I'm not sure if I can ask this question clearly, but anyway... We have this collection of 17 or so particles. Some of them have been known to exist for a very long time and were found when physics was still in its relative infancy, because they were trivial to detect and distinguish. I'm talking abundant, everyday particles like protons, electrons. Then we have the more 'fancy' particles like top quarks and tau neutrinos whose discovery came later. We also have this labrynthine body of mathematics that gets called the Standard Model, which I believe falls under the wider umbrella of being a collection of 'quantum field theories'. Every now and then when I get a chance, I sit down and chip away at some introductory QFT and QED material. However, I have not gotten deep into it enough yet to answer a question that I've wondered about it: Do you start off by guessing at the nature of your particle before fiddling around with a bit of mathematics & some physical laws, and seeing if some testable equation crops up? Or do you start off with the mathematics & the physics and find that lo and behold, some particle pops out that should have certain properties? The real reason I'm asking this is that I want to understand why each standard model particle (I'll worry about any extras later) should exist based on theoretical understanding rather than, for example, just postulating that something neutrino-ish must exist based on observing the energy spectrum of electrons in beta decays or whatever and determining its properties experimentally. Should I be able to look up derivations of every standard model particle? Or am I misunderstanding something?
We can't mathematically "derive" the existence of a particle. The proof of the existence of a certain particle must be experimental. The formulation of QFT starts from the need to include Special Relativity into Quantum Mechanics (you can already see that this automatically implies that we are assuming that some relativistic particles following the rules of Quantum Mechanics exist). From that you can derive a set of equations, fields, rules, etc. that allow you to describe the behavior of these particles and how they interact with each other. However, as I said, we are not deriving the existence of the particle itself by saying, for example, that the Dirac field has certain properties. I'll make you an example. The first field that one usually studies is the Klein-Gordon field which describes scalar, spin 0 particles. The Higgs boson is the first elementary particle observed with these properties. Therefore, up to a few years ago, we had this field but no observed elementary particle with its properties. However, there are cases in which you can deduce (or maybe guess) the existence of a certain particle by looking at inconsistencies in your theory when related to experimental results. I'll make you another example. At a certain point, experimentalists observed that a certain type of current (called Flavor Changing Neutral Currents (FCNC)) were highly suppressed. The theory developed by that time wasn't able to explain this suppression. Therefore three theorists (Maiani, Glashow and Iliopoulos) proposed the existence of a new particle, the charm quark, whose presence was just fine to explain the experimental result. Later on this quark was actually observed, however, as you can see, the prediction of its existence came from an experimental result. There is no equation that can tell you if a particle exists or not. I hope this answer your question.
It does seem as physics is composed of a long list of patches, I assume our theories are still incomplete, waiting for yet another patch, another particle...
Physics itself is not composed of a long list of patches (at least not a very long list). For example, the Standard Model of particle is actually in general very consistent and coherent. The point is that the discovery itself goes on by adding new patches, until at a certain moment all the patches make the whole sense. However, yes, we already know that the Standard Model is not the final theory but it's incomplete. Now, one usually imagine that this incompleteness comes from one missing piece (an unknown particle, ad unknown new interaction, etc.). However, in my opinion, there is nothing that ensures that the incompleteness is due to a missing piece. If you think about it, assuming that, for example, we need a new particle means to assume that the scheme we have been using so far it's still valid. This could not be the case. Maybe we need some kind of "breaking" between old and new theories, like the one happened between Classical and Quantum Mechanics, when the whole way of viewing physical phenomena change, in a way that was hardly predictable before.
QFT is unable to predict which particles actually exist. Why we have the set of particles that we do is currently a mystery, and we can only hope that some day a more fundamental theory will explain it. Currently we take the particle content of the Standard Model as given, and then work from there. However, a very attractive aspect of QFT is that it provides a well-defined and tightly constrained framework for constructing theories like the Standard Model. For example, there are only a few general types of particles that are allowed, classified by the particle's spin, and the interactions between these particles are constrained to take certain simple forms.
hmmm, so to clarify: Yes, I understand that "The proof of the existence of a certain particle must be experimental." You can have the most elegant-sounding theory ever but if you don't go looking for what it predicts then it's nothing more than a curiosity, and if you do go looking and find nothing then it goes in the bin. What I was getting at is, people don't just build experiments by accident, they suspect that if they go looking, they'll find this specific thing that they've painstakingly tailored their experiment towards searching for. Ie. It's motivated by theory? "There is no equation that can tell you if a particle exists or not." I agree, some squiggles on a piece of paper don't have that power, but they CAN tell you what might not be so surprising to find if you go looking, right? "I'll make you an example. The first field that one usually studies is the Klein-Gordon field which describes scalar, spin 0 particles. The Higgs boson is the first elementary particle observed with these properties. Therefore, up to a few years ago, we had this field but no observed elementary particle with its properties." OK, so, I'm familiar with the Klein-Gordon equation yeah. So the Higgs field can be plugged into that equation with pleasing results. Fair enough - but there was good reason to go looking for the Higgs in the first place, it was a particle that was expected to turn up. Where did that come from? It wasn't required to explain experimental discrepancies was it? It's been a long time since I attended any course covering it but vaguely I remember mexican hats and non-zero VEVs etc. Was a particle with some of its properties at least necessary to prevent other theories from falling apart, and if so, wouldn't that count as a theoretical prediction of sorts?
No, of course they don't. Take for example LHC. One of the most popular extension to the Standard Model is Supersymmetry (SUSY). There no proof that SUSY is right, but it's a beautiful, elegant theory, and it makes sense, therefore a lot of the LHC experiments were built keeping in mind that we where looking for SUSY. So, yes, we don't simply go blind, we have an idea of what the possible new theories are and we go search for their experimental consequences. Of course it could happen that we are going to find something completely unexpected and that no theorist predicted. That would be a pretty happy discovery! Absolutely yes. That's exactly, for example, what happened with the charm quark. It's basically the same thing. The need for the Higgs field arises from the need to provide elementary particles with a mass. This may not seems like that, but it's actually an experimental need. We empirically observe that each particle has a mass and therefore we need something that can produce this mass. The Higgs mechanism (i.e. the introduction of the Higgs field in the Standard Model) is just one possible idea to assign mass to particles. Many other possibilities were proposed and no one could have a priori said anything about any of them being the right one. The Higgs mechanism is just the simplest idea physicists were able to find. And it turned out to be the right one!
If Standard Model was to be a consistent theory, it had to have the Higg's particle within. The Higgs particle would be an excitation of the Higg's field that is responsible for massless particles to acquire a mass... Standard Model is a Yang Mills (the Lagrangian is one of Maxwell type) gauge theory* SU(3)xSU(2)xU(1) and for simplicity you can take the SU(2)xU(1) part... In such a theory, the gauge bosons are massless... Only after the SSB you can get the (experimentally confirmed) massive bosons of weak interactions. I am not sure whether there are alternative ways to give mass.... The only alternative is to break down the Standard Model group explicitly by adding mass terms to your gauge bosons and matter fields. But that's not a nice way (for me) since it is arbitrary and cannot explain how the 3 bosons acquire a mass while the 4th (photon) doesn't... (*There's a lot of geometry within this... The gauge group is the structure group G of a fiber bundle. The connection forms of a principle fiber bundle are identified as the gauge potential and the curvature forms are identified as the gauge fields. Intersections of the associative vector bundles are called mater fields. After a spontaneous symmetry breaking in a subgroup H of G, a universal intersection of the fiber P/H is called the Higgs field.)
I think there are. For example in the case of neutrinos they could be Majorana fermions and, in that case, they don't need the Higgs field to generate their mass. However, still remaining in the Higgs mechanism, there are many variant with more than just one Higgs doublet that still generate fermion and gauge masses. Of course they also predict a large number of other phenomena that are never been observed so far.
There are alternative ways to do the symmetry breaking. The Higgs mechanism is just the easiest one. Technicolor is another one, for example. And supersymmetry gives multiple Higgs bosons instead of just one. The Higgs boson is one of the examples that you can expect particles based on experimental results from other particles. The third generation quarks (top and bottom) are another example - they were predicted based on CP violation within the other quarks.