quantum mechanics - What does a Field Theory mean?

What exactly is a field theory?

How do we classify theories as field theories and non field theories?

EDIT:

1. After reading the answers I am under the impression that almost every theory is a field theory. Is this correct?


2. What is difference between QM and QFT?

Answer

Update to address new questions.

1. 
   

   The answer to this question is no. At least if you take the question purely formally. Only theories such as classical field theory, quantum field theory and continuum mechanics are field theories (you generally recognize them by having continuous degrees of freedom; also they usually have the word field in the title :-)). But physically, lots of different theories may be equivalent, or may be approximations of some other theory, so there are many connections among them (this is the point I was trying to illustrate, but maybe I overemphasized it).

   
   


2. 
   

   Difference between QM and QFT is essentially the same as between classical mechanics and classical field theory. In the mechanics you have just a few particles (or more generally, small number of degrees of freedom), while fields have an infinite number of degrees of freedom. Naturally, field theories are a lot harder than the corresponding mechanics. But there is a connection I already mentioned: you can see what happens when you let the number of particles grow arbitrarily large. This system will then essentially behave as a field theory. So in a sense, we can say that field theory is a large $N$ (number of degrees of freedom) limit of the corresponding mechanical theory. Of course, this view is very simplified, but I don't want to get too technical here.

   
   

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Field theory is a theory that studies fields. Now what is a field? I suppose everyone should be familiar with at least some of them, e.g. gravitational or electromagnetic (EM) field.

Now, how do you recognize that object is a field? Well, essentially, you look at how complicated the object is. To make this more precise: main objects of study of classical mechanics are point particles. All you need to keep track of them is just few parameters (position, velocity). On the other hand, consider the EM field: you need to keep track of the data (electric and magnetic field vector) in every point of the universe, so there is infinitely many parameters of this system! This is what I meant by system being large: you need a lot of data to describe it.

Now, it might seem that something is amiss. You do need a lot of data to describe real objects (just think of how many atoms there are in the grain of sand). So are ordinary objects fields? Yes and no, both answers are correct depending on your point of view. If you consider a massive object as essentially being described by few parameters (like center of mass velocity and moment of inertia) and completely ignore all information about atoms then it's clearly not a field. Nevertheless, at the microscopic level, atoms wiggle around and even the grain of sand really is as complicated object as any EM field (not to mention that atoms themselves produce EM field), so it's certainly correct to call them that.

Now let us see where our definition of field takes us. Let's talk about quantum mechanics for a while. What about two quantum particles? Is it a field? Well, clearly not. What about three? Still not. And what if we keep adding particles so that there will be a huge number of them? Well, it turns out that we'll get a quantum field! This is precisely the correspondence between e.g. photons and quantum EM field. You can either look at EM field as being described by vector of electric and magnetic field at every point as in the classical case, or you can instead reorganize your data so that you keep track of what kind of photons you have. It's useful to carry both pictures in head and use the more appropriate one.

There is also a subject of continuum mechanics. There you can also start with particles (describing atoms in some real object, e.g. water) and because there are so many of them, you can again reorganize your data, consider the object as being essentially continuous (which real objects surely are at least unless you look at them with a microscope), and instead describe them by parameters such as pressure and temperature at every point.

To summarize: the field theory is essentially about dealing with large objects. However, when we are looking at the problem with particle hat on, we usually don't say it's a field. For instance, when describing real objects as consisting of atoms, we are usually talking about statistical mechanics, or condensed matter physics. Only when we move to the realm of continuum mechanics, we say that there are fields.

There is much more to be said on the topic but this post got already too long so I'll stop here. If you have any questions, ask away!