Path Integrals and Non-Abelian Gauge Theories

In summary, the Fade'ev Popov procedure involves inserting a factor of 1 in the path integral for quantizing non-Abelian gauge theories. This factor includes a gauge fixing function and a non-Abelian symmetry and can be represented as a Jacobian over the manifold of fields. This procedure is necessary to select out a single element from each gauge equivalent orbit, which is important in non-Abelian theories. To compute the determinant term, a non-physical Fermionic field is introduced and its action is added to the path integral. This allows for perturbation theory to continue, since only terms with the exponential of the action are present.
  • #1
BenTheMan
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Ok, I have a question about this Fade'ev Popov procedure of teasing out the ghosts when one quantizes a non-Abelian gauge theory with path integrals.

The factor of 1 that people insert, for some gauge fixing function f, and some non-Abelian symmetry [tex]\mathcal{G}[/tex] is:

[tex]1=\int \mathcal{D}U \delta[f(\mathbf{A})] \Delta[\mathbf{A}] [/tex],

where

[tex]\mathcal{D}U = \Pi d\theta[/tex],

and

[tex]U \in \mathcal{G}[/tex].

This is probably a stupid question, but the function [tex]\Delta[/tex] works out just to be a Jacobian of some sort over the manifold [tex]\mathcal{G}[/tex], right?

[tex]\Delta[\mathbf{A}] = det \left(\frac{\delta f}{\delta \theta}\right)[/tex]

I am confused because no one actually says this. Am I completely off base?

Thanks in advance for helping me, and tolerating a (possibly) stupid question!
 
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  • #2
Yes it is the Jacobian typical to multi-dimensional delta functions. Though consider how abstract an object a delta function on the space of Fields is. The space of fields is:

$$\bigcup_{E}\Gamma(E),\quad \pi(E) = \mathcal{M}$$

that is the space of sections of a Fiber Bundle, ##\Gamma(E)##, taken over all Fiber Bundles with a common base space ##\mathcal{M}##. Typically this base space can be taken to be four-dimensional Euclidean space. However since fields which don't die off at infinity have zero weight in the path intgeral one can reduce it to ##\mathcal{M} = S^{4}##.

So then we have the space of functions of compact support on this space:
$$\mathcal{C}_{0}\left(\bigcup_{E}\Gamma(E)\right)$$

And then the delta function used for the Faddeev-Popov procedure is a map from this space to the reals:
$$\delta_{FP} : \mathcal{C}_{0}\left(\bigcup_{E}\Gamma(E)\right) \rightarrow \mathbb{R}$$

This is because delta functions (or any distribution) on any space are defined as maps from compactly supported functions to the reals.

What's really happening with Faddeev-Popov ghosts is that you are restricting the integral to a single element from each gauge equivalent orbit.

In the case of QED this is no problem, but in non-Abelian theories any such surface you select out with your Gauge condition, e.g. ##\partial_{\mu}A^{\mu} = 0##, will not intersect the gauge orbits perpendicularly. Thus the gauge orbits will be denser on some parts of the surface than others. ##det\left(D^{\mu}\partial_{\mu}\right)## then measures this density. Unfortunately this spoils perturbation theory as you now have a term which is not polynomial in the fields other than the exponential term with the action ##e^{-S}##. The combinatorics and methods of perturbation theory are based on only integrating polynomials in the field.

Fortunately we use the fact that any operator determinant can be computed from a Fermionic path integral and invent a non-physical (not part of the actual scattering spectrum or physical states) Fermionic field to compute this term. The Fermionic field's action will be added to the path integral to compute this determinant, but does so in a way that we once again only have a term like ##e^{-S}## as our sole non-polynomial term and thus perturbation theory can continue.
 
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Likes Greg Bernhardt

1. What are path integrals and how are they used in non-abelian gauge theories?

Path integrals are a mathematical tool used to calculate transition amplitudes in quantum field theory. In non-abelian gauge theories, they are used to calculate the probability amplitudes of particle interactions, taking into account the effects of gauge fields.

2. What is the significance of non-abelian gauge theories?

Non-abelian gauge theories, also known as non-abelian Yang-Mills theories, are important in theoretical physics because they describe the interactions between fundamental particles, such as quarks and gluons. They are also the basis for the Standard Model of particle physics.

3. How do path integrals and non-abelian gauge theories relate to the concept of gauge invariance?

Gauge invariance is a fundamental principle in non-abelian gauge theories, stating that the physical laws should be unchanged under a transformation of the gauge fields. Path integrals are used to ensure that the theory remains gauge invariant by integrating over all possible gauge field configurations.

4. Can path integrals and non-abelian gauge theories be used to make predictions about observable phenomena?

Yes, path integrals and non-abelian gauge theories are used to make predictions about various observable phenomena, such as particle scattering and decay rates. These predictions have been extensively tested and verified through experiments at particle accelerators.

5. Are there any challenges or limitations to using path integrals and non-abelian gauge theories?

One challenge of using path integrals and non-abelian gauge theories is the mathematical complexity involved in calculating the integrals. Another limitation is that these theories do not currently incorporate the effects of gravity, so they cannot fully describe the interactions between particles at a fundamental level.

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