"Virtual" particles

Lockdown reading

My lockdown reading list consists of just one book. This might last a long time I thought, so it's my opportunity to make a 2nd stab at understanding Quantum Field Theory. Last time, I bought "Quantum Field Theory for the gifted Amateur" and I learned a lot from it. Mainly that I am not gifted! I got three chapters through it and then gave up on the book, and on quantum field theory.

This time round I did my research better and found a much more gentle book: Student Friendly Quantum Field Theory, by Robert D. Klauber. It covers the same material, but takes pains not to lose the reader, by spelling out every ambiguity and subtlety. I'm half way through and feeling quite chuffed with myself. Here I am studying hard, on a sunny day in Lockdown Britain:

No, the weights are not mine

In this book, and every other in the QFT literature there is a concept of some particles being virtual. What's this about?  Why are some virtual and others not?  Before answering that we need a quick intro to the theory.

Quick intro

Like any other useful theory, QFT has testable outputs.  In the case of QFT the testable outputs are probabilities.  The theory allows you to calculate the likelihood of getting a particular outcome of an experiment given some starting conditions.   How do you use QFT do this?

Feynman diagrams

All you need to do is come up with a narrative for how the outcome might be achieved.  Here's an example of a starting condition and an outcome: you start off with an electron $e^-$ and a positron $e^+$ and end up with a muon $\mu^-$ and anti-muon $\mu^+$.  And here's an example of a narrative for how that outcome can be achieved.

Figure 1

Narratives like the one above are known as Feynman diagrams, and in them space is represented by the vertical axis and time by the horizontal one.  So the positron and the electron come together, annihilate, generate a (virtual) photon, which then decays into a muon/anti-muon pair.  Note that in Feynman diagrams the direction of motion for anti particles like the positron and anti-muon is represented by a backwards pointing arrow!

Each Feynman diagram is associated with what is known as a probability amplitude.  This is a complex number $z$ whose magnitude squared $\lvert z \rvert^2$ is the probability we are interested in: the probability that the initial conditions lead to the final conditions.  A large part of QFT is concerned with how, exactly, you work out these probability amplitudes, and it's complicated by the fact that it depends on a bunch of things like the spin and momenta of the initial state particles, the spin and momenta of the final state particles, and the polarization and momentum of the photon.  But the diagram conveys the gist of it pretty well.

However, there's never just one narrative,  For example, you could also explain the outcome with a Feynman diagram like this one:

Figure 2

... in which the photon is replaced by a Z-boson.  Or this one ...

Figure 3

 
... in which the photon decays half way through into a muon/anti-muon pair , which recombine to generate a second photon.

So, how do you get the probability for the particular outcome (muon/anti-muon pair) given the initial state (electron/positron pair)?  Intuitively one would expect the answer to be that you sum over the probabilities for each narrative.  However, the odd thing that QFT tells you to do is to find the sum of the probability amplitudes for each narrative, and then obtain a probability from that by squaring it's magnitude.

To illustrate the difference, imagine that the probability amplitudes for the first, second, and third diagrams are $\sqrt{p}$, $\sqrt{p} e^{\frac{2\pi i}{3}}$, and $\sqrt{p} e^{\frac{-2\pi i}{3}}$, respectively.  This means the probability (note, not probability amplitude) corresponding to each diagram is the same number $p$.  If you had to add together probabilities for each diagram you would get the result that the probability for getting the particular outcome given the initial state is $3p$.  (Or, rather, at least $3p$, since there may be other narratives.)  However, since you have to add together the probability amplitudes first and then square the magnitude, the answer is $\lvert \sqrt{p} + \sqrt{p} e^{\frac{2\pi i}{3}} + \sqrt{p} e^{\frac{-2\pi i}{3}} \rvert^2 = \lvert 0 \rvert ^2 = 0$.  What this says is that although each individual narrative has a non-zero probability they add together in a strange way to produce an overall probability of zero.

Note that we obtained a probability of zero in the toy example above because the phases of the 3 probability amplitudes all pointed in different directions.  Had they all pointed in the same direction we would have got a result of $9p$!  These are examples of destructive and constructive interference.  Reflection of photons by a mirror gives a good example of both of these phenomena.  If a photon is emitted from a bulb and received by a camera pointing at a mirror then there are infinitely many narratives which could account for the transition from the initial to final states, corresponding to all the different places on the mirror that could have reflected the photon.  However, the phase of each probability amplitude is proportional to the flight time and so varies from one narrative to another.  This means that most of the narratives cancel out, leaving just a handful where the flight time is minimized.

Virtual particles

Having laid down the groundwork we can finally get to the issue of "virtual" particles.  In Figure 3 above there were some particles which weren't part of either the initial state $\lvert e^+e^-\rangle$ or the final state $\lvert \mu^+ \mu^-\rangle$.  Specifically there was a photon, a muon/anti-muon pair, and another photon which, in a sense, never saw the light of day.  These particles are known as "virtual" particles. But why not just call them "particles"?

Mathematically the "virtual" particles are created by the same mathematical operators, from the same input states, as their non-virtual counterparts.  If I could physically represent the Hilbert space of all quantum states I wouldn't be able to lean in and pick out a state representing a muon and anti-muon $\lvert \mu^+\mu^-\rangle$ and then pick out a different state representing a virtual muon and virtual anti-muon. So, what is it that motivates the prefix "virtual" for some of the particles in the Feynman diagrams and not others?

One answer is that virtual particles don't conserve energy and momentum, relative to the particles they are created from in the Feynman diagram.  (Although they may conserve energy and momentum relative to the particles in the initial state.)  In real world experiments the total energy and momentum going in always matches that coming out.  For this reason, if you cut Figure 1 in half you get a diagram with amplitude zero, even though the amplitude for the whole diagram is non-zero.

However, I think it goes deeper than that.  Calling the particles "virtual" suggests that actually this whole Feynman diagram thing is really just a mathematical trick and not in any way an interpretation.  Unfortunately no alternative interpretation is offered up.  Instead the whole difficult subject is just ... avoided.  (And many of the current generation of physicists like it that way.)

Whatever the nature of virtual particles is, the fact that there are a superposition of histories for each observed outcome is, I believe, a real phenomenon and not a mathematical trick.  I say this because this is what the Schrodinger equation tells us.  It also tells us that the future is a superposition of states, and in fact the combined probability amplitude for each outcome is merely the weight for that state in the superposition.  The premise that the state "collapses" on inspection is an unnecessary adjunct.  If we ditch that we're left with a nice deterministic theory, instead of one in which an unknown hand picks an option from a list of alternatives at unknown intervals.  And this deterministic theory does not in any way contradict our subjective experience of QM being probabilistic because each of those final states includes a version of the experimenter, to whom the result looks random.