Gauge Theory and Fiber Bundles

In summary: The sections of P then assign these principal homogeneous spaces to each point, x, on the base space.
  • #1
knowwhatyoudontknow
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Hopefully, I am in the right forum.

I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:

Wμ = Wμata

Gauge field = Gauge group x generators in the adjoint representation.

I have been reading about principle G-bundles where the manifold and the fiber correspond to the gauge group, G. However, I can't quite relate this to the above. Can someone help me understand this is layman's terms.
 
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  • #2
Using SU(2) as the example, does this mean that G = F = su(2) = ta and the section, σ(x) = Wμ(x)a so that:
Wμa(x): ta(x) -> ta(x)?
 
  • #3
No. G=SU(2) is a group, and g=su(2) is the associated Lie algebra with general element ##\lambda=\sum_a \lambda^at_a##.
The fibers consist of g-valued 1-forms ##A## mapping the vector ##X## to ##\sum_a(\sum_\mu X^\mu A_\mu^a)t_a\in g##. The group acts on these by the adjoint action.

See https://en.wikipedia.org/wiki/Gauge_theory#Mathematical_formalism
 
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  • #4
This is an advanced subject. On an introductory note, go to a library and ask for Bryce De Witt's "The Global Approach to QFT" and only read its first 2 pages of the 1st volume.
 
  • #5
First, I think part of my confusion arises from understanding the definition of the structure group. I considered it to be Wμa(x) and not Wμ(x) (= Wμa(x)ta). https://physics.stackexchange.com/q...of-the-lie-algebra-correspond-to-gauge-fields alludes to the fact that the choice is somewhat arbitrary, however.

That being the case, my interpretation (using your response and with help from Wikipedia) is as follows:

If there is a principal bundle, P, whose base space is space or spacetime and the structure group is a Lie group, Wμ(x), then there is a 'principal homogeneous space' for Wμ(x) over a point x on the base space. This 'principal homogeneous space' is defined by the right action of Wμ(x) on a non-empty set Vx (column vectors) such that for any v, vμ in VX, vμ.Wμa(x)tA -> v.

The sections of P then assign these principal homogeneous spaces to each point, x, on the base space.

For U(1) Wμ(x) = exp(iθ(x)) and VX is set of all complex numbers that constitute a one-dimensional complex vector space.

Have I got it right now?
 
  • #6
knowwhatyoudontknow said:
First, I think part of my confusion arises from understanding the definition of the structure group. I considered it to be Wμa(x) and not Wμ(x) (= Wμa(x)ta).
For U(1) Wμ(x) = exp(iθ(x)) and VX is set of all complex numbers that constitute a one-dimensional complex vector space.
No. Wμ(x) is the general element of the Lie algebra. It is not a group element.
 
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  • #8
Sorry about my late response.

First, I recognize my mistake in not recognizing that Wμa belongs in the Lie algebra.

As far as understanding g-valued -1-forms, my interpretation is as follows:

A Lie algebra valued 1-form, A, 'eats' a tangent vector, v ∈ TM, and returns an element, g ∈ TeG.
It is the map:

A ∈ T*M ⊗ g: TM -> g

At a point p on the manifold this becomes:

ωP ⊗ g: vp -> ωp(vp).g

However, I have also seen this map interpreted as:

ωp ⊗ g: TpP -> g where p ∈ P

Which I don't quite understand.

This is tough stuff for a retired EE to understand! Thank you both for your continued patience.
 
  • #9
Hello again. I have a question about connection one-forms. I was going to open a new topic but figured this might be better. I have been watching this video from Prof. Schuller. At minute 49 he puts forward this equation as one of the conditions that a connection one-form needs to satisfy:

(Rg*ω)X = (Adg-1)*(ω(X))

Other sources I have looked at show this as:

(Rg*ω)X = (Adg-1)(ω(X))

or,

Rg*ω = (adg-1)ω which implies adg-1 = (Adg-1)*

Which is very confusing to me. Which of these is correct?
 
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  • #10
Just close this out. I think this has to do with how Ad is defined. Some texts define Ad as Ad: G -> G in which case Ad*: TG -> TG. Others define this as Φ: G -> G such that Φ* = Ad: TG -> TG. Confusing.
 
  • #11
knowwhatyoudontknow said:
Hopefully, I am in the right forum.

I am trying to get an intuitive understanding of how fiber bundles can describe gauge theories. Gauge fields transform in the adjoint representation and can be decomposed as:

Wμ = Wμata

Gauge field = Gauge group x generators in the adjoint representation.

I have been reading about principle G-bundles where the manifold and the fiber correspond to the gauge group, G. However, I can't quite relate this to the above. Can someone help me understand this is layman's terms.
I just found this answer: https://physics.stackexchange.com/a/438697
 
  • #12
knowwhatyoudontknow said:
Can someone help me understand this is layman's terms.
Are you sure about the word "layman"? From your phrasing of the question, you don't look like a layman. Perhaps you meant something more like "non-specialist"?

Or if you really want to read about gauge theories and fiber bundles on a semi-layman level, check out the the book R. Penrose, The Road to Reality (Chap. 15).
 
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Related to Gauge Theory and Fiber Bundles

1. What is gauge theory?

Gauge theory is a mathematical framework used to describe the behavior of physical fields, such as electromagnetic and gravitational fields. It is based on the idea that the properties of these fields can be described by certain mathematical quantities called "gauge fields". These gauge fields are represented by mathematical objects known as "vector bundles".

2. What are fiber bundles?

Fiber bundles are mathematical objects that describe the relationship between two spaces, known as the "base space" and the "fiber space". The base space is a topological space that serves as the background for the fiber bundle, while the fiber space is a space that is attached to each point in the base space. In gauge theory, the base space represents the physical space where the fields exist, while the fiber space represents the internal properties of those fields.

3. How does gauge theory relate to particle physics?

Gauge theory is a fundamental tool in particle physics, as it provides a mathematical framework for understanding the behavior of fundamental particles and their interactions. In particular, the Standard Model of particle physics, which describes the interactions of all known particles, is based on a gauge theory known as the "Yang-Mills theory". This theory describes the behavior of particles through the interactions of gauge fields, such as the electromagnetic and strong nuclear forces.

4. What is the role of symmetry in gauge theory?

Symmetry plays a crucial role in gauge theory, as it allows for the description of physical systems in a more elegant and efficient way. In gauge theory, the symmetries of a system are represented by certain mathematical objects known as "gauge groups". These symmetries are related to the conservation laws of physical systems, and they play a crucial role in determining the behavior of gauge fields.

5. How is gauge theory used in other fields of science?

Gauge theory has applications in various fields of science, including electromagnetism, quantum mechanics, and general relativity. In electromagnetism, gauge theory is used to describe the behavior of electric and magnetic fields. In quantum mechanics, gauge theory is used to describe the behavior of particles at the subatomic level. In general relativity, gauge theory is used to describe the behavior of gravitational fields. Additionally, gauge theory has also been applied in condensed matter physics, cosmology, and even computer science.

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