[This post is a continuation of this one from Monday]
Coming to Terms
Before we continue, a little terminology — trivial, yet crucial and slightly subtle.
Think about the distinction between the words “humanity” and “a human” and “humans”; or “higher education”, “university” and “universities”; or “royalty”, “king” and “kings”. In each case, the three words refer to the general case, the specific case, and a group of specific cases. Sometimes you even have to use context to figure out whether you’re dealing with the general case or a group, because “humans” or “kings” is sometimes used in place of “humanity” or “royalty”.
In a similar way:
- The words “quantum field theory” (with no word in front) refers to the general case: the general type of equations that we use for particle physics and many other fields of study.
- The words “a quantum field theory” refers to a particular case; an example of a set of equations drawn from the general case called “quantum field theory”.
- The words “quantum field theories” refers to a group of particular cases: a set of examples, perhaps ones that share something in common. We might talk about “some quantum field theories”, or “all quantum field theories”, or “a few quantum field theories”.
Before I go on, I should probably point out that string theory/M theory is different, for a weird (and perhaps temporary) reason — it appears (currently) that there’s only one such theory that has fully consistent mathematics and contains a quantum version of Einstein’s theory of gravity. That was figured out in the 1990s.
However, in that theory you need another, similar distinction. Let me postpone a proper definition until later, but roughly, a “string vacuum” is a particular way that string/M theory could manifest itself in a universe; if you changed the string vacuum that we’re in (says string theory), we’d end up with different particles and forces and somewhat different laws governing those particles and forces. Many such vacua would have particles and forces described by a quantum field theory, along with Einstein’s gravity; many (perhaps most) other vacua might not have this property. And so, as before, we need to distinguish “string vacua in general”, “a string vacuum”, and “string vacua”, plural.
Now it’s time to talk about quantum field theory — the general case — in more detail.
Quantum Field Theory and Its Predictions
Does quantum field theory as a whole predict anything? In short, if you knew a universe had physics described by a quantum field theory, but you didn’t know which particular quantum field theory, what could you predict?
The answer is: almost nothing.
If you’ve been following this blog with care, or have read some books or articles about particle physics, you might well think: quantum field theories have fields; and fields have particles; so here’s a prediction: there should be some particles of some type.
Nope. You’ve learned your lesson well — everything I’ve told you on this website suggests that it is true — but the conclusion is false. I’ve been white-lying to you this whole time, and I have to apologize for that. Quantum fields do often have particles. But many quantum field theories are scale-invariant. A scale-invariant thing looks and behaves more or less the same in a microscope no matter how strong a lens you use. The wikipedia article on scale invariance actually has a nice animation of a scale-invariant process. Another example of something scale-invariant is a fractal. And in a scale-invariant quantum field theory (except for one in which there are no forces at all), there are no particles. I’ve told you that particles are well-behaved ripples in fields… well, it’s true, but in a scale-invariant quantum field theory with at least one force, any such ripples die away and turn into several ripples, which die away and turn into several ripples, which die away and turn into several ripples, which die away and turn into several ripples, which…
In short, no particles. This has been known for many decades (I’m actually not sure how far it goes back in the form I just described.) We observe this behavior in experiments — not in particle physics, where it isn’t relevant (at least currently), but in many solids, liquids, electrical conductors, magnets etc. that are undergoing “phase transitions”, such as melting, or spontaneous magnetization. In some cases we can also calculate this behavior, exactly or approximately. In some cases we can see this behavior emerge in computer simulations of real or imaginary materials. So we know there are many quantum field theories that don’t have particles.
[This might make you wonder if there are string/M theory vacua that don’t have strings. Good thinking… though actually there are other, unrelated ways to end up without strings… more on this later in the month.]
But If There Are Particles… A Prediction!
But here’s something important that we can predict. If we are studying a quantum field theory that does have particles (now we are talking not about all quantum field theories, but an interesting subset of them) then
- the particles will come in types (in our world, electrons are examples of a type of particle, Higgs particles are an example of another, etc.); and
- two particles of the same type will be identical in all their intrinsic properties. They will have the same electric charge, spin, mass [i.e. rest mass] etc.; if somebody swaps one for another, you won’t be able to tell the difference. (In our world, all electrons are identical.)
- Furthermore, in a world with three (or more) spatial dimensions in which Einstein’s special relativity is true, then particles of each type will be either fermions or bosons. (The Pauli Exclusion Principle, which determines all of atomic physics and chemistry, is a consequence of the facts that all electrons are identical and that electrons are fermions.)
Again, these are general prediction, not of quantum field theory as a whole, but of the subset of quantum field theories that have particles in the first place. The Standard Model, the quantum field theory that seems to describe much of our world very well so far, is an example of one that has particles.
That these are the most important predictions of quantum field theory with particles was pointed out to me, when I was just out of graduate school 20 years ago, by none other than Freeman Dyson, who helped develop quantum field theory back in its infancy.
Moving Toward Particular Quantum Field Theories
Now, what about specific quantum field theories. What can we predict about them?
The answer, hardly surprising, is: it depends.
- In some quantum field theories, we can predict an enormous amount, by doing some straightforward type of calculation.
- In others, brute-force computer simulation of the quantum fields makes it possible to study the particles and forces described by that quantum field theory, and allows us to make some predictions.
- In some quantum field theories, we can predict a smattering of things very well, using fancy math methods (for example, using fancy geometry, or supersymmetry, or string theory).
- In still others… unfortunately, many of the most interesting… we have absolutely no idea what’s going on. Sometimes we have the equations we’re supposed to use, but none of our methods for calculation work for those equations. Occasionally we can make a good guess, but we can’t always check it.
- In yet others, we don’t even know what the equations are that we should use to start studying them. In fact, the existence of some of these theories was, until relatively recently, unknown. We didn’t even suspect they existed. Maybe there are even more that we still don’t know about.
I’ll start describing these categories of quantum field theories in my ensuing posts. Fortunately for particle physicists, the Standard Model is mostly in the first category, with a little leakage into the second.
The Standard Model: A Quantum Field Theory Of Particular Significance