This is the third part of a series of posts introducing the basics of quantum field theory, previous installments are here (part 1) and here (part 2).

Suppose we have a continuous medium, an elastic band or the surface of a pond, we can then have traveling waves propagating in the medium. Just throw a stone into a calm pond and observe those waves. As long as the equations governing the medium are linear, we can superpose solutions and create pulses: localized bundles of energy which look a lot like particles, almost. So, it looks like we can throw away the problematic point particle, replacing it by a special kind of field, the one that obeys linear equations. Those are called linear fields, or free fields. However, we are not yet there…

Two things seem at first to distinguish particles from wavepackets, the ones you form from free fields. First, the energy in a wavepacket can be anything at all, whereas particles come in discrete units. You can have one electron, two electrons, but not 2.3 electrons. Secondly, electrons are microscopic: they are tiny, almost pointlike, and individual electrons carry minuscule amount of energy. Waves are usually thought of as macroscopic phenomena involving lots of moving parts.

Both these issues call for quantization. When quantizing the waves of a free field we get a discrete, equally spaced spectrum. The problem is actually identical mathematically to the familiar harmonic oscillator. We are then free to call those excitations of the field particles: they carry small amounts of energy, and that energy comes in multiples of some basic quantity, which is then naturally interpreted as the energy of a single particle. This is the miracle of particle-wave duality: quantization of waves in a linear field give rise to particles. We don’t really have to have particles as separate from fields, all we need is fields.

Suppose now the equations governing the motion of the fields are not exactly linear. If they are nearly linear, we will get nearly the same picture as before: we will have excitations looking like one particles, 2 particles, 3 particles…but the energy is not going to be precisely equally spaced. In that case we can just say the particles are not precisely free: when putting a few of them together the total energy is slightly different than the sum of the individual energies. This is because they feel each other’s presence, they interact. As long as the energy associated with that interaction is small compared to that of individual particles, we have a recognizable picture of small particles moving about and interacting.

But, as before, fields are much more complicated and interesting than particles. In some circumstances, when their equations are linear, or nearly so, they can give rise to particles. In other circumstances they do not: think about the pond on a windy day, or on a cold January day in Ontario… If the fields are strongly interacting, or are under unusual circumstances, they exhibit a rich variety of phenomena, none of which could be interpreted in terms of particles. The study of QFT is the study of those phenomena.

on September 17, 2008 at 4:45 amandy.sI see these waves being referred to in some books as scalar, vector, pseudovector and pseudoscalar.

I’ve had some exposure to these concepts in Hestenes’ Clifford Algebra books. Are they one an the same?

on September 17, 2008 at 3:04 pmRobertWhat exactly do you mean by “When quantizing the waves of a free field we get a discrete, equally spaced spectrum”? When I quantize the free scalar field of mass m, with a single creation operator I can create states with any eigenvalue of the time evolution generator (aka the energy) greater or equal m.

Maybe you talk about eigenvalues of P^2. Then it’s true, with a single creation operator you get a discrete m^2, but again with two or more anything greater or equal than 4m^2 is possible.

But I admit that so far was nit picking. What I am more concerned about is that I believe that the particle-field relation is here once more being described as a property of a quantum theory. I am convinced that is a misconception. Strictly speaking, particles are properties of free fields (your linear field equations) and that notion extends to weakly coupled theories in a perturbative treatment (including quasi-particles of any sort, having those just means you have an effective description in terms of weakly coupled fields). I tried to take that point of view in the QFT class I taught a year ago. Especially, I tried to explain that when solving a classical field equation of motion you already have all properties of a particle theory with collisions and all that stuff. See my lecture notes here:

[The lecture notes disappeared with our old webserver. I should have them on my old laptop as well which sits at home. I will supply those later]

The basic idea is that classical fields are in some sense already quantum: There is no difference between a classical field and a first quantized particle. Often, the Klein-Gordon equation is introduced (wrongly as I think) as the relativistic version of the Schrödinger equation which is quantum and is supposed to show both particle and wave behavior. But of course you can as well see it as a classical equation just as usually Maxwell’s equations are viewed as classical equations and not as the wave function equations of the photon. the only difference is you only consider tree graphs.

In those lecture notes I describe in detail how that restriction comes about. But of course from a quantum path integral perspective this is obviously the leading order in h-bar and thus the classical contribution.

on September 17, 2008 at 3:50 pmMosheHi Robert,

You are correct about the nitpicking point: the spectrum of free fields when quantized is isomorphic to that of any number of free particles. That spectrum is discrete and equally spaced only when you fix the spatial momentum, say discuss particles at rest.

Classical wavepackets by themselves do not look like fundamental particle. For one things it is a matter of scales: to get something as small as an electron you get beyond the region of validity of classical mechanics and have to quantize the field.

I decided not to nitpick too much on first and second quantization, but maybe I should. What is conventionally called first quantization is a procedure that takes you from a classical particle to the corresponding classical free field. To call it quantization is a confusion in the same universality class as to call the Klein-Gordon equation a relativistic wave equation. In all that there is no trace of h-bar, the Planck constant, no probabilistic interpretation, or anything else associated with quantum mechanics. Those are what David called “legacy” terms, coined before we had complete understanding of how things are organized.

Let me also say that it would be good if a lot of the machinery of QM were taught separately: Hilbert spaces and path integrals for example are useful mathematical tools in the theory of differential equations, whether it is the Schrodinger equation, the Klein-Gordon equation, or even equations that have nothing to do with physics. Quantum mechanics is often confused with the mathematical machinery used to solve the Schrodinger equation, just because typically you learn both at the same time.

on September 17, 2008 at 4:05 pmMosheAndy: probably, I am not sure. Those words describe how things change as you rotate in space, could be waves or anything else.

on September 18, 2008 at 6:59 amRobertI uploaded the lecture notes to the new web server, here they are:

http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf

Not to confuse things too much (well maybe I do it still even more) I even avoid talking about wave packets and simply equate plane waves with classical particles. That must sound completely moronic from a localization point of view but only until you realize that for scattering problems you usually take the point of view of momentum space and then both have sharp momentum (that can even be represented by an arrow or associated to a line in a Feynman diagram).

on September 18, 2008 at 3:27 pmMosheThanks Robert, I agree that the plane wave viewpoint is cleaner, but maybe less intuitive at first. Also, I agree with your first paragraph: Feynman diagrams are tools for solving differential equations perturbatively, and are not specific to quantum mechanics or even to physics. Ditto for the path integrals, canonical quantization, Hilbert spaces, etc. etc. I find it interesting to think about what is special about quantum mechanics, after we take out the mathematical machinery needed to solve linear differential equations approximately. So far the only thing I see is the probabilistic interpretation of the wavefunction.

on September 18, 2008 at 11:10 pmDavid EdwardsHow’s the following for a definition of electrons?

The classical limit of the mathematically undefined QED is the mathematically well defined Maxwell-Dirac System which does admit of solitonic solutions which can be viewed as “classical” electrons. The “analogous” solitons in QED would be what electrons are.

on September 19, 2008 at 6:32 amHaelfix“So far the only thing I see is the probabilistic interpretation of the wavefunction.”

Yea, it really comes down to how you interpret the wave, moreso than any huge difference in the mathematics. Already in classical wave mechanics you can have superposition. The difference is here you take the square of that, normalize it and call it a probability.

Still there is perhaps more of a subtle difference from the point of view of the phase space. In Dirac quantization you replace the classical Poisson bracket with the Dirac bracket, however you also crucially deal with the constraints somewhat differently.

Another arguable difference is you are forced, at least for relativistic field theory, to deal with particle creation and can no longer make sense of simply a finite amount of fields.

on September 28, 2008 at 5:30 pmThe basic dichotomy of quantum gravity « Shores of the Dirac Sea[…] 28, 2008 by Moshe Last time I had time to write a detailed physics post, I got to the point of declaring fields as the fundamental entities of nature. Since we can get […]

on October 28, 2008 at 9:15 pmHow many dimensions? « Shores of the Dirac Sea[…] of my post about duality can guess what is coming next. There, I was describing how (some of the) particles in a field theory can be visualized as localized excitations of a continuous medium, and how that […]