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FQXI ARTICLE
July 22, 2018

Quantum Dream Time
Defining a ‘quantum clock’ and a ’quantum ruler’ could help those attempting to unify physics—and solve the mystery of vanishing time.
by Kate Becker
November 8, 2017
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Quantum Clock
How do you measure time at the quantum
scale?

Credit: Mark Rasmussen
Picture a man dreaming. On the outside, he’s totally still. But inside is a world in motion: moving, growing, changing, evolving.

Could our universe be like the dreaming man—static seen from outside, but alive on the inside?

Maybe, according to a new definition of "quantum time" from Lorenzo Maccone, of the University of Pavia in Italy, Seth Lloyd at MIT in Cambridge, USA, and Vittorio Giovannetti at the Scuola Normale Superiore, in Pisa, Italy. With support from a grant of almost $50,000 from FQXi, the three physicists are reviving a long-abandoned approach to quantum mechanics. They hope their strategy may make it possible to solve one of the biggest problems in physics: the apparent incompatibility of quantum mechanics, which governs the physics of the very small, and general relativity, which describes the motion of stars and planets.

For the better part of a century, physicists have been trying to reconcile the contradictions between quantum mechanics and general relativity. Individually, each has withstood every experiment and mathematical stress test it has been put to. General relativity, Einstein’s theory of how gravity emerges from the warping of space and time, makes bullseye predictions of phenomena happening on the far side of the cosmos. Quantum mechanics, meanwhile, is the consummate subatomic oddsmaker, delivering impeccable predictions on events at the tiniest size scales.

Yet the theories offer clashing worldviews. Quantum mechanics is seeded with randomness, but general relativity is ruled by cause and effect. In general relativity, space and time are woven together into a pliable thing called spacetime, but quantum mechanics runs on quaintly separate, classical notions of space and time. And when physicists try to apply the equations of general relativity to the realm of quantum mechanics, those equations spit out nonsense. Like puzzle pieces from two different picture puzzles, quantum mechanics and general relativity just don’t fit, and paradoxes peer out through the gaps.

The Problem of Time

The nature of time is one of those paradoxes. In the 1960s, physicists John Wheeler and Bryce DeWitt made a major push toward mathematically unifying quantum mechanics and general relativity, deriving a new version of the equation that describes the evolution of quantum systems, the Schrödinger equation, so that it also includes gravity. The problem: time drops out of the equation entirely, leaving physicists who believe that quantum theory is fundamental with a conundrum.

"Most physicists regard the quantum state as a complete description of the state of the universe," says Matthew Leifer, a quantum physicist at Chapman University, who is not involved in the project. "Then the Wheeler-DeWitt equation implies that there is no change in the state of the universe, which contradicts our everyday observations."

This notion of a timeless universe jars with our everyday experience. "It’s clearly meaningless because we see things evolving all the time!" says Maccone.

Solving the conundrum may require appreciating ambiguities about the notion of time in quantum theory. Although time is an essential ingredient in the equations of quantum mechanics, there is no definition that’s natively quantum, says Maccone. Instead, time is whatever the clock on the lab wall says it is. That’s an "ugly" way to treat time, Maccone said in a 2015 Google Hangout talk, and it sets the results of quantum equations on a potentially flawed foundation.

The state of the
universe as seen
from the outside
is static, but from
the inside, it is
not.
- Lorenzo Maccone
A few years ago, Giovannetti hit on a promising solution: What if he just swapped the classical clock for a quantum one? The quantum clock could be something linear, like the position of a particle on a wire, or cyclical, like the polarization of a photon: anything that could be entangled with another system. (Entanglement, the bizarre quantum effect most famous for the "spooky action at a distance" phenomenon, links the quantum states of two or more systems.) The second, entangled, system could be as simple as a single photon or as intricate as the whole world. In this view, the probability equations at the core of quantum mechanics are conditional on the state of the quantum clock; the approach is therefore called "conditional probability amplitudes."

Conditional probability amplitudes seem to solve Wheeler and DeWitt’s problem of vanishing time, says Maccone. An "inside" observer, correlated with the quantum clock, would see the system change over time, but someone watching from the outside could only observe the static properties of the combined systems. "The state of the universe as seen from the outside is static, but from the inside, it is not," says Maccone. Like the dreamer, the quantum system looks to be at a standstill, while it’s actually alive with action on the inside.

Though the idea was new to Giovannetti, it was actually first hatched in the early 1980s by theorists, and FQXi members, Don Page, now at the University of Alberta, Canada, and William Wootters, at Williams College. When Giovannetti and his colleagues did a literature search, they quickly turned turned up about a half-dozen papers outlining it. "They had even used the exact same notation," marveled Maccone. So why had Giovannetti, Lloyd and Maccone never heard of it?

The decades-old papers revealed that the approach had been quickly abandoned due to criticisms that, at the time, seemed fatal. But today, Maccone argues, those objections can be overcome using insights from the quantum information theory, a field which was still in its infancy when Page and Wootters’ idea was nixed. Quantum information theory has introduced new ways to think about measurement devices, and has turned once-abstract ideas, like quantum entanglement, into practical tools.

Quantum Time

Juan León, who studies quantum information at the Instituto de Fisica Fundamental in Madrid, Spain, met Maccone in Italy in 2015, and in 2017 they coauthored a paper showing that "quantum time" can also help resolve a longstanding disagreement about how time and energy are related at the quantum level (Leon, J. & Maccone, L. Found Phys (2017)).

The FQXi grant has also made it possible for Maccone, Lloyd, and Giovannetti to attend workshops outside the borders of their usual fields, says Maccone, sparking fortuitous connections that would not have been made otherwise. Maccone discovered one such connection with theorist Francesca Vidotto, who studies an approach to quantum gravity called Loop Quantum Gravity, at Radboud University in the Netherlands. (See "The Spacetime Revolutionary.")

Loop Quantum Gravity states that geometrical quantities, such as area and volume, have a discrete rather than continuous spectrum on small scales. For instance, an entity could have an area of one unit, or two units, but not half a unit, or two-thirds. "Namely when I measure them I can obtain only certain values in ’jumps,’" says Vidotto. She is currently working to extend this principle to time.

But the quantum clock is only half of the project. To create a quantum version of spacetime, Giovannetti, Maccone and Lloyd must devise a "quantum ruler" to complement the clock. That is turning out to be a thornier problem than they had anticipated. "I thought that the time part would be the most complicated one and that the rest would follow more easily," says Maccone. "I was wrong."

Though Maccone has derived the formulas that describe quantum space, the researchers are still grappling to understand exactly what those formulas mean. They must continue working through the equations to see if the promising hints pan out: to see if the dream is real, or just a mirage that disappears upon waking.

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My own experiment recently succeeded in uncovering the mathematical sub-structure of pi, which is not the motionless and infinite, irrational, and...


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