The dream of a particle theory of quantum physics may finally be realized
Adding a 5th dimension to the universe radically alters vacuum fluctuations
Richard Feynman published the first diagram that would bear his name ever after in Physical Review in 1949. It was a stick figure of particles interacting that looked like this:
This diagram describes two electrons exchanging a photon. The electrons are the solid lines on left and right. The photon is the wavy line in between them. This process is fundamental to how all forces work by exchange of what are called gauge bosons. The particle exchange creates a potential between the electrons that is equivalent to a āhillā in energy space between them, causing the electrons to move away from one another down the energy slope.
Part of the reason for creating these diagrams was, according to Feynman, to return discussion of quantum field theory, the relativistic theory of forces and matter developed through the 1930s and 40s, away from talking about ghostly field interactions and back to interactions between particles.
After all, we never directly measure fields, only particles. If you think about bringing two magnets together, you can feel them attract or repel one another depending on how their North and South poles are aligned. The magnetic field in between is real by virtue of the forces it creates, but you can only measure it by looking at the behavior of the magnets themselves. Fields are, themselves, almost like mathematical conveniences.
Feynman diagrams have been incredibly successful at both making predictions and, perhaps even more importantly, helping physicists understand what is going on in complex interactions between forces and particles. They are a language, all of their own, to talk about quantum experiments in a precise, mathematically formal sense.
Indeed, they are far from just being pictures. You can actually calculate predictions directly from the diagrams in a very turn-the-crank sort of way, building up contributions from particles to a particular experimental outcome and completely avoiding a lot of the infinite dimensional calculus that is inherent to quantum fields. Rather, you mostly get four dimensional integrals over space and time, which are, if not easy, easier.
One question I have about Feynman diagrams, having worked with them over the years, has always been whether they really stand for particles. Are they a particle interpretation of quantum field theory or not?
Adherents of that strict, particle-based interpretation of quantum mechanics, Bohmian mechanics, would say no. Bohmian mechanics doesnāt really have fields in it. Instead it has particles which are all being mediated by wavefunctions that give rise to quantum predictions. If that is confusing, imagine it like this: each particle is like an iron filing under the influence of some magnetic field. Everything we can detect and know about is created by those filings. And yet, the magnetic field guides them so that they form particular patterns. We never detect the magnetic field. Rather it is a kind of underlying probabilistic force, but at least everything from electromagnetic to gravitational forces all are just particles moving in the probability field.
So, why arenāt Feynman diagrams about particles? Itās because ultimately Feynman diagrams are a direct, diagrammatic translation of fields interacting. Hence, every particle in the diagram is actually a stand-in for a field that covers all of space and time. Those particles donāt exist anywhere or anywhen. They are simply representative of a particular kind of interaction. So, if I have an electron field that has two incoming electron particlesāāāreal particles that I know about. That electron field then interacts with itself via the photon field (also called the electromagnetic vector potential).
So, really, the diagram is a lie of sorts. Instead, what it means is that I am tossing two electrons into a fog of electronness. Somewhere inside that fog there is a photon fog causing electronness to interact with electronness via photonness. That I threw in the two electrons creates a disturbance that allows something else to emerge from the fog (two electrons) by conservation of charge and so on, but a particle theory it is not. It is a field theory that allows particles to exist sometimes.
Feynman eventually, with disappointment, acknowledged that his diagrams were not a particle representation of quantum field theory but an approximation method, and few saw a reason to turn their backs on field theory which was so successful in describing what happens in particle accelerators, the Sun, distant galaxies, and the Big Bang.
Feynman diagrams are more than just a description of quantum processes as well. You could create Feynman diagrams for any physical process based on field theory in fact, even classical. If you have a model for how water interacts with itself, for example, you could create diagrams for that. You would just show how fluid fields interact in your diagrams and all the ways they can combine.
Counter-intuitively, those diagrams would not represent how actual water molecules interact. Instead, they would show how infinitesimal points in a field of water interact. This idealized interaction might discard a great deal about real particles, retaining only what matters to the field theory.
Still, it makes you wonder if Feynman diagrams do represent particles in some idealized, smoothed out way, like fluid fields, and it is the field description that gives rise to them that is an approximation. After all, fluid dynamics experts regularly represent water or air as a continuum despite knowing that to be false. Might all these quantum fields be likewise just a mishmash of particles?
There is some difficulty with that approach because quantum field theory allows its particles to act like waves that have no particle nature. Indeed, if you take something like an electron, the quantum description for it includes an electron field that has electrons with all possible masses, but only one is a legitimate mass. (The measured mass is a function of energy level of your experiment as well, but, if you keep the energy level fixed, there is just one mass.) All the others, while contributing to the predicted outcomes of experiments, are called āvirtualā because they have no reality. Their correlations with each other are imaginary numbers. How can you have a particle that is an imaginary number?
This may not be that big a deal as it could just be the result of the odd hyperbolic relationship between time and space. Unlike ordinary statistics, quantum field theory is a 4-D statistical theory that includes time. This makes it a wave theory with oscillations and cancellations. Imaginary numbers tend to crop up in all wave theories including in communications and radar. These are certainly real. Partly, it is just mathematical convenience. Waves have amplitude, frequency, and phase. It is much easier to get rid of phase and replace it with amplitude and frequency in a complex plane than to deal with phase directly in the real numbers. (This is why we often represent signals as samples of in-phase and quadrature or I/Q data.) Hence, the imaginary numbers in quantum theory can be thought of as a convenience related to the existence of phase in waves.
It turns out that there are some bigger differences between water and quantum field theory. One of the biggest is what are called vacuum loops. These are when particles are created out of nothing and then destroyed but never detected. We know they are there because we can detect their outcomes.
For example, light that has high enough energy, in the Petawatt or Exawatt range as with ultrapower lasers, can produce electron-positron pairs out of the vacuum. These pairs create non-linearities in the light that cause it to refract (bend) through vacuum. This light bending can be measured as an effect of vacuum loops.
Vacuum loops appear in statistical processes (not only quantum but ordinary statistical mechanics) but not in classical, deterministic processes. In classical processes, Feynman diagrams always follow a tree structure with particles going in, interacting, and exiting. Nothing is ever just created in the middle and vanishes.
Partly because it doesnāt allow for fields, Bohmian mechanics has always had a problem with representing particle creation and annihilation, and the way of doing it in that theory, called a Bell-type field theory after John Bell, is hardly realistic, but more like something stolen from a computational algorithm. If it has trouble, how can we deal with it in a particle interpretation of field theory?
The answer lies in statistics.
The way you do it is to look at how loops appear when you average random processes over time. It turns out that loops are actually the result of statistical correlations between different parts of a field at different times. They arenāt real in the sense that particles are actually being created out of nothing. Rather, the uncertainty of the statistical process creates correlations that result in loops.
Hereās an example from one of my papers:
In this example, on the left side, you see three particles for a generic field coming together at a point y that is uncertain. You can think of the point y as being different at different times according to some random process. They are propagating between that and a point x1 that is known. Their interaction and propagation, meanwhile, is being correlated, statistically, to a fourth, separate particle that exists at another known location x2.
The result of this statistical correlation is something that appears to be a particle, propagating from x1 to x2 and interacting with a vacuum loop in between. The loop, however, doesnāt exist. It is just the result of averaging over many statistical configurations in which particles exist at different places, represented by y. At each moment in time, y has a particular value but, and this is important, at each moment in time, x1 and x2 never change.
Hereās another example with electrons and positrons. Take two examples similar to those that Feynman drew and correlate them together.
This diagram is in momentum space instead of position space so we are interested, not in where the particles are, but how much their velocity is. In this case, there are two random, uncertain momenta p and pā, and two certain ones p1 and p2. These are correlated to reveal:
This diagram shows an electron, far left, interacting with another electron, far right, mediated by a photon, but the photon produces an electron-positron loop in the middle. The electrons and positrons that had uncertain momenta in the correlation now appear in a loop. The loop is not real. It is just a representation of uncertainty. Even the existence of the electron and positron in the middle is uncertain, but at some times they exist and at other times they donāt.
The one caveat is that when I say time I donāt mean ordinary time but a second time dimension. Once you remove loops from a theory, it is much easier to represent it as a sea of particles instead of mysterious fields, but in order to do so with quantum field theory you have to add an additional dimension. Most researchers consider this additional dimension unphysical, but it may not be.
In any case, this approach to quantum field theory might finally achieve what Feynman hoped for: a particle-based representation of quantum field theory, without all the drawbacks of Bohmian mechanics. If so, it would be a triumph both for physics and meta-physics, the theory of what is real.
Andersen, Timothy D. āQuantization of fields by averaging classical evolution equations.ā Physical Review D 99.1 (2019): 016012.
Namiki, Mikio. Stochastic quantization. Vol. 9. Springer Science & Business Media, 2008.