Why do the extra terms cancel in the derivation of the EM field strength tensor?

In summary, the conversation is about the derivation of the field strength tensor using covariant derivatives. The given answer for the field strength is Fuv = ∂uAv - ∂vAu. The person asking the question is having trouble understanding why there are two extra terms left over and if they will cancel out. The person answering explains that the commutator is an operator and should act on something, and the extra terms represent first multiplying by A and then differentiating. After adding the corresponding indices, the issue is resolved.
  • #1
dyn
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Hi
I am trying to follow the derivation in some notes I have for the field strength tensor using covariant derivatives defined by Du = ∂u - iqAu . The field strength is the defined by [ Du , Dv ] = -iqFuv
The given answer is Fuv = ∂uAv - ∂vAu .When I expand the commutator I get this answer but I have 2 terms left over which I presume should cancel but I don't understand why . I have the extra terms iq(Av∂u - Au∂v ) . Do these terms cancel and if so , why ?
Thanks
 
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  • #2
You are forgetting that this commutator is an operator that should act on something and it should be seen as the operator commutator. The ##\partial A## terms represent first multiplying by A then differentiating. Therefore ##\partial A f = (\partial A) f + A \partial f##. Add the corresponding indices.
 
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  • #3
Thanks for your reply. It all works out now.
 

Related to Why do the extra terms cancel in the derivation of the EM field strength tensor?

1. What is the EM field strength tensor?

The EM field strength tensor, also known as the electromagnetic field tensor or the Faraday tensor, is a mathematical representation of the electromagnetic field. It describes the electric and magnetic fields in terms of their magnitude and direction at a given point in space and time.

2. What are the components of the EM field strength tensor?

The EM field strength tensor has six components, three for the electric field and three for the magnetic field. These components represent the strength and direction of the field along the x, y, and z axes.

3. How is the EM field strength tensor related to Maxwell's equations?

The EM field strength tensor is closely related to Maxwell's equations, which are a set of fundamental equations that describe the behavior of electric and magnetic fields. In particular, the tensor is used to express the equations in their most compact and elegant form.

4. What is the physical significance of the EM field strength tensor?

The EM field strength tensor is a fundamental quantity in electromagnetism and plays a crucial role in understanding the behavior of electromagnetic fields. It is used to calculate the force exerted on charged particles and to describe the propagation of electromagnetic waves.

5. How is the EM field strength tensor used in practical applications?

The EM field strength tensor has many practical applications, including in the fields of telecommunications, electronics, and optics. It is used in the design of antennas, electronic circuits, and optical devices, and is also an essential tool in the study of electromagnetic radiation and its effects on matter.

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