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As far as my knowledge goes, Higgs field is only currently formulated in term of classical gauge theory. What is the importance of Higgs field being formulated in term of quantum gauge theory? In other words, why do we need to formulate Higgs field in term of quantum gauge theory?

Reference: "However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of quantum gauge theory, though somebody treats it as sui generis a condensate by analogy with that of Cooper pairs in condensed matter physics."

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    $\begingroup$ [citation needed] (In other words, I have no idea what you are asking, because the Higgs is only formulated in terms of quantum field theory) $\endgroup$ – ACuriousMind Jul 28 '14 at 14:19
  • $\begingroup$ I know that Wikipedia is a bad reference, but: en.wikipedia.org/wiki/Higgs_field_(classical) "However, no adequate mathematical model of this Higgs vacuum has been suggested in the framework of quantum gauge theory, though somebody treats it as sui generis a condensate by analogy with that of Cooper pairs in condensed matter physics." $\endgroup$ – user56220 Jul 28 '14 at 14:24
  • $\begingroup$ i think the wiki page is in need of attention. $\endgroup$ – innisfree Jul 28 '14 at 14:30
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    $\begingroup$ Okay, that's weird. I suspect that the author of that article (which seems to be the same guy having written a lot of stuff on "gauge gravitation theory") has an understanding of "classical gauge theory" vs. "quantum gauge theory" that is a bit different from what most others would say. However, since the Higgs field appears in the Standard Model formulation, it is part of a quantum theory already, and I've no idea what the author might mean. $\endgroup$ – ACuriousMind Jul 28 '14 at 14:31
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The Higgs mechanism has been understood in the framework of quantum field theory since the very beginning i.e. since the 1960s. Quantum field theory may be constructed as a quantization of its classical limit, i.e. of a classical field theory, so that's what's important for understanding some basic properties of the Higgs mechanism, too. But many other things, such as the existence of the Higgs boson itself, require one to study the quantum field theory, not just the simpler classical field theory. Classical field theory doesn't really imply the existence of any particles. To have particles described by fields, one usually needs quantum field theory, too.

The Higgs mechanism is a mechanism in quantum gauge (field) theory using a scalar field whose vacuum expectation value breaks a gauge symmetry which makes some gauge bosons massive. By definition, the Higgs mechanism is inseparable from quantum field theories and from gauge theories. Everything that is called "Higgs something" in physics is inseparable from gauge theories (a subset of quantum field theories), too.

The importance of all these things in particle physics (and even condensed matter physics, sort of) is very high.

Concerning the Wikipedia page: in condensed matter physics, the objects that break the symmetry are often composite – Cooper pairs (of electrons) in superconductivity. All available evidence in particle physics indicates that the field breaking the electroweak symmetry of the Standard Model is elementary, not composite. Some substructure may ultimately be found but there is absolutely nothing wrong or incomplete about elementary scalar fields – at the level of quantum field theory – and the Wikipedia page sentence was entirely incorrect.

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  • $\begingroup$ Can you have a look at my edited question? $\endgroup$ – user56220 Jul 28 '14 at 14:25
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    $\begingroup$ Yup, I have erased the insanely wrong sentence from the Wikipedia page. I am sure that the same incompetent "author" will try to restore it but what I can do. $\endgroup$ – Luboš Motl Jul 28 '14 at 14:54

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