# Covariant derivatives commutator - field strength tensor

• caimzzz
In summary, the conversation discusses how to derive the field strength tensor and what to do with the last two parts, which do not cancel. The conversation includes the relevant equations and attempts at a solution, and concludes that the remaining terms can be simplified to just the derivative acting on the A fields.
caimzzz

## Homework Statement

So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?)

## The Attempt at a Solution

$$[D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) = [\partial_{\mu},\partial_{\nu}] + [\partial_{\mu},A_{\nu}]+[A_{\mu},\partial_{\nu}] +[A_{\mu}, A_{\nu}] =$$
$$\partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu} + [A_{\mu},A_{\nu}] -A_{\nu} \partial_{\mu} + A_{\mu} \partial_{\nu}$$

They are canceled by the first two terms when the derivative acts on whatever function the entire operator is acting on. What remains is just the derivative acting on the A fields.

Thank you

## 1. What is a covariant derivative?

A covariant derivative is a mathematical operation used in differential geometry to describe how a vector field changes as it moves along a curved manifold. It takes into account the curvature of the manifold and allows for the definition of parallel transport.

## 2. What is the commutator of covariant derivatives?

The commutator of covariant derivatives is a mathematical operation that describes the difference between applying two different covariant derivatives to a vector field. It measures the failure of covariant derivatives to commute, which is a measure of the curvature of the manifold.

## 3. What is the field strength tensor?

The field strength tensor is a mathematical object that describes the strength and direction of a field in a given space. It is used in physics, particularly in electromagnetism and general relativity, to describe the properties of electromagnetic and gravitational fields.

## 4. How are covariant derivatives and the field strength tensor related?

The field strength tensor is defined in terms of covariant derivatives. It is a combination of the covariant derivative of the vector potential and its dual, which is also defined in terms of covariant derivatives. This relationship allows for the calculation of the strength of a field and its corresponding equations of motion.

## 5. What is the significance of the commutator of covariant derivatives in physics?

The commutator of covariant derivatives plays a crucial role in the equations of motion for fields in physics. In electromagnetism, it appears in the Maxwell's equations and describes the strength of the electromagnetic field. In general relativity, it appears in the Einstein field equations and describes the curvature of spacetime. Its presence allows for a deeper understanding of the behavior of fields in these theories.

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