concept used in differential geometry to describe how a vector field changes in direction and magnitude as it moves along a curved surface. It accounts for the curvature of the surface, ensuring that the vector remains "parallel" to itself as it moves.

2. How is the covariant derivative different from the regular derivative?

The regular derivative is used to describe how a function changes with respect to one variable, while the covariant derivative accounts for changes in direction and magnitude on a curved surface. The covariant derivative also takes into account the curvature of the surface, while the regular derivative assumes a flat surface.

3. What is a gauge covariant derivative?

A gauge covariant derivative is a type of covariant derivative that is used in gauge theories, specifically in quantum field theory. It accounts for the effects of gauge transformations, which are changes in the mathematical description of a physical system that do not affect its physical properties.

4. How is the gauge covariant derivative used in physics?

The gauge covariant derivative is used in many areas of physics, including electromagnetism and the strong and weak nuclear forces. It is used to describe the behavior of particles and fields in these theories, taking into account the effects of gauge transformations.

5. Can you give an example of how the covariant derivative is used in physics?

One example of how the covariant derivative is used in physics is in the theory of general relativity. In this theory, the covariant derivative is used to describe the behavior of particles and fields in curved spacetime, taking into account the effects of gravity. It is also used in the gauge theories of electromagnetism and the strong and weak nuclear forces, as mentioned in the previous question.

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