The question was really difficult for me. Not because nothing came to my mind but because too much came to my mind! Throwing out the Heisenberg uncertainty principle, Lorentz-invariance, the positivity of gravitational mass, or the speed of light limit – been there, done that. And that’s only the stuff that I did publish...
At our 2010 conference, we had a discussion on the topic “What to sacrifice?” addressing essentially the same question as the FQXi essay, though with a focus on quantum gravity. For everything from the equivalence principle over unitarity and locality to the existence of space and time you can find somebody willing to sacrifice it for the sake of progress.
So what to pick? I finally settled on an essay arguing that the quantization postulate should be modified, and if you want to know more about this, go check it out on the FQXi website.
But let me tell you what was my runner-up.
“Physical assumption” is a rather vague expression. In the narrower sense you can understand it to mean an axiom of the theory, but in the broader sense it encompasses everything we use to propose a theory. I believe one of the reasons progress on finding a theory of quantum gravity has been slow is that we rely too heavily on mathematical consistency and pay too little attention to phenomenology. I simply doubt that mathematical consistency, combined with the requirement to reproduce the standard model and general relativity in the suitable limits, is sufficient to arrive at the right theory.
Many intelligent people spent decades developing approaches to quantum gravity, approaches which might turn out to have absolutely nothing to do with reality, even if they would reproduce the standard model. They pursue their research with the implicit assumption that the power of the human mind is sufficient to discover the right description of nature, though this is rarely explicitly spelled out. There is the “physical assumption” that the theoretical description of nature must be appealing and make sense to the human brain. We must be able to arrive at it by deepening our understanding of mathematics. Einstein and Dirac have shown us how to do it, arriving at the most amazing breakthroughs by mathematical deduction. It is tempting to conclude that they have shown the way, and we should follow in their footsteps.
But these examples have been exceedingly rare. Most of the history of physics instead has been incremental improvements guided by observation, often accompanied by periods of confusion and heated discussion. And Einstein and Dirac are not even good examples: Einstein was heavily guided by Michelson and Morley’s failure to detect the aether, and Dirac’s theory was preceded by a phenomenological model proposed by Goudsmit and Uhlenbeck to explain the anomalous Zeeman effect. Their model didn’t make much sense. But it explained the data. And it was later derived as a limit of the Dirac equation coupled to an electromagnetic field.
I think it is perfectly possible that there are different consistent ways to quantize gravity that reproduce the standard model. It also seems perfectly possible to me for example that string theory can be used to describe strongly coupled quantum field theory, and still not have anything to say about quantum gravity in our universe.
The only way to find out which theory describes the world we live in is to make contact to observation. Yet, most of the effort in quantum gravity is still devoted to the development and better understanding of mathematical techniques. That is certainly not sufficient. It is also not necessary, as the Goudsmit and Uhlenbeck example illustrates: Phenomenological models might not at first glance make much sense, and their consistency only become apparent later.
Thus, the assumption that we should throw out is that mathematical consistency, richness, or elegance are good guides to the right theory. They are desirable of course. But neither necessary nor sufficient. Instead, we should devote more effort to phenomenological models to guide the development of the theory of quantum gravity.
In a nutshell that would have been the argument of my essay had I chosen this topic. I decided against it because it is arguably a little self-serving. I will also admit that while this is the lesson I draw from the history of physics, I, as I believe most of my colleagues, am biased towards mathematical elegance, and the equations named after Einstein and Dirac are the best examples for that.