Timeless geometry and what we say time is

Another article about physics and mathematics by Natalie Wolchover, published in both Wired and Quanta Magazine, got my attention because it began like this:

In late August, paleontologists reported finding the fossil of a flattened turtle shell that “was possibly trodden on” by a dinosaur, whose footprints spanned the rock layer directly above. The rare discovery of correlated fossils potentially traces two bygone species to the same time and place.

Cosmologist Nima Arkani-Hamed makes the connection:

Paleontologists infer the existence of dinosaurs to give a rational accounting of strange patterns of bones…We look at patterns in space today, and we infer a cosmological history in order to explain them.

I doubt my 12-year old son has ever thought that the existence of dinosaurs is inferred. For him, the facts are clear. The dinosaurs are just not here anymore. But Arkani-Hamed’s observation caused a few things to go through my mind quickly. First I thought, this is cool – corresponding a tactic in paleontology to one in physics. And then, I realized what very little thought I have given to how we have come to know so much about creatures whose lives occurred completely outside the range of our experience. We have fully life-like images of them, and treat their existence as an unquestionably known quantity. Thinking about the labor it took to transform fossil discoveries into these convincing images highlighted the need, as I see it, to make the labor of science as apparent to non-science audiences as the results of that labor have been. The creativity involved in all of our inquiries is as important to see as the outcomes of those inquiries.

As a species, it seems that we are very good at piecing things together. Some facet of our reasoning and cognitive skills is always on the hunt for patterns with which our intellect or our imagination will then build countless structures – from the brain’s production of visual images created by the flow of visual data it receives, to the patterns in our experience that facilitate our day-to-day navigation of our earthbound lives, to the patterns in the sky that hint at things that are far beyond our experience, and the purely reasoned patterns of science and mathematics. We use these structures to capture, or harness, things like the detail of astronomical events billions of light years away, or the character of particles of matter that we cannot see, or species of animals that we can never meet. The reach or breadth of these reasoned structures likely rivals the extent of the universe itself or, at least our universe. I would argue that it is useful to reflect on how our now deep scientific knowledge is built on pattern and inference because, in the end, it is the imagination that has built them. By this I do not mean to discredit the facts. Rather, I mean to elevate what we think of the imagination and of abstract thought in general.

Wolchover’s article describes how Arkani-Hamed and colleagues have worked on schemes that use spatial patterns among astronomical objects to understand the origins of the universe. (Based on the paper, The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities). Physicists have considered simple correlated pairs of objects for some time.

The simplest explanation for the correlations traces them to pairs of quantum particles that fluctuated into existence as space exponentially expanded at the start of the Big Bang. Pairs of particles that arose early on subsequently moved the farthest apart, yielding pairs of objects far away from each other in the sky today. Particle pairs that arose later separated less and now form closer-together pairs of objects. Like fossils, the pairwise correlations seen throughout the sky encode the passage of time—in this case, the very beginning of time.

But cosmologists are also considering the possibility that rare quantum fluctuations involving three, four or more particles may have also occurred in the birth of the universe. These would create other arrangements, like triangular arrangements of galaxies, or objects forming quadrilaterals, or pentagons. Telescopes have not yet identified such arrangements, but finding them could significantly enhance physicists’ understanding of the earliest moments of the universe.

Wolchover’s article describe physicists’ attempts to access these moments.

Cosmology’s fossil hunters look for the signals by taking a map of the cosmos and moving a triangle-shaped template all over it. For each position and orientation of the template, they measure the cosmos’s density at the three corners and multiply the numbers together. If the answer differs from the average cosmic density cubed, this is a three-point correlation. After measuring the strength of three-point correlations for that particular template throughout the sky, they then repeat the process with triangle templates of other sizes and relative side lengths, and with quadrilateral templates and so on. The variation in strength of the cosmological correlations as a function of the different shapes and sizes is called the “correlation function,” and it encodes rich information about the particle dynamics during the birth of the universe.

This is pretty ambitious. In the end, Arkani-Hamed and colleagues found a way to simplify things. They borrowed a design from particle physicists who found shortcuts to analyzing particle interactions using what’s called the bootstrap.

The physicists employed a strategy known as the bootstrap, a term derived from the phrase “pick yourself up by your own bootstraps” (instead of pushing off of the ground). The approach infers the laws of nature by considering only the mathematical logic and self-consistency of the laws themselves, instead of building on empirical evidence. Using the bootstrap philosophy, the researchers derived and solved a concise mathematical equation that dictates the possible patterns of correlations in the sky that result from different primordial ingredients.

Arkani-Hamed chose to use the geometry of “de Sitter space,” to investigate various correlated objects because the geometry of this space looks like the geometry of the expanding universe. De Sitter space is a 4-dimensional sphere-like space with 10 symmetries.

Whereas in the usual approach, you would start with a description of inflatons and other particles that might have existed; specify how they might move, interact, and morph into one another; and try to work out the spatial pattern that might have frozen into the universe as a result, Arkani-Hamed and Maldacena translated the 10 symmetries of de Sitter space into a concise differential equation dictating the final answer.

It is significant that there is no time variable in this analysis. Time emerges within the geometry. Yet it predicts cosmological patterns that provide information about the rise and evolution of quantum particles at the beginning of time. This suggests that time, itself, is an emergent property that has its origins in spatial correlations.

It should be clear that confidence in the geometric calculations is coming from how they square (no pun intended) with empirical measurements that we do have.

By leveraging symmetries, logical principles, and consistency conditions, they could often determine the final answer without ever working through the complicated particle dynamics. The results hinted that the usual picture of particle physics, in which particles move and interact in space and time, might not be the deepest description of what is happening. A major clue came in 2013, when Arkani-Hamed and his student Jaroslav Trnka discovered that the outcomes of certain particle collisions follow very simply from the volume of a geometric shape called the amplituhedron.

I wrote about this discovery in March.

Arkani-Hamed suspects that the bootstrapped equation that he and his collaborators derived may be related to a geometric object, along the lines of the amplituhedron, that encodes the correlations produced during the universe’s birth even more simply and elegantly. What seems clear already is that the new version of the story will not include the variable known as time.

An important aspect of the issues being discussed is the replacement of time-oriented functional analyses with time-less geometric ones. As I see it, this raises questions broader than how the structure of the universe itself is mathematical. This work highlights the relationships between physical things, abstract or ideal objects, and the constraints of logic. It says as much about us, and what we do, as it says about the origins of the universe or what we say that time is. I’ll stress, as I often do, that these issues are relevant to people, not just to science. This shift from one kind of organization of concepts (dynamic change) to another (geometric relationships) should encourage us to consider where these conceptual structures are emerging from and how are they connecting us to our reality.

I’m convinced that paying more attention to how we participate in building our reality will clarify quite a lot.

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