A Little Patch of Spacetime

Recently there’s been a lot of buzz around the idea that the universe is a big simulation. The idea is pretty out there, right?

What if I was to tell you that us humans have been creating universes on computers, taking into account the most fundamental of physics, detailed to some of the smallest length scales that we understand? They’re not quite the size of our universe, or even something smaller like a planet, current computers would struggle somewhat. They’re only about 10 femtometers across, smaller than an atom. But it’s a start!

They’re called Lattice simulations, and belong to a subgenre of particle physics called Lattice Gauge Theory.

To illustrate what this is and the drive behind it, let’s consider a very simple and general problem in physics. Working out the trajectory of, say, an electron. Deducing a trajectory is to be able to say where the electron is at any given point in time.

In classical mechanics (a.k.a how the world looked before quantum mechanics became a thing), given all the forces acting on the electron, along with the initial conditions (i.e. its position and velocity) there exists one unique trajectory the particle can take. One can plug the initial conditions into an equation of motion (like Newton’s 2nd law, F=ma) and solve it to deduce with certainty the position of that particle after some arbitrary period of time.

Taking quantum mechanics into account the water becomes muddied. The electron is no longer bound to follow the unique trajectory, but can take other trajectories which disobey its classical equation of motion. Before, the probability of the electron following the classical path was 1, and following any other path was 0. Now, each path has some non-binary probability between 0 and 1.

Not even the most versed physicist can predict with certainty where the particle will be after a period of time. As a consolation prize, it is possible to deduce the probability of the particle arriving at a certain point in space at a given time.

To do this, a physicist would basically sum up the probabilities of each of the many trajectories that result in the electron arriving at your chosen location at your chosen time. This is called a path integral, the sum of all probabilities of a particle taking each possible path between two points. In general there is an infinite number of possible paths. The classical path is always the most likely, paths that are close to the classical path have a smaller probability but still contribute, and completely deviant paths that go to jupiter and back are incredibly unlikely and basically don’t contribute.

One of the reasons quantum mechanics is ‘hidden’ at sizes bigger than an atom is that the perturbed paths become so unlikely that the classical path is basically the only path the particle can take.

fig1.jpgFig.1: Particle travelling from A to B. Rightmost- a particle on the quantum scale, e.g. an electron. Leftmost- a particle on the classical scale, e.g. a baseball. Solid lines are “very likely paths”, and dotted are less likely.

Now let’s complicate the picture further by moving from quantum mechanics to quantum field theory. This takes into account the possibility of the electron emitting and absorbing particles, or decaying into different particles only to reappear somewhere down the line before reaching its destination. Things become more complicated, but the principle of the path integral still holds, with the new feature that the bunch of paths we need to add up now include all combinations of emissions and decays. I’ll refer to a ‘path’ as a trajectory + any specific interaction including other particles.

fig2.jpgFig.2: Similar to figure 1, now with the possibility of other particles being created and destroyed.

Once we’re in quantum field theory we are getting into some real fundamental shit. The standard model of particle physics, containing the recently discovered Higgs boson, is expressed in the language of quantum field theory.

In practice it’s not possible to work out probabilities for an infinite number of paths.  Happily, as I discussed, there are a small number of dominant paths which account for the majority of the probability, the classical path and small perturbations of it. In particle physics, the way we usually work out the probability of some process is to consider only these dominant paths, and we get to a result which is pretty close to the ‘true’ answer. It can be done with just a pen, paper and the knowhow. The method is referred to as perturbation theory.

This doesn’t work for everything. Namely, if we were trying to compute the path integral for a quark rather than an electron. The electron interacts mostly with the electromagnetic force (electricity+magnetism). Quarks feel not only the electromagnetic force, but also the strong nuclear force, it’s horrendously more complicated cousin. The strong force is ‘purely quantum’ in the sense that there isn’t really a dominant path and subdominant perturbations of the path, there are many different dominant paths and there is no good way to order them in terms of probability.

It’s possible to use perturbation theory on quarks, but since it’s difficult to find all the dominant paths, uncertainties usually lie at around 10% (i.e., the true answer could be the answer we worked out give or take 10% of that answer). Compare this with what one can achieve with the electron, with uncertainties dipping well below 1%.

The solution: lattice simulations!

Simulate a small period of time playing out on a small patch of space on a powerful computer, give the patch a little prod so it has enough energy for a quark to appear, and let your tiny universe play out all the possible paths, decays, interactions, whatever.

fig3.JPGInside the patch it’s necessary to approximate space and time as a lattice of discrete points. Each point has some numbers attached to it, signifying the probability of a quark being at that point, the probability of the presence of other quarks, and the strength of the strong force. With this little universe, we can forget about which paths are dominant and which aren’t since all paths occur automatically in the simulation.

Like perturbation theory, it is also an approximation. Spacetime is not discrete (as far as we know), and not inside a finite patch. People often will perform the simulation at many different ‘lattice spacings’ (the distance between each discrete point), and look at the trend of these numbers to extrapolate the answer to a zero lattice spacing, representing a continuous space. Similarly in the real world there are no ‘walls’ like there are on the edges of the patch. So  folk will use a range of sizes of patch, and extrapolate results to an infinite size where there are no walls.

Uncertainties in lattice simulations are in many cases a lot smaller than perturbation theory, at about the 1% level. The method has proven shockingly effective in understanding how quarks bind together to form mesons. A meson is like the little brother of the proton and neutron, while these contain 3 bound quarks, a meson contains 2. Lattice people have their sights on making a simulation big enough that a whole proton can fit inside its walls, but we’re not quite there yet.

I think lattice gauge theory is still only in its ‘calibration phase’. The motivation of a lot of the work lattice people do is to show it works, by matching its predictions to experiments. As computers become faster, our methods become more efficient, and our understanding of the physics improves, the lattice could end up being the tool which uncovers the next big discovery in particle physics. Watch this space.

Path integrals in quantum mechanics

Lattice Gauge Theory

What if spacetime really is discrete?

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One thought on “A Little Patch of Spacetime

  1. Thank you for this post, you wrote it very clearly and I appreciate that. I have been meaning to read up a bit on QFT and the techniques used in it, but I never moved beyond a few QM courses during my studies so there is a knowledge gap there. It’s great to read a little about this field without getting clobbered with Hamiltonians 🙂

    I’ll follow this blog, and I’ll eagerly await any future content!

    Like

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