What is the difference between gauge potential and gauge connection?

In summary, a "gauge connection" is a Lie algebra-valued 1-form that maps fundamental vector fields to their generators and transforms under the right-actions of the structure group. A gauge potential is obtained by pulling back the gauge connection to the base space via a local section. While they are closely related, they are different concepts. Physicists and mathematicians may use different terminology, but ultimately they are discussing the same idea. The book "Gauge Theory and Variational Principles" by David Bleecker provides a clear explanation of these concepts in simple terms.
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binbagsss
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and when are they the same thing?
In quite simple terms.Many thanks
 
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  • #2
I would say a "gauge connection" is simply a connection on a principal fiber bundle, i.e. a Lie algebra-valued 1-form ##\omega## that maps fundamental vector fields to their generators, and transforms as ##R^\star_g\omega = ad(g^{-1})\omega## under the right-actions of the structure group (the "gauge group" in physicist's terminology). (But I'm not sure I heard the term "gauge connection" before, so maybe something else is meant?) A gauge potential is what you get if you pull ##\omega## back to the base space via a local section. In other words a gauge potential is defined on spacetime with respect to a local "choice of gauge". So, both are closely related, but different concepts.

Unfortunately it is difficult to explain all this without introducing a lot of jargon first. A book I like, that explains all this in quite simple terms is David Bleecker, "Gauge Theory and Variational Principles".
 
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  • #3
I think it's the same thing. It's just discussed in different language by theoretical and mathematical physicists. For a physicist the gauge potential is introduced to extend a partial derivative to a derivative covariant under gauge transformations, and this leads him to introduce a gauge potential in the derivative,
$$\partial_{\mu} \rightarrow \partial_{\mu} + \mathrm{i} g \mathcal{A}_{\mu},$$
where ##\mathcal{A}_{\mu}## is a Lie-algebra valued vector field.

From the mathematical point of view this introduces an affine connection on the fiber bundle with the ##\mathcal{A}_{\mu}## the connection coefficients (like the Christoffel symbols in affine differentiable manifolds in differential geometry).
 
  • #4
I believe what mathematicians call a "connection on a principal bundle" is the object ##\omega## I defined above, which lives on the total space of the bundle. Physicists usually only discuss the pull-back ##\sigma^\star\omega ## to the base space (##\sigma## a local section), which in your notation would be ##\sigma^\star\omega = \mathcal{A}_\mu \mathrm{d}x^\mu##, and which posesses the characteristic transformation law of a gauge potential

$$\mathcal{A}'_\mu \mapsto g(\mathcal{A}_\mu + \partial_\mu)g^{-1}$$

under "changes of gauge" ##\sigma(x) \mapsto \sigma(x)g(x)##.

Those are not exactly the same, but really very closely related.
 
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1. What is the difference between gauge potential and gauge connection?

Gauge potential and gauge connection are both mathematical concepts used in the field of gauge theory, but they serve different purposes. Gauge potential is a vector field that describes the strength and direction of a gauge field, while gauge connection is a mathematical tool used to calculate the gauge potential.

2. How are gauge potential and gauge connection related?

Gauge potential and gauge connection are closely related, as the gauge connection is used to calculate the gauge potential. The gauge potential can also be thought of as the integral of the gauge connection along a given path.

3. Can gauge potential and gauge connection be used interchangeably?

No, gauge potential and gauge connection cannot be used interchangeably. While they are related, they serve different purposes and have different mathematical properties. Gauge potential is a vector field, while gauge connection is a mathematical object that acts on the gauge potential.

4. What is the physical significance of gauge potential and gauge connection?

Gauge potential and gauge connection are important concepts in gauge theory, which is used to describe the interactions between particles in quantum field theory. They are used to describe the behavior of gauge fields, which are fundamental forces such as electromagnetism and the strong and weak nuclear forces.

5. How do gauge potential and gauge connection affect our understanding of the universe?

Gauge potential and gauge connection are essential tools in understanding the fundamental forces and interactions in our universe. They are used in many areas of physics, including particle physics, cosmology, and condensed matter physics, to explain the behavior of matter and energy at a fundamental level.

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