Hi everyone. With lots of exciting successes with the LHC startup, I thought it would be good to teach everyone a bit about Feynman diagrams. These are the funny squiggly lines that one will often see on particle physicists’ chalkboards (or whiteboards if they do experimental physics…) that describe what’s going on when particles interact. The diagrams are very simple to draw and can actually be interpreted very straightforwardly, but like many things there’s a lot of very elegant physics going on “under the hood.” Thus, in order to build a bit of foundation for that, I’d like to discuss something even simpler: the double slit experiment.
Actually, we’ll do even more: we’ll do the triple, quadruple, and infinite-slit experiment! Take that quantum mechanics textbook! I’ll then discuss why this is the basis for Feynman diagrams. This discussion (and the images) borrows heavily from Tony Zee’s excellent textbook, Quantum Field Theory in a Nutshell (there will be a new edition next spring).
I’m not going to discuss “wave-particle duality,” the idea for which the double split experiment is often invoked. Those who are unfamiliar with this can look it up in popular physics books or on Wikipedia, but it won’t be necessary for our purposes. In a double-slit set up, a photon travels from some point A to some point B. In between those points however, is an impenetrable barrier that has two slits (S1 and S2) cut into it, allowing the photon two paths to get to the point B.
You might ask about why the photon’s path has a kink in it at S1 and S2, since it seems strange that it takes a bent path. My answer: don’t worry about it, this is quantum mechanics: weird things happen. More precisely, the probability for the particle to go from A to S1 to B is a well defined and non-zero quantity.
The point of the set up is this: we’ve constructed a system with a well defined initial state (photon at point A), a well defined final state (photon at point B) and well defined intermediate states:
- The photon goes from A to B via slit S1 or
- The photon goes from A to B via slit S2.
The rules of quantum mechanics tells us that if all we can measure are the initial and final states, i.e. we can’t tell which intermediate process occurred, then the physics of the process is determined by the “sum” of both possible intermediate states. Now you might ask what I mean by “sum.” Without going into the details, quantum mechanics assigns a [complex] number to each intermediate process. By summing these numbers for unmeasured possible intermediate states we get a number associated with the entire process. We call this number the probability amplitude. It’s not important how we determine these numbers; what is important is that the square of the amplitude is the probability for the initial state to turn into the final state, i.e. a rate that we can measure in the lab. This is the actual “result” of the double slit experiment, though the actual experiment isn’t important for us right now! All you need to understand is that these paths are summed.
Since the double slit experiment is somewhat abstract, an example is in order. Consider Uranium: this is a radioactive element made up of 92 protons and a bunch of neutrons (140-ish, depending on the isotope). Uranium will emit an alpha particle (two protons and two neutrons) to decay into thorium. Which two protons and neutrons did it emit? That’s not measurable (all the protons and neutrons are identical). So what’s happening is that the uranium atom is taking all possible paths to becoming a thorium atom, summing over all possible pairs of protons and neutrons to emit.
Ok, good. Still with me? So far we’ve been talking about quantum mechanics. We’d now like to transition to quantum field theory, or relativistic quantum mechanics. Now that you understand how the double slit experiment works, I can ask you the following clever question:
Well… what if the wall had not just two slits, but three slits?
The answer is easy: you have to sum over all three possible paths. If you had four slits, you’d have to sum over four paths. Simple!
Ok… now what if you had two barriers, each with some number of slits? What if there are three barriers?
Alright, now it gets harder to keep track of the combinatorics of how many paths we have to sum over… but we know what the heuristic answer is: the photon takes every possible distinct path, and so we have to sum over all possible combinations of slits from A to B. Here’s a nice picture:
Indeed, it’s starting to look messy, but you can see what’s going on. In this set up I say that the photon can go from A to B following paths A-Sa1-Sb1-B, A-Sa1-Sb2-B, A-Sa1-Sb2-B, … and so forth for all twelve paths. It’s clear that I can add more barriers and add more slits as much as I want. The number of possible diagrams gets huge, but the principle remains the same: sum over all paths!
Now here’s the key step: suppose I have an infinite number of barriers, each with an infinite number of slits. What would this look like? This would look like there are no barriers at all, just an infinite number of paths from A to B. This is really neat. The corresponding diagrams would then look something like:
If this were The Matrix the appropriate response would be a Neo-esque “Whoa.” Or maybe, “There is no spoon.” But the principle always remains the same:
If a system is observed in state A and then in state B [with no intermediate observations], then the system must take all possible intermediate states between the two.
This is known as the “sum over states” or “sum over all paths.” Those with some calculus background know that a continuous sum is called an integral, so this is also called the path integral formulation of quantum mechanics. It was pioneered by Richard Feynman and is, in fact, the topic of his PhD thesis. This is intimately tied to the “principle of least action” in classical mechanics.
Alright, where has this taken us? The above principle is one to keep ingrained in the back of your mind whenever you think about particle physics. For example, “state A” could be two protons barreling towards each other at the LHC and “state B” could be two highly energetic electrons passing through the CMS detector (along with some other “hadronic junk”). The principle above tells us that the protons must take all possible paths to go from state A to state B. What are “all possible paths”? Certainly some of these paths are boring: the quarks in the proton can interact via the W and Z bosons to produce energetic electrons. This is well-understood physics from earlier experiments. There could be, however, other paths that contribute. For example, maybe the quarks interact via a new particle state which later decays into electrons.
This is a generic “problem”: most signals for new particles don’t actually involve the new particle passing through our detectors! They tend to be heavy and so decay rather quickly. The point is that since we understand the “boring” physics, we can calculate the expected probability that we’d observe state B. We can check this against the actual measured rate of state B at our detector to see if the two results match. If not, then we suspect that there are more paths that need to be summed, i.e. possibly new particles. Here’s an example of a null-result from the D0 detector in Fermilab:
This is a plot of “dielectron” events, i.e. “State B” is two energetic electrons as we used in our example above. (State A is a proton anti-proton pair colliding at 2 TeV.) On the x-axis we have the “invariant mass” of the electrons. Each x value is a distinct final state B. The precise meaning of this number isn’t really important at the moment, but if there’s a bump at some value it means there’s a particle at that mass. Thus the bump in the picture above represents the Z-boson (mass = 91 GeV). The area under the red line (the brown, green, and white space) are what we would expect from adding together known Standard Model “paths” from A to B. The blue dots are data. You can see that the data agrees very well with the Standard Model prediction, and so we don’t see any signature of new particles.
Ok great. If you were only interested in looking at experimental plots, then this summarizes the main points. But if you’re more theoretically inclined, you’re wondering what I really mean by “paths.” In the double slit experiment, it was clear that a path was a physical trajectory through space and time. When I’m talking about colliding particles, what is the “path” from two protons (state A) to two electrons (state B)? This sum over paths becomes a sum over possible interactions. Feynman came up with a very nice diagrammatic way of understanding this sum over paths, and that’ll be the topic of an upcoming post.
– Flip
