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?
• I am not sure about better tag (I think this one is fine), but someone should edit the question to "What does a Field Theory mean?" or "What is meant by Field Theory?". – Marek Nov 15 '10 at 18:29
• I've heard that Zee's book "Quantum Field Theory in a nutshell" is an excellent introduction to (quantum) field theories, but there are many other books out there. – j.c. Nov 16 '10 at 0:16

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.

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!

• You mentioned two pictures. Are there ways to transform one picture to another? – Pratik Deoghare Nov 20 '10 at 13:30
• @Charmer: yes. In general this works kind of like a transformation of coordinates. Analogously to the way you can change from cartesian to polar you can also change from field to particle (this is very roughly said) and vice versa. To make it a little bit more precise: consider quantum mechanical harmonic oscillator. You can either investigate the wave function or you can investigate its energy spectrum. The first case corresponds to (0+1)-dimensional field theory (via Feynman path integral) while energy levels correspond to number of particles in the system. (continued) – Marek Nov 21 '10 at 22:20
• Similar ideas can be carried out for (3+1)-dimensional QFT and you'll obtain a descriptions which tells you how many particles there are at any given point of the space. So this is one (more or less) precise correspondence. Another correspondence is via lattice description of the space: you create some lattice on the finite volume of space. Let us suppose the chart contains $N$ points. You can then approximate the field description by this mechanical $N$-DoF theory. By letting the lattice size go to zero and the size of the volume to infinity, you recover the field theory. – Marek Nov 21 '10 at 22:26
• @centralcharge QFT does not always refer to relativistic QFT. – DanielSank Apr 1 '18 at 7:33
• It would be nice to see this answer include the mathematical definitions of fields and vector spaces. – Galen Jan 17 at 21:55

A field theory is a physical description of reality in which the fundamental entities are fields, i.e. objects having no definite spatial location but a certain value or intensity at each place.

Examples of fields are the temperature in a room, for each location in the room, a temperature can be specified, although in most cases temperature will be pretty uniform, unless for instance if you just turned on a heater, then there will be a temperature gradient.

The gravitational field in Newtonian mechanics is a description of what the force of attraction on a test particle is as generated by a large mass. This field is vector-valued.

Another example of a vector-valued field is the velocity field in a fluid. It gives the velocity of each infinitesimal piece of fluid at some instant t.

The electromagnetic field is specified by giving the value of an antisymmetric rank-2 tensor at each space-time location.

The novelty of quantum mechanics with respect to classical mechanics is that it has to incorporate the discreteness of the action. That's what we call quantizing. In particular, this means that energy will be quantized under certain circumstances (namely due to some boundary conditions or potentials limiting the amount of possible states). In classical mechanics, systems typically have a finite amount of degrees of freedom. For instance, the 1D harmonic oscillator has two degrees of freedom, the position and the momentum of the oscillator. In quantum mechanics, the energy of the oscillator (which is a combination of position and momentum) becomes quantized.

Quantum field theory takes this one step further, instead of quantizing systems with a finite amount of degrees of freedom, it tackles systems with an infinite amount, in other words fields. The way non-interacting fields are quantized is reminiscent of the way the harmonic oscillator is quantized except you now have an infinite amount of oscillators. This brings with it a lot of technical complications.

A non field theory is a theory where effectively there are fixed number of point particles or rigid bodies. A field theory is a theory where there are so many particles or body that they form a density or distribution.

Let me illustrate with a simple example. Take a given electric field in a one dimensional space. Imagine two point charges. To determine the motion of these two charges, we calculate the force from the electric field and the force between the two charges. The energy for example is the sum of the energies of the particles plus the sum of the interaction energies between the particles.

Imagine the same thing, but now with a billion charges. Instead of treating the charges as points, we treat them as one entity, the charge density. The energy is integral of the energy at each point due to the charge there, plus the (2 dimensional) integral over the interaction energy between the charge density at $x$ and at $y$.

So for non field theories, the object of interest are discrete whereas in field theories the objects of interest are densities or as we call them fields.

One particular point of interest in this difference. For a discrete field, the particles are fixed. You will always have $n$ particles each with a fixed charge. For the particle number to change dynamically you have to introduce some mechanism for it to change.

Looking at the line charge example, with one electric field, the charge density in most regions is zero i.e. the negative and positive charges cancel, but in one spot there is a net positive charge. Change the electric field and you might get overall charge mostly zero but two regions where the charge is positive and one where the charge is negative. So the total charge is the same but there is a larger total positive charge and a larger total negative charge. This is effectively particle creation, which you get for free in the field theory.