Since at least Einstein we have seen spacetime as fundamental. But modern physics, from quantum field theory to gravity, now suggests spacetime is doomed. So, what lies beyond spacetime? We, ourselves, might be part of the answer, writes Donald D. Hoffman.

Who am I? If I glance in a mirror, I appear as a body, as one object among scores in space and time. I feel myself to be immersed in space and time. When I gaze at countless stars on a crisp night, I feel myself shrink to a mere speck that is trekking through space and coasting through time. My immersion is total: space and time are my perceptual reality, yes, but also my conceptual cage. If I challenge myself to imagine something—anything—outside of space or beyond time, I’m stymied. I may as well try to imagine a new color I’ve never seen before. Nothing happens. My confinement within space and time appears complete.

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It is no wonder then that, for centuries, science has taken space and time as fundamental. With Isaac Newton they were fundamental and distinct. With Albert Einstein, and his 1905 theory of special relativity, they are fundamental and united. [1] As Hermann Minkowski announced in 1908: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” [1]

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David Gross, a 2004 Nobel Laureate in physics, predicted in his tribute to Einstein that spacetime is “doomed”, that it is not fundamental.

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That union, that fundamental and independent reality, is spacetime. We feel its curves as gravity and see its singularities as black holes. The quantum fields of the acclaimed Standard Model of particle physics are defined over spacetime. It is the foundation for the edifice of science and the spin-offs of technology.

In 2005 physicists celebrated the centenary of spacetime. Then they prophesied its demise. David Gross, a 2004 Nobel Laureate in physics, predicted in his tribute to Einstein that spacetime is “doomed,” that it is not fundamental. He quoted prominent physicists who agreed.[2]

Why do physicists say that spacetime is doomed? Because, they argue, it has no operational meaning below the “Planck scale,” roughly 10^-33 centimeters and 10^-43 seconds. 

For instance, to measure the position of a subatomic particle with higher resolution, we must use radiation of smaller wavelength. Quantum theory tells us that as wavelengths shrink their energy grows. Einstein tells us, in a famous equation, that energy and mass are the same. Thus, as we increase resolution we pack more energy, and therefore more mass, into less space. When the resolution approaches the Planck scale, the density of mass grows so large that gravity spoils the party, creates a black hole, and destroys our measurement.

Moreover, quantum theory tells us that a measuring device is a quantum system, subject to quantum uncertainties. Therefore a more precise measurement requires a device with more degrees of freedom, and thus more mass. So, as I upgrade my lab to make my device more precise, its mass grows to the point where gravity again creates a black hole, destroying my lab and measurement.

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Well, why not take the device out of the lab? With endless room, can’t we make the device arbitrarily precise without creating black holes? Nice idea, but it has a problem: Our universe is expanding. Most galaxies are receding from us faster than light. We may see them, but we can never reach them. The total matter within our reach is finite. Thus our device can only have finite degrees of freedom and finite resolution.

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The physics of spacetime must arise, precisely, as a special case of our new theory. Quantum field theory can’t say what objects lie beyond spacetime, but it can veto bad hunches.

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Well, why not take the device out of the lab? With endless room, can’t we make the device arbitrarily precise without creating black holes? Nice idea, but it has a problem: Our universe is expanding. Most galaxies are receding from us faster than light. We may see them, but we can never reach them. The total matter within our reach is finite. Thus our device can only have finite degrees of freedom and finite resolution. 

So quantum field theory and gravity together warn us that spacetime is not fundamental. But they cannot tell us what is fundamental, what lies beyond spacetime. How can we possibly figure this out?

We must take a creative leap. We must propose a new theory, with new structures and processes beyond spacetime. This is fun, exciting and speculative. But there is a rule. Any theory we propose must project onto spacetime. The physics of spacetime must arise, precisely, as a special case of our new theory. Quantum field theory can’t say what objects lie beyond spacetime, but it can veto bad hunches.

Wait. Objects beyond spacetime? What could that possibly mean? Everything I’ve ever seen has been patently inside space and time.

Help with this puzzle comes from an unexpected source: evolution by natural selection. Darwin informs us that our senses evolved to keep us alive until we raise kids. But what if selection did more? Could it shape our senses to report truths about objective reality?

Yes, it could. But the probability that it does is zero. This is the startling verdict of evolutionary game theory, a mathematical formulation of Darwin’s insight. Yes, our senses guide adaptive behavior. No, they are not a window on reality. [3]

Then what are they? An adaptive fiction. A helpful metaphor is virtual reality. Suppose you’re playing Grand Theft Auto in VR. Seated in your ride, you see a dashboard, steering wheel, and the road ahead. You turn your headset to the right and see a red Ferrari. To the left you see a green Porsche. In reality, as you play the game you’re toggling millions of voltages each second in an unseen supercomputer. There is no green Porsche in that computer. The Porsche, your dashboard, and all that you see, is a fiction that helps you to play GTA.

And that’s what evolution did for us. Spacetime and physical objects are just our VR headset. They let us play the game of life, blissfully ignorant of the nitty-gritty of a reality beyond.

So evolution agrees with physics that spacetime is not fundamental. But evolution offers a metaphor: Spacetime is just a headset by which we interact with an unseen reality. Science, until now, has only studied the contents and format of our headset. It’s time for science to remove the headset and venture beyond.

And it has. In 2005, as physicists pronounced on the doom of spacetime, they caught a glimpse of something beyond. Ed Witten, at the Institute for Advanced Study at Princeton, and his collaborators Ruth Britto, Freddy Cachazo, and Bo Feng, discovered a new method to compute “scattering amplitudes.” [4] These amplitudes describe what happens when subatomic particles collide and scatter, and are essential for research at particle colliders, such as the Large Hadron Collider in Geneva. Their new method, dubbed “BCFW recursion relations,” greatly simplified the calculations of scattering amplitudes by eliminating the ghostly “virtual particles” that pop up if the computations are done within spacetime.

The BCFW recursion relations hinted that there may be structures beyond spacetime. This hint was pursued by a colleague of Witten at Princeton, Nima Arkani-Hamed, who with his graduate student Jaroslav Trnka discovered in 2013 a remarkable geometric object, the “amplituhedron.” [5] It is not an object in spacetime. It is beyond spacetime and quantum theory, and projects down to spacetime and quantum theory. Its volumes are scattering amplitudes, and its faces encode relativistic and quantum properties of spacetime.

But what could this mean? Particles are in spacetime. Yet physicists say the amplituhedron is beyond spacetime. In what sense is it ‘an object’ if it is beyond spacetime? How is it ‘projecting down’ onto spacetime and ‘encoding’ properties of spacetime?

An analogy may help. Suppose you watch a video in which one race car clips another and spins out of control. If you focus just on the pixels you see a hot mess: millions of pixels changing color and brightness. But if you focus beyond the pixels to 3D, it’s simple: a car spins. Its shape does not vary as it spins: its shape is an invariant not easily seen in the pixels. So, in this analogy, the simple motion of a car projects to a complex mess of pixels.

Similarly, when physicists compute scattering amplitudes using spacetime and quantum theory, the result is a hot mess. The interaction, for instance, of six particles called gluons takes hundreds of pages of algebra. But when physicists drop spacetime, and instead use the volume of the amplituhedron, the computation is simple: just one term. And as a bonus, they see a new invariant of the dynamics, the “infinite Yangian,” that can’t be seen in spacetime. Another reason that spacetime is doomed.

But how is the amplituhedron beyond spacetime? And how can such an object have a volume? The idea is simple. Spacetime has 4 dimensions, 3 of space and 1 of time. In certain string theories it might have as many as 11 dimensions. But the amplituhedron is a geometric object that can have trillions of dimensions and more, and these dimensions are not about space and time, but about something else that physicists have not yet figured out.

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The amplituhedron is a monolith beyond spacetime. Who ordered that? And why? Like the monolith in 2001: A Space Odyssey, the amplituhedron is imposing, mute, yet pregnant with meaning.

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Computing the volume of such an object is conceptually straightforward, although for complex shapes the formulas can be complex. But, just to get the idea, let’s take a simple case, the volume of a cube. In a 3D Euclidean space, the volume of a cube with sides of length r is r³. In a 4D space it is r⁴, and in an n-dimensional space it is rⁿ. So, for instance, the volume of the 100-dimensional cube with sides of length 2 is 2¹⁰⁰, which is about 10³⁰.

The amplituhedron is a monolith beyond spacetime. Who ordered that? And why? Like the monolith in 2001: A Space Odyssey, the amplituhedron is imposing, mute, yet pregnant with meaning. And like the apes in Kubrick’s film, we seem clueless to decipher its message.

The amplituhedron is static. Physicists love dynamical theories, but there is no obvious dynamics in the amplituhedron. It encodes the dynamics of particles in spacetime, but is itself a motionless Platonic form.

But there may be a hint of dynamics. Behind the amplituhedron physicists find, to their surprise, that “decorated” permutations capture much, and in some cases all, of the invariant information about particle interactions.[6] Standard permutations are like shuffles of cards. Let’s number the positions of cards: the top card is in position 1, the next card is in position 2, and so on. A shuffle changes positions of cards. For instance, a shuffle of two cards might send the card in position 1 to position 2, and vice versa. We could write this (12) → (21) . In a deck of 62 cards, there are 62 possible choices for the first position, 61 for the second, and so on. So there are 62 x 61 ... x 2 x 1= 62!, or about 3 x 10⁸⁵ permutations, which is roughly the number of atoms in the observable universe.

Decorated permutations never shuffle a card to a lower-numbered position. Consider, for instance, the standard permutation (12) → (21). The card in position 2 goes to position 1, which is a lower number. A decorated permutation rewrites this as (12) → (23), so that the card in position 2 goes to a virtual position 3. But here we think of virtual position 3 as the real position “3 modulo 2,” that is, as the remainder when you divide 3 by 2, which is 1. If there are n cards, then the real position is the virtual position modulo n. This little trick somehow gives decorated permutations just the oomph needed to capture the essence of particle interactions.

Why should tricky shuffles beyond spacetime encode particle interactions within spacetime? No one knows. But there is an intriguing new hint: decorated permutations also capture the behavior of a particular class of dynamical systems, called Markov chains. In this case, the decorated permutation attached to a Markov chain encodes the “communicating classes” of its dynamics. Consider, for instance, a social network such as the Twitterverse. There are millions of users, with a complex web of connections: each user follows and is followed by other users. A communicating class of users is a largest group of users in which everyone eventually sees the tweets of everyone else in the group. So if Bob follows Alice, but Alice doesn’t follow Bob, and doesn’t follow anyone else who eventually sees Bob’s tweets, then they are not in the same communicating class. But if Alice starts following Bob, or if she starts following someone who eventually sees Bob’s tweets, then they are in a communicating class. Communicating classes can differ in “inertia,” in how quickly, on average, tweets percolate through a class to reach all of its members. [7] They can also differ in “spin,” the directions in which tweets flow through the class.

So decorated permutations hint at a dynamics beyond spacetime, in which one communicating class, seen as “the vacuum,” morphs into several distinct communicating classes, seen as “particles.”  But a dynamics of what? And to what end? Perhaps it’s a dynamics of conscious units, with some units interacting via a spacetime interface; then the metaphor of social networks is apt. [8] Again, no one knows.

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Fortunately, the search for what lies beyond spacetime has been engaged by a growing band of intrepid explorers. Their forays may yet stumble across an answer to our first question: Who am I?

Not an object in spacetime.

A special thank you to Chetan Prakash and Robert Prentner for their comments on an earlier draft of the article.

References

1. Lorentz, H., Einstein, A., Minkowski, H., Weyl, H. (1952). The principle of relativity. Mineola, New York: Dover.

2. Gross, D. (2005). Einstein and the search for unification. Current Science, 89 (12): 2035-2040.

3. Hoffman, D. (2019). The case against reality: How evolution hid the truth from our eyes. London: Allen Lane.

4. Britto, R., Cachazo, F., Feng, B., Witten, E. (2005). Direct proof of tree-level recursion relation in Yang-Mills theory. Physical Review Letters. 94 (18): 181602.

5. Arkani-Hamed, N., Trnka, J. (2014). The amplituhedron. Journal of High-Energy Physics. doi:10.1007/JHEP10(2014)030.

6. Arkani-Hamed, N., Bourjaily, J., Cachazo, F., Goncharov, A., Postnikov, A., Trnka, J. (2016) Grassmannian geometry of scattering amplitudes. Cambridge, UK: Cambridge University Press.

7. Levin, D., Peres, Y. (2017). Markov chains and mixing times. Providence, Rhode Island: American Mathematical Society.

8. Hoffman, D., Prakash, C. (2014). Objects of consciousness. Frontiers in Psychology, https://doi.org/10.3389/fpsyg.2014.00577.