Physicists are challenging the traditional concept of space-time by proposing that the universe’s fundamental nature could be understood through abstract geometric shapes, like the “cosmohedron.” This new approach simplifies complex calculations and could eventually lead to a theory of everything, possibly reshaping our understanding of the cosmos.
A cosmic shape could explain the fundamental nature of the universe
Author: Karmela Padavic-Callaghan
WHAT is the structure of our physical reality? Physicists have long imagined space and time interweaving into “space-time”, the metaphorical fabric that underlies the cosmos. But there may be something even more fundamental. Instead of space-time’s three spatial dimensions and one of time, the physics of our universe could be encoded into a set of odd geometrical shapes – and studying them may chart a new, space-time-free path towards a theory of everything.
“The idea is that space-time somehow has to go, that it has to be replaced by something more primitive and deeper,” says Nima Arkani-Hamed at the Institute for Advanced Study in New Jersey. “The notion of space-time has got to emerge out of some more abstract objects.”
Space-time has to go, it has to be replaced by something more primitive and deeper
Now, working with Carolina Figueiredo at Princeton University and Francisco Vazão at the Max Planck Institute for Physics in Germany, he has uncovered one such abstract object. They call it the cosmohedron.
The researchers have been exploring similar shapes that fit into a kind of family tree of fundamental mathematical objects. For example, shapes called associahedra encode the way particles may collide with, or scatter off, each other – without having to use equations that involve space and time. Physicists have known how to calculate so-called scattering amplitudes that predict what will happen in these collisions for decades, but even when armed with their best mathematical tricks they often end up with thousands of lines of tedious and difficult calculations, says Arkani-Hamed.
In 2013, Arkani-Hamed and his colleagues discovered one specific shape that had many slanted sides and sharp edges, similar to a cut diamond, that encoded some of those answers, but with less maths involved in getting there. Instead of thousands of lines of equations, the researchers had to tackle the somewhat simpler problem of drawing an associahedron for each collision and then calculating its volume. Figueiredo says that this method draws on the mathematical structure of physicists’ usual equations, but doesn’t require that laws of physics be explicitly referenced at every step.
This is because the method is more like a geometrical building project than traditional equation solving. In the simplest example, the researchers imagine a collision between three particles, leading to the creation of three new particles. It is a core tenet of physics that the momentum of the original particles has to match the momentum of those created after, and this is often expressed by writing two formulae, one for the particles before and another for after, and setting them mathematically equal.
An alternative approach sees the momentum of the particles represented as an arrow, with six arrows in total – three before and three after. In this representation, momentum being conserved means that these arrows, when joined, must connect to form a six-sided shape. Arkani-Hamed and his team go several steps further, using a set of geometrical and combinatorial rules to transform these flat shapes into jewel-like, three-dimensional objects – the associahedra.
Abstract thinking
Working in this manner allows the researchers to temporarily set aside particles and their paths in space-time. The 3D associahedron shapes are far more abstract, but, remarkably, formulae that describe the volume of each such shape turn out to match the scattering amplitude formulae that we would traditionally use to predict the outcomes of a particle collision.
“You want to somehow find some objects that [ensure] your answer obeys the fundamental principles which underlie the physical theories like quantum mechanics and relativity. You don’t put them in automatically, the object kind of just knows about them,” says Figueiredo.
She and Arkani-Hamed, along with their colleagues, demonstrated that this method could be used for many particles, but they have now set their ambitions beyond simple collisions. Instead, they are attempting to describe the entire universe – hence the “cosmo” in cosmohedron.
The transformation from an associahedron into a cosmohedron is both surprisingly simple and incredibly powerful. It involves shaving each of the shape’s edges to create new surfaces, adding extra parameters into the formulae that correspond to associahedra and endowing them with more meaning. Specifically, by turning a given associahedron into a cosmohedron and then studying the more complex shape, the researchers could reconstruct a quantum mechanical wave function – the formula that summarises all properties and possible behaviours of a quantum object.
“[Their] cosmological wave function attempts to describe the whole universe, in this case a theoretical one, all at once,” says Sebastian Mizera, also at Princeton, who wasn’t involved in the work.
Surprising results
Jaroslav Trnka at the University of California, Davis, says that the fact the method works at all is extremely surprising because it starts with such different concepts to those physicists usually employ. “You don’t put in physical properties like locality or even some particles or fields,” he says.
It is too early to say that the idea of space-time is completely obsolete, or that space-time isn’t meaningful, but the new work does point to the possibility of expressing the laws of physics in a completely different language that doesn’t refer to it, says Trnka. In other words, this could rewrite the way we understand the cosmos and take space-time out of the dictionary.
We feel that something remarkable is going on, and this is really the first step
Exactly what this could mean for our particular universe also remains unclear because the cosmohedron method doesn’t currently work for every particle that we know exists. For instance, the researchers aren’t certain how to use their method for any particles with an electric charge, such as electrons, because electromagnetic interactions make the mathematics of their collisions more complex. Trnka says the team’s task now is to find more examples where it does work.
“At the moment, we’re reformulating things. In the distance there is a world of physical theories, and there are these fascinating mathematical structures over here, and what we’re seeing is that there are these subterranean links between some of them,” says Arkani-Hamed.
“There’s a lot that we’d like to understand better. We feel that something remarkable is going on, and this is really the first step,” he says.
Credits: TCA, LLC.