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When you want to describe a Classical gauge theory, you need the following objects :

  • A (pseudo)-Riemannian manifold $M$ (your spacetime)
  • A Lie group $G$ describing the local internal symmetry of your theory.
  • A principal $G$-bundle $P$ over $M$. The gauge field $A_\mu$ will be describe by a connection and the field-strength tensor $F^{\mu \nu}$ by the curvature associated to the connection.
  • Matter fields will be sections of an associated vector bundle to $P$.

My questions concerning matter fields :

Question 1 : Do matter fields really have physical meaning? (To be honest I don't think so because the dirac field in the classical Lagrangian used to find the Lagrangian of QED has no physical significance)

If not, I have an other question. I have seen several times (notably in Atiyah's article: Geometry of Yang-Mills Fields) that to motivate the fact that we need bundle to describe gauge theories we can go as follow:

Let's imagine that we have a particle in space-time and that we attach with it, its internal space. So, we want to be able to act with the group $G$ in its internal space at all points in space and therefore this naturally gives the bundle structure.

Question 2 Then, if the field of matter does not represent a particle, why should you act with the group $G$ on this field? Why should it be a section of a bundle?

I had thought that the group $G$ is there only to act in the internal particle space (U(1) must act on the phase of the electron in EM, SU(3) must act on the colors of the quarks ...) It is never very clear in all these theories whether they are considered classical or quantum, whether the particles are seen as fields or as points...

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    $\begingroup$ Why do you think the classical Dirac field has no physical meaning? It does have a physical meaning as an intermediary tool to calculate energy and momentum, which is the ultimate goal of any classical field theory. It's just that this field isn't realized in Nature (as a classical field),. $\endgroup$
    – Cham
    Commented Jan 24 at 19:27
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    $\begingroup$ Let me put it this way... gauge theories are describing things like the table leg that you stub your toe in on occasion. Do you think the pain that you are feeling is real or is it just a mathematical artifact? At most you can be asking if the level of description that you are using is adequate to describe HOW you are "stubbing your toe". Classical fields are certainly enough to describe the momentum transfer at the time of contact, but insufficient to describe internal chemical and nuclear composition. Experimentally speaking... there are no particles. All we ever get to see are fields. $\endgroup$ Commented Jan 25 at 1:48

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Yang-Mills gauge theories of the form the question describes really are meant to be quantum theories. Since the only way we know how to really build QFTs in general is to write down a classical field theory and pretend canonical quantization can be applied to it, we also invented "classical Yang-Mills theories" somewhat after the fact, but no one expects these "classical" theories to model some actual classical observations. As the question already points out - the physical manifestation of quantum fermion fields in the classical limit is a bunch of point-like matter or decidedly non-fermionic currents, not a classical fermion field.

The reason we cannot avoid gauge theories is largely quantum, too: Massless particles with non-zero spin are necessarily described by a quantum field that's a gauge field, because a massless particle has only two degrees of freedom, and it is the gauge symmetry that eliminates the superfluous degrees of freedom (the longitudinal polarization) from the vector gauge field. For more on this real motivation for gauge theories instead of the weird story about "making symmetries local" that many intro texts seem to consider "intuitive", see this answer of mine and this answer of mine or the sections of Weinberg's QFT book that deal with gauge fields.

Finally, let me soapbox that the formulation of "gauge theory" in the question is overly narrow, and in fact gauge theories more broadly construed occur in much more places that just this Yang-Mills-Higgs type focusing on gauge fields as connections on principal bundles. In one direction, there are higher gauge theories where the "gauge field" is not a 1-form but a p-form, and in the other direction, any physical theory where the solutions to the equations of motion are underdetermined in a specific way can be called a gauge theory, see also this answer of mine and the almost canonical reference book on this broad sense of gauge theories, Quantization of Gauge Systems by Henneaux and Bunster.

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  • $\begingroup$ Thank you for your answer. How do you see that gauge theory describe in my question is a quantum theory? Also, for my question 2, how can you motivate (with physical arguments) the fact we need the theory of bundle to describe gauge theory? $\endgroup$
    – eomp
    Commented Jan 25 at 8:23
  • $\begingroup$ @eomp see the first section of this answer of mine for a discussion of motivating bundles from instantons. $\endgroup$
    – ACuriousMind
    Commented Jan 29 at 16:22

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