|“Infinite Grid” by Georg Koch.|
To begin with quantizing gravity isn’t all that difficult. Gravity can and has been quantized much like the other interactions by promoting gravitational waves to quantum waves. That is called perturbative quantization and technically somewhat annoying but perfectly doable. The problem starts only after this because the so quantized theory does not make sense anymore when gravity becomes strong. It delivers infinities as results, no matter what, and that makes it not only useless but also meaningless as a fundamental theory.
You might shrug shoulders on the infinites because you have also probably heard that quantum field theory has a problem with infinities anyways. But that too is not true...
Yes, the occurrence of infinite results was historically a big issue. But quantum field theory development hasn’t stopped in the 1930s and we know today very well how to do these calculations. Whenever you get an infinite result, you need to use a measurement to fix a physical parameter. We call it renormalization and it’s a mathematically clean procedure.
However, if you need an infinite amount of measurements to fix parameters, then you have a real problem because now your theory is no longer predictive: You need infinitely many measurements before you can make a prediction about the next measurement. A theory with that problem is said to be perturbatively non-renormalizable and on that diagnosis most physicists will not resuscitate the patient. Perturbatively quantized gravity has exactly this disease.
As long as gravity is weak, for all practical purposes you need only a finite amount of parameters to get to a desired precision. You can use the theory there. But once gravity gets strong, the theory becomes useless. This means it is not a candidate for a fundamental theory.
The use of quantum field theories and their properties thus depend on the energy scale. Interactions can for example become weaker or stronger depending on the energy of the interaction. Quantum Chromodynamics famously becomes weaker at high energies; it is “asymptotically free” and that insight was worth a Nobel Prize in 2004.
But how theories in general depend on the energy scale has only been really understood within in the last two decades or so. It has been a silent development that almost entirely passed by the popular science press and goes under the name renormalization group flow. The renormalization group flow encodes how a theory depends on the energy scale, and it is at the basis of the idea of effective field theory.
The dependence of a quantum field theory on the energy scale that is used to probe structures is much like Ted Nelson’s idea of the stretch-text, a text in which you can zoom or click into layers of more detail. The closer you look, the more new features, new insights, new information you get to see. It’s the same with quantum field theory. The closer you look, the more new layers you get to see.
Based on this, Weinberg realized in 1976 that perturbative renormalizability is not the only way for a theory to remain meaningful at high energies. It is sufficient if, at high energies (technically: infinitely high), you need to fix only a finite number of parameters. And none of these parameters should become infinite itself in that limit. These two requirements: A finite number of finite parameters that determine the theory at high energies are what make a theory asymptotically safe.
This then raises the question of whether quantum gravity, though perturbatively nonrenormalizable, might be asymptotically safe and meaningful after all. That this might be so is the idea behind “asymptotically safe gravity”. While the general idea has been around for almost four decades, it has only been in the late 1990s, following works by Wetterich and Reuter, that asymptocially safe gravity has caught on.As of today, it has not been proved that gravity is asymptotically safe, though there are several arguments that support this idea. The problem is that doing calculations in an infinite dimensional theory space is not possible, so this space has to be reduced. But then the result can only deliver a limited level of knowledge. The other problem is that even if the theory is asymptotically safe, it might be physically nonsensical at high energies for other reasons.
Another criticism on asymptotically safe gravity has been that it does not seem to take into account that space-time fundamentally might be described by degrees of freedom different from those used in general relativity. While that arguably is so in existing approaches, the idea of renormalization group flow is in principle perfectly compatible with changing to different – more ‘fundamental’ – degrees of freedom at high energies, as Percacci and Vacca have pointed out.
That is to say, this approach towards quantum gravity has its problems, its friends and its foes, as has every other approach towards quantum gravity. But it is a strong competitor. What makes this approach so appealing is its minimalism: Maybe quantum gravity makes sense as a quantum field theory after all! Depending on your attitude though you might find exactly this minimalism unappealing. It’s like at the end of a crime novel the murder victim comes back from vacation and everybody feels stupid for their conspiracy theories.
Whatever your attitude, asymptotically safe gravity has made some contact to phenomenology, mostly in the area of cosmology, though for all I know these studies haven’t yet resulted in a good observable. Most interestingly, asymptotically safe gravity has been shown to also lead to dimensional reduction and it has recently been argued that it might be related to Causal Dynamical Triangulation. It seems to me that whatever quantum gravity ultimately looks like, asymptotically safe gravity will almost certainly be part of the story.
So the next time somebody tells you that we don’t know how to quantize gravity, keep in mind the many layers of details underneath that statement.