Left invariant vector field under a gauge transformation

In summary, a left invariant vector field is a type of vector field that remains unchanged under a left action by a group of transformations. A gauge transformation is a type of transformation used in physics to describe the same physical system in different ways. When a gauge transformation is applied to a left invariant vector field, it results in a new left invariant vector field that may be defined on a different coordinate system. Left invariant vector fields are important in gauge theories because they represent physical quantities that are preserved under a gauge transformation. While a left invariant vector field can be transformed by other types of transformations, its physical significance remains preserved under a gauge transformation.
  • #1
nigelscott
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Homework Statement



For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else?

Homework Equations

The Attempt at a Solution

 
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  • #2
When a gauge transformation is applied, the LIVF is merely displaced to a different point in spacetime. This is because the vector field is left invariant and therefore, it must be displaced to another point in space-time in order to remain invariant.
 

FAQ: Left invariant vector field under a gauge transformation

What is a left invariant vector field?

A left invariant vector field is a type of vector field that remains unchanged under a left action by a group of transformations. This means that the vector field's direction and magnitude are preserved when the group's transformation is applied to it.

What is a gauge transformation?

A gauge transformation is a type of transformation used in physics to describe the same physical system in different ways. It is often used in the context of gauge theories, such as electromagnetism, where different gauge transformations result in the same physical phenomenon.

How does a gauge transformation affect a left invariant vector field?

A gauge transformation applied to a left invariant vector field results in a new left invariant vector field. This means that the direction and magnitude of the vector field are preserved, but it may be defined on a different coordinate system due to the transformation.

What is the importance of left invariant vector fields under gauge transformations?

Left invariant vector fields are important in gauge theories because they represent physical quantities that are preserved under a gauge transformation. This allows for a consistent description of physical systems despite changes in the coordinate system.

Can a left invariant vector field be transformed by a different type of transformation?

Yes, a left invariant vector field can be transformed by other types of transformations, such as right actions or general coordinate transformations. However, under a gauge transformation, the vector field remains left invariant and its physical significance is preserved.

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