The use of mathematics in science in general and physics in particular is often described as a language, raising the impression that it’s mainly a secret code to discourage the outsider, more nuisance than necessity. As much as I appreciate and support science popularizations, the careful evasion of technical terms and equations often makes mathematics appear somehow optional, a shorthand notation in the best case, or an instrument of torture in the worst case. But mathematics is so much more than that.
Mathematics is in the first line a discipline of thought. Cleaned of the vagueness of human language, mathematics is a tool to deduce consequences from assumptions. It is incorruptible by human fragilities and knows no pity, it’s the gatekeeper of objectivity.
The way theoretical physics works today is by the construction of theories based on a set of assumptions or axioms, though they might not be explicitly stated and are sometimes only tacitly known. Regardless, when formulated in mathematical terms, these assumptions will imply a much larger set of conclusions that are then forced upon physicists. For a theory to be viable as applicable to the Universe, all these conclusions must be both internally consistent, meaning they must not lead to contradictions, and they must agree with observations.
The theories that we presently have to describe nature at the most fundamental level are General Relativity and the Standard Model of particle physics. General Relativity is a classical theory whereas the Standard Model of particle physics is a quantum field theory. The former is not subject to Heisenberg’s uncertainty principle, the latter is. The two theories together can describe all the observations we presently have, though some aspects of this description are unsatisfactory, such as the missing microscopic structure of dark matter. The combination of the two theories is consistent with observation, but the big trouble is that it is internally inconsistent.
This inconsistency is most vividly demonstrated by the black hole information loss problem. Combining general relativity with quantum field theory leads to what is known as quantum field theory in curved space. It’s a partly classical, partly quantized theory which is referred to as “semi-classical gravity.” One can calculate in this combined theory that black holes emit radiation, called “Hawking radiation” after its discoverer.
Hawking radiation is a featureless blackbody spectrum of radiation with no defining parameters save one: its temperature, which depends on the initial mass of the black hole. This means that all black holes of the same initial mass evaporate into the exact same final state of thermal radiation, regardless of what they were initially formed out of. The process of black hole formation and subsequent evaporation is thus not reversible: even if we know the final state completely, we cannot determine what the initial state was. Information is lost. The problem is that such a fundamentally irreversible process is incompatible with the quantum field theory that we used to derive the process: it’s an internal contradiction, an inconsistency — and therefore this cannot be how nature works. Mathematics forced this conclusion upon us.
The semi-classical combination of General Relativity and the Standard Model leads to other problems. We do not know, for example, what happens to the gravitational field of an electron that passes through a double slit. We know that the wavefunction of the electron is in a superposition and goes through both slits, creating a statistical distribution on the screen upon measurement. We also know that the electron carries energy. And we know that energy creates a gravitational field. But since the gravitational field is classical, it cannot be in a superposition and go through both slits like the electron itself. What happens to the electron’s gravitational field? Nobody really knows since it’s far too weak to be measured. So simple, yet so frustrating!
A third reason which convinces physicists that the combination of General Relativity and the Standard Model is an incomplete description of nature is that it leads to the formation of singularities under fairly general circumstances. Singularities are instances of infinite energy density and infinite curvature. They are unphysical and should not appear in a meaningful theory. Singularities also appear in hydrodynamics, for example when a drop of water pinches off. In this case however we know that the singularity is an artifact of using an approximation — hydrodynamics — which is no longer valid on subatomic distances. At very short distances we should be using more fundamental theories (e.g., theories of quantized, discrete particles) to describe the drop of water, and these contain no singularities, as expected.
It is generally believed that a quantization of gravity will resolve these three problems by revealing the structure of space-time on very short distances. Unfortunately, gravity cannot be quantized like the other interactions in the standard model. If one applies the same methods to gravity, one arrives at a theory known as “effective quantum gravity” that fails to solve these problems: it still breaks down at strong curvature. This naively (“perturbatively”) quantized gravity is not good to solve the problems with singularities and black hole evaporation because it only works when gravity is weak. It does not make sense as a fundamental theory. What physicists normally refer to as “quantum gravity” is a theory that will still be good no matter how strong gravitation gets.
There are presently several theoretical approaches to quantum gravity. The best known are probably asymptotically safe gravity, loop quantum gravity, string theory, and causal dynamical triangulations, as well as ideas that take seriously the hydrodynamic analogy and treat gravity as an emergent phenomenon. So far, we have no way of telling which of these approaches is the correct description of nature.
In the wake of the BICEP measurement of polarization in the cosmic microwave background (CMB) that was now shown to be due to foreground dust, it has been claimed that such a measurement would constitute evidence for the quantization of gravity. This isn’t quite correct. First, note that this only concerns weak gravitational fields and thus not the fundamental theory of quantum gravity. Besides this, one also has to be careful about the assumptions that were made for the argument. It is true that quantum gravitational fluctuations in the early universe would leave an imprint in the CMB that is potentially observable. However, it is much more difficult to demonstrate that quantum gravity is the only way to produce the observed fluctuations. What one would need to draw this conclusion is something like Bell’s theorem, a proof demonstrating that a classical theory could not have done it, and such a proof is missing.
Quantum gravity is not a large research area if you compare it to, say, condensed matter physics or cancer research. It is a fairly small community, yet it attracts a lot of public interest. It does so for a reason. Without quantum gravity we do not know how space and time really behave, and we also cannot understand how our universe began. We need quantum gravity to tell us how the cosmos holds together and how it came to be.
I do, moreover, believe that this theory will teach us important lessons about quantization in general that will have practical impacts. If you listen to string theorists they might tell you that this is already happening, regardless of whether string theory will eventually turn out to solve the black hole information loss problem!
What makes up much of the appeal of quantum gravity is the cleanliness of the problem, the inevitability of the mathematical logic leading to the conclusion that we are missing a piece of the puzzle. It remains to be seen whether purely mathematical pursuits will ultimately be sufficient to find the missing piece. If not, our conclusions will remain ambiguous without more data to lead the way.
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