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I'm studying the Higgs field and I encountered some problems. My book says that the Higgs field has four components which are conveniently arranged into a two-component vector as: $$\phi=\binom{\phi^{+}}{\phi^{0}}=\frac{1}{\sqrt{2}} \binom{\phi_{3}+i\phi_{4}}{\phi_{1}+i\phi_{2}}$$ My question is that why does Higgs field have 4 components? If it has 4 components, why don't we just write it like: $$\phi=\begin{pmatrix} \phi_{3}\\ \phi_{4}\\ \phi_{1}\\ \phi_{2} \end{pmatrix}$$ A more genral problem I've been having is that how do I know if something can be written in a multiplet and what kind of multiplet should I use to represent a state?

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To encapsulate the SU(2) properties of the Higgs field, it is convenient to choose a 2D representation, since the fundamental representation of SU(2) is of that dimension. This is how you get the complex 2-component vector, which means we have two independent complex fields. This is 4 independent real fields.

In your suggestion of a 4-component vector it would be harder to write down matrices for SU(2) transformations.

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  • $\begingroup$ So how do I know that the Higgs field is invariant under $SU(2)$? $\endgroup$ – LY3000 Jul 27 '16 at 8:17
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    $\begingroup$ @Daniel a field can't be invariant under SU(2). A Lagrangian can. And SU(2) is one of the fundamental symmetries we observe in nature, particularly when considering the weak force (W+, W- bosons). Their Lagrangians are invariant under SU(2) transformations. $\endgroup$ – Wolpertinger Jul 27 '16 at 8:28
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If you were to read Frank Close's amazing history, THE INFINITY PUZZLE, it details the torturous discovery of the Higgs, (et al.) mechanism. So, why does the Higgs field have 4 components? Why are they arranged in a 2x2 matrix? I think because it allowed physicists to prove that the electroweak theory is mathematically consistent; to show why the W & Z have a mass; and to show why the Higgs Boson exists at all. The point is, we observe nature, but at first we don't understand it. Then, Physicists work like crazy to "figure it out". A theory that explains observations also predicts something crazy no one has every seen...like the Higgs Boson. Group theory was a big idea, and ultimately, SU(2) solved the problem of the weak interaction. But it was certainly LESS than obvious to even the greatest minds. There's a good reason it's hard to understand--you're asking a question that the legendary Richard Feynman grasped but couldn't answer! This was not a problem for some Eisenstein working in solitude. Many physicists labored many years and went down many theoretical dead-ends. My recollection is that it took (among many others) Nambu, Gell-Mann, Yang, Mills, Schwinger, Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble, Glashow, Salam (dubious), Weinberg, Ward (unsung hero) and a grand finale by 't Hooft, to understand Gauge Invariance, the Higgs field, spontaneous symmetry breaking, the weak interaction and its renormalization. Jeffry Goldstone threw a monkey wrench into the process of understanding with his theorem that predicted unobserved (and unwanted) Goldstone Bosons implied by theories of electroweak spontaneous symmetry breaking. But after much consternation, it turned out that Goldstone Bosons observable in the breaking of global symmetries, are NOT directly observed in the spontaneous breaking of LOCAL gauge symmetries. In the weak interactions, the Goldstone Bosons combine with mass-less gauge bosons (the W and Z) to create a zero-momentum minimum energy--that is "mass". This effectively HIDES the Yang-Mills gauge invariance we would expect to observe if the Higgs 4-component field did not exist. Much pain & suffering was required to give birth to the concept of the 4-component Higgs Field! But before that, infinite probabilities cropped up when earlier physicists, like Feynman, tried to introduce a mass term by hand into the normally mass-less Yang-Mills theory (which had so beautifully described the local U(1) symmetry of the electromagnetic field). In summary, a 4-component 2x2 complex-valued matrix is used to represent the Higgs Field because...it works: 3 components lend a minimum energy to 3 bosons, and the 4 component is observable by itself as the Higgs Boson.

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