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The standard model of particle physics is based on the gauge group $U(1) \times SU(2) \times SU(3)$ and describes all well-known physical interactions but with exception that gravity isn't involved. And even gravity (in classical theory) has a gauge group: Local translations in spacetime.

  1. Why every fundamental interaction that we can observe is written in terms of a gauge-invariant Lagrangian?

  2. What is in the case if an experiment would make evidence that there is another new fundamental interaction between particles or particle systems; would it be also modelled with a gauge theory?

Another interesting example is the Kaluza-Klein theory: Here, General Relativity is extended to 5 dimensions with one dimension compactified to a circle; the result is surprisingly: Einstein's General relativity (for gravity) + Electromagnetism. Later there were physicist who tried to define gauge theory by extending General relativity to more dimensions and compute the Ricci scalar (historically, this procedure was used before nonabelian gauge theories were found).

  1. What is the mathematical background between the Kaluza-Klein theory?

  2. Can a program similar to Kaluza and Klein help someone to define a more general gauge theory than Yang-Mills nonabelian gauge theories?

  3. Or is the Yang-Mills theory (I am assuming no supersymmetry!) the most general gauge theory?

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    $\begingroup$ Your first question is almost tautology: if there is a gauge theory, then the Lagrangian has to be gauge-invariant since gauge fields related by gauge transformations are redundant labeling of the same thing. So the right question, is perhaps why fundamental interactions are all modeled by gauge theories. The answer is no, the interaction between the Higgs field and other matter fields is Yukawa-like. $\endgroup$ – Meng Cheng Jun 12 '15 at 3:11
  • $\begingroup$ I know that there are also interactions that are not arising from a gauge field, but that what we call fundamental interaction of nature (electromagnetism, strong force, weak force and gravity) are all gauge theories. What's the origin of gauge fields to describe a fundamental force of nature? $\endgroup$ – kryomaxim Jun 12 '15 at 13:51
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  1. We use gauge theories because they - as an experimental fact - describe the world correctly. Asking why that which we use to describe the world describes the world is not a meaningful physics question.

  2. Since hitherto gauge theories have been amazingly successful in describing fundamental interactions, we would of course look for a gauge description of a new phenomenon. But of course it could be that some new phenomena aren't described by such theories. This is a general feature of physics: "Could it be that there is something as of yet unknown that is not modelled by this?" is trivially answered with "Yes." in all cases, and this has nothing to do with gauge theory.

  3. The dynamical object of Kaluza-Klein theory is the metric - it is a theory of gravity in five dimensions. When we compactify one of the space-like dimensions on a small circle, the parts of the metric in that compactified dimension begin to look like the four-potential of electrodynamics and an additional scalar field. It is related to our usual gauge theories by observing that the 5D Kaluza-Klein manifold is a $\mathrm{U}(1)$-principal bundle over a 4D spacetime, explaining why a 4D gauge theory might be expected to come out of it.

  4. A "program similar to Kaluza-Klein" exists, it is called string theory, and the compactification processes that are used there to obtain 4D theories from the 10D superstring theories are, in essence, nothing more than higher-dimensional versions of the circle compactification of Kaluza-Klein.

  5. There are gauge theories which are not Yang-Mills theories. For instance, the action of Chern-Simons theories is a topological action which is not the usual Yang-Mills action. More generally and from an entirely different point of view, one might call a gauge theory any description of a system which has unphysical degrees of freedom in the phase space - commonly called a constrained Hamiltonian system - where the constraint algebra closes on the constraint surface to form a Lie algebra, which is then the algebra of the gauge group.

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