Trespassing on Einstein's Lawn (50 page)

My dad tags a chalkboard at the Kavli Institute for Theoretical Physics
A. Gefter

We don't live in a box, I thought, but we do live inside a cone. I still didn't understand why we didn't just surrender to the incoherency of eternal inflation and stick a cosmic hologram on each observer's boundary in a de Sitter space that would no longer decay. Sure, we'd lose every last invariance, but who were we kidding? Weren't we headed there already?

“What about the observer-dependent de Sitter horizon?” I ventured.

Polchinski thought for a moment. “Have you talked to Tom Banks? He has a rather different take, which I think emphasizes more the point you just made. That somehow each individual observer should have their own version of holography.”

As it happened, Tom Banks was en route to Santa Barbara.

The blaring ring of the telephone woke me from a sound sleep.

“Dad's having a kidney stone attack.” It was my mother calling from the room next door.

“Shit!” I said, snapping awake. “Is he okay?”

“I don't know. We might end up going to the hospital. We need to keep the car here in case. You should take a cab to the campus.”

I threw on sneakers and ran to their room. My father was lying on the bed, doubled over in pain, moaning. This wasn't the first time this had happened. I knew it would eventually pass, that he'd be fine. Still, he was the doctor. He was the one we all turned to when
we
were sick. Seeing
him
sick was like seeing the world turned inside out.

“I don't have to go,” I told him. “I'll cancel it and stay here.”

He cringed, clutching his stomach. “You … should … go,” he stammered, pushing the words through the pain. “Kick … physics … ass.”

“Go,” my mother reassured me. “There's nothing you can do. I'll call you if anything changes.”

I wasn't sure how long it would take for the cab to get there, so I called one right away. It came immediately, and I found myself arriving
at the Kavli Institute forty-five minutes early to meet Banks. I headed into the common room to wait.

The place was empty except for one man who was pouring himself a cup of coffee. Was it Banks? I had glanced at a photo of him online a few weeks back, but I'd never had a good memory for faces. I offered up a big smile and said hi, figuring that if it was him, he'd react accordingly. The man looked at me, gave a slight nod, and returned to his coffee. Surely not the reaction of someone awaiting a blind interview. I sat down on the couch to wait.

The man took his coffee and headed outside, sitting down at a table in the courtyard just behind me, perhaps ten steps away. I waited, patiently biding my time.

I was excited to talk to Banks. I didn't know much about his work, but I did know that he and his collaborator Willy Fischler had a theory of holographic spacetime—one that, as Polchinski suggested, was even more observer-dependent than the other ideas we'd encountered, one that embraced our current de Sitter situation rather than holding out for billions of years in the hopes of transitioning to flat FRW space. Then again, Susskind had said that formulating physics in a single observer's patch in de Sitter space “probably can't be right because the de Sitter space can decay.” I was curious to know what Banks would make of that.

The more time passed, the more I began to suspect that the man with the coffee was in fact Banks—only now it was way too late to say anything without making the situation even more awkward.

So I just sat there. And he just sat there.
For thirty minutes.

Finally I decided to email him to say that I had arrived early and that I was in the common room whenever he was ready. He emailed me back. I could hear him typing:
Hi, Amanda. I'm in the courtyard.

With an awkward smile, I joined Banks in the courtyard. I felt like an idiot. I sat down at his table, shaking his hand, apologizing for the confusion and explaining why my father couldn't join us. Banks nodded. He was quiet and reserved, but as soon as we launched into a physics discussion his demeanor transformed and he became friendly and animated.

I figured I'd dive right in and ask what I now knew was the million-dollar
question: “How can we apply the holographic principle to our de Sitter universe?”

“The thing about de Sitter space,” Banks began, “is that no matter how long you live, you can only access a finite region. Your horizon will always have a finite area. In the conventional picture of spacetime from general relativity, you'd say that there's more universe out beyond what we can see, beyond our horizon. But the holographic principle tells us that, no, there's a complete description of everything beyond the horizon right here in stuff that's causally connected to us. Each observer's universe, their causal diamond, is finite but complete.”

Susskind had been hesitant to apply complementarity to the de Sitter horizon, but Banks wasn't hesitating at all. “This idea just follows black hole complementarity to its logical conclusion,” he said. Information never leaves the observer's light cone; it just piles up at the horizon, scrambled and burnt by the Hawking radiation.

A causal diamond is the diamond formed by the intersection of an observer's past and future light cones, the total region of spacetime with which that observer could ever interact. A complete but finite universe.

I already knew the problem with a finite universe: there's not enough room for invariance. Invariant definitions of an S-matrix, of particles and strings, required an infinite boundary, infinitely far away. Finite boundaries a finite distance away couldn't cut it. As I had learned a long time ago, particles—and, in turn, strings—were irreducible representations of the Poincaré symmetry group. But horizons break Poincaré symmetry. It was precisely that fact that led to Hawking radiation, to the holographic principle, and to horizon complementarity. In a world with horizons, observers can't agree on what's a particle and what's empty space. What's more, neither one of them is more right than the other. In a de Sitter universe like ours, even the most stable building blocks of reality are rendered observer-dependent.

“What happens to the S-matrix?” I asked. “Don't you need an infinite region to get any kind of invariance?”

“You're right,” Banks said. “If the causal diamond can ever become infinite, then all observers agree and there are gauge-invariant observables like the S-matrix in asymptotically flat space. But in de Sitter
space this never happens.… Susskind et al. want to define some kind of observables in zero cosmological constant, approximately supersymmetric FRW. They want de Sitter to be unstable to decay into FRW.”

“You don't think it will?”

“No,” Banks said. “That idea is based on eternal inflation and the string landscape, and those are ideas that I think are just wrong.” One reason they're wrong, he explained, is that they rely on the notion that spacetime can undergo quantum fluctuations.

“It can't?” I asked, shocked.

“Not in the holographic picture of spacetime,” he said.

Banks explained that thanks to the holographic principle, it was now possible to encode all the properties of spacetime into the language of quantum mechanics. That, of course, was the holy grail. Quantum gravity.

The properties of spacetime fell into two categories: causal structure and scale. The causal structure tells you which points can communicate with each other and which can't—that is, the positions of light cones. Scale tells you how big things are.

I was surprised to hear that causal structure could be encoded into quantum language. Given the conceptual gulf between relativity and quantum mechanics, you'd think that light cones would have nothing to do with anything remotely quantum.

But the key, Banks explained, was commutativity.

I already knew a little something about commutativity. I knew, for instance, that certain pairs of measurements—certain “operators”—can't both be specified to arbitrary precision. One of these pairs was position and momentum, and another was time and energy; these were conjugate pairs linked by uncertainty. Uncertainty tells you that the order in which you perform the measurements matters. Measure position first and you obliterate information about momentum; measure momentum first and you smear out the position. The order matters, which is to say, the operators don't commute.

“If you have two spacetime regions that can't communicate—that is, points in spacetime so far apart that light couldn't possibly have traveled between them—then the quantum operators associated with those regions commute with each other,” Banks said.

That made sense. After all, it wasn't quite accurate to simply say that position and momentum operators don't commute—the true statement was that position and momentum operators don't commute
within a single reference frame.
Within a single light cone. If you came across position and momentum operators that
do
commute, you'd know you were talking about causally separated events—events that lie outside each other's light cones.

“That commutation expresses the lack of causal connection between them,” Banks said. “When operators
don't
commute, they interfere with each other. If you didn't have quantum interference, you could send signals faster than light. So the causal structure of spacetime tells you which quantum operators commute and which ones don't. But you can also read that backward. You can start with the algebra of quantum operators, which tells you what commutes and what doesn't, and from that you can deduce the causal structure of spacetime.”

Voilà—quantum spacetime. Well, almost. In addition to causal structure, you still needed scale. Quantum commutation relations can tell you that two points are too far apart to communicate, but they can't tell you how far apart they are.

The holographic principle, however, can. “The holographic principle tells you that the number of quantum states—the entropy—measures an area,” Banks said. “The area of the region's boundary. So if the holographic principle is true, we now have a way to state completely all the properties of spacetime in the language of quantum algebra.”

“And that tells you that spacetime can't fluctuate?” I asked.

“That's right. Fluctuating spacetime is an old idea, but it's wrong. The holographic principle tells you that the properties of spacetime are encoded into which quantum operators commute with each other and how big the space of states is, the Hilbert space, the entropy. Those are things that don't fluctuate. It's the values of variables that fluctuate in quantum mechanics, not the size of the Hilbert space or the commutation relations. When you talk about the uncertainty principle for position and momentum, what fluctuates is how much you know about either one. But the commutation relation between them is just there.
It's exact; it doesn't fluctuate. So holographic spacetime tells us that geometry doesn't fluctuate. And that has a very, very profound influence on how we should think about string theory.”

That was putting it lightly. If geometry—that is, spacetime—didn't fluctuate, that meant no eternal inflation and no string landscape. No decay to FRW. No infinite, flat space. No invariance. No reality. Just us, sitting here in de Sitter space. Finite. Stuck.

“Okay,” I said. “So you have an observer sitting in a causal diamond in de Sitter space, surrounded by a finite horizon. But the horizon is observer-dependent. It's not like in AdS/CFT, where you have one boundary for the whole universe. Here there's a cosmic hologram for each observer?”

“AdS/CFT is a special case,” Banks said. “The areas of the causal diamonds are taken to infinity in a special way. Observers are related by a symmetry transformation of the space, so they're all equivalent to each other. That's not true in de Sitter space. As a result, it's much more observer-dependent. People trying to squeeze everything into the AdS/CFT paradigm remind me of movie producers who are trying to make the fifth sequel to Saw.”

“What happens when you have more than one observer?”

“There's a beautiful galaxy called the Sombrero galaxy, which is not gravitationally bound to our local group. If we're really in de Sitter space, we will eventually see the Sombrero galaxy's light redshift farther and farther away. It will appear to approach our horizon, and as it does, it redshifts more and more until it disappears from sight. The only trace of it left will be the uniform background of radiation coming from the horizon. But if you were sitting on the Sombrero galaxy, you would see
us
approach the horizon and become absorbed into that temperature while you're just sitting there having a cup of coffee and thinking you're perfectly okay. Those are two completely equivalent descriptions of the same physical system, but they use degrees of freedom that can't be measured by a single observer.”

I nodded. “They only contradict each other if you try to take a God's-eye view.”

“That's right,” Banks said. “You can't say that our story or the Sombrerinos' story is the right one. Neither is more real than the other. In
quantum mechanics I can talk about a particle's position or its momentum but I can't talk about both simultaneously, even though either description is equally valid. Likewise, I can't be both the accelerated and the inertial observer.”

You can't be inside and outside a horizon. Safe and Screwed.

“Holographic spacetime is constructed observer by observer,” Banks continued. “If I think about two observers, like you and me, there's a large region of spacetime which we can both explore for long times. So you have some description of that spacetime and I have some description of that spacetime. Those descriptions are individually complete.”

“So a second observer is … what? A copy?”