Taming infinity

by Anil Ananthaswamy

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WoodardRichard Woodard


But then he took his first course in quantum field theory, being taught by Nobel laureate Steven Weinberg. "It was one of the seminal experiences of my life," says Woodard. "The idea that you could write down the DNA of the Universe in just a few statements — a brush stroke on a piece of paper — was just awesome. I recall lying in my bed in my dormitory room, just staring up at the ceiling to look at something blank, because I was overwhelmed by the power of the ideas that were being put into my mind by those lectures."


Well, almost perfect.

Quantum mountains



While quantum field theory does a fantastic job of describing electromagnetism, and the strong and weak nuclear forces, it doesn't work for Einstein's theory of gravity. No matter how hard you work at applying quantum field theory to gravity, you get the same, dramatically wrong, answer: infinity.

"If you take the theory seriously, it says that when I wave my arm, every being in the solar system gets fried by hard gravitational radiation. That's obviously not true," says Woodard. "So, right now we just have nonsense."

Something is clearly going wrong — but where? Almost everyone agrees that quantum mechanics is not the culprit. Most particle theorists think that the problem lies with general relativity. But Woodard thinks they are in danger of throwing the baby out with the bathwater. He argues that just because the calculations don't work, that doesn't mean general relativity is wrong, or incompatible with quantum mechanics. And it doesn't mean that we need to introduce string theory, loop quantum gravity, or any other exotic new physical theory. Instead, the blame could lie with the approximate techniques used to carry out quantum field theory calculations.

In an ideal world, physicists wouldn't have to make approximations. But we don't live in an ideal world. Anyone who has taken high-school physics will remember having to simplify complicated problems in order to have any chance of applying textbook equations — pretending that cars are perfect cuboids sliding along frictionless roads, say.

Sweeping away infinity

Similarly, physicists are often forced to make approximations as they try to solve equations that describe things that can be measured, such as the total energy of a quantum system. Except for the simplest systems, these equations are impossible to solve exactly. That's when physicists start approximating — using a technique known as perturbation theory — and the trouble begins.

The trick with perturbation theory is to start with one of the few ideal quantum systems that you can solve completely and then perturb them, so they look approximately like the situation that you actually want to study. For gravity, this is where things get nasty. Specifically, you end up with an answer in the form of a power series expansion or a never-ending string of numbers, each multiplied by higher powers of Newton’s constant for the strength of gravity, $G$, times even powers of the energy or mass, $E$, of the thing being computed ($1 + aGE^2 + bG^2E^4+ cG^3 E^6 + ..., $ where $a,$ $b,$ $c, ...$are pure numbers). Because $G$ times any energy or mass that can be accessed in particle physics is such a small number, just a few terms of the expansion would be wonderfully accurate — if only the pure numbers $a,$ $b,$ $c,$ and so on were finite. But they aren’t, leading to an infinite answer.

Computer simulationPhysicists make their best guess when simulating quantum systems on computers. But is the guess good enough for gravity?



Woodard is scrutinising perturbation theory’s power series expansion. Could the series expansion for gravity be wrong? Could the terms of the series involve, for example, the logarithm of $G$? He thinks so. "What I am speculating is that general relativity is the right theory of quantum gravity and that there is a very good series expansion for it," says Woodard. "It just isn’t conventional perturbation theory."


Woodard is also helped by clues that any corrections to general relativity must either be tiny or else disguised as something that we don't recognise as a quantum correction. That's because we do not recognise any large quantum gravitational effects around us.

Light reflections




Still, quantum gravity remains an enigma, thanks in part to the lack of experimental evidence for it. "It's like figuring out what is in a room behind a locked door, and we do not even know if there is a room there," says Casadio. "Just because of this, any approach to quantum gravity remains an act of faith."