Groucho Marxism

In a previous blog post I questioned the existence of irrational numbers. So I was interested to see an article in the most recent edition of New Scientist which suggests that removing irrational numbers from quantum mechanics could eliminate its weirdest features. Irrational numbers are firmly embedded in the standard theory of quantum mechanics. However according to Tim Palmer, a physicist at the University of Oxford, it doesn’t need to be this way. Palmer argues that banishing irrational numbers from quantum mechanics would remove oddities such as entanglement an superposition. If true, this would revolutionize our understanding of quantum mechanics and have a huge impact on quantum computing.

To be clear, Palmer does not have a problem with irrational numbers per se (unlike me, and unlike Pythagoras, whose followers allegedly drowned his contemporary Hippasus for suggesting √2 is irrational). Rather, he questions the formalism that underpins the standard mathematical exposition of quantum mechanics. To analyse the behaviour of a quantum particle such as an electron, physicists assign it a mathematical object that lives in something called a Hilbert space, named after the German mathematician David Hilbert. In Palmer’s view, not all such objects should be allowed. In particular, Palmer argues that those with length equal to an irrational number should be banished. We may refer to this view of quantum mechanics as ‘rational quantum mechanics’.

Palmer claims that rational quantum mechanics is consistent with all the experimental findings of quantum mechanics, but does not suffer from the oddities that come with the conventional interpretation. To exemplify this, Palmer focuses on a famous experiment referred to as the Bell test, named after the Irish physicist John Stewart Bell. Suppose that a pair of entangled particles are produced, with one sent to an experimenter called Alice and the other to another faraway experimenter called Bob. Bell determined that Alice’s and Bob’s measurements of the entangled particles will be correlated in a non-local way, somehow influencing each other across large distances.

Some interpretations of quantum mechanics simply accept this non-locality, whereas others argue that some law of physics must pre-determine the measurements (this view is known as superdeterminism; see my previous blog post on this for an introduction). Rational quantum mechanics apparently cuts through all this by stipulating that some measurements are impossible to make in principle. In Palmer’s view, carefully thinking through which quantum states are theoretically possible can resolve the problem of non-locality, plus all manner of odd quantum scenarios, including Erwin Schrödinger’s famous thought experiment in which a cat is both dead and alive at the same time.

Rational quantum mechanics also has practical significance for quantum computing (see my previous blog post on quantum computing for an introduction to this). There is currently just a small number of problems that mathematicians have proved can be solved by a quantum computer but not a standard computer. Moreover, no quantum computer currently exists that is powerful enough to solve these problems. In rational quantum mechanics, qubits – the things that are manipulated by a quantum computer – have a fixed information capacity. This suggests that the problems that could theoretically be solved by a quantum computer but not a standard computer could not in fact be solved by a quantum computer after all.

If a future quantum computer is invented that solves these problems, rational quantum mechanics goes up in smoke. But this is a strength rather than a weakness of the theory, as it not only shows that it is falsifiable but demonstrates exactly how it could be falsified. Another strength of rational quantum mechanics, in my view, is that it is a discrete rather than a continuous theory. I have argued in several previous blog posts that aspects of mathematics which rely on the existence of a continuum are not as rigorous or ‘real’ as those made up out of finite or discrete parts. This is because the continuum relies of the existence of infinite sets, and infinite sets do not exist in the physical universe.

Palmer is not the first to suggest that reality might be discrete rather than continuous. Several physicist have argued for the existence of a discrete rather than a continuous space-time. However this view is marginal and the majority of physicists still work under the assumption that space-time is continuous. It is well-known that physicists have not been able to find a theory that successfully combines quantum physics and gravity, despite having searched for one now for well over 100 years. My hunch is that the assumption of a continuous space-time is clouding their judgement and hindering their search. Of course the only way to prove that is to develop a discrete theory of quantum gravity.  

Palmer thinks that gravity ought to play a role in the mathematical space that dictates the reality of quantum objects, and has built this assumption into his theory. However at present this assumption raises more questions than it answers. For rational quantum mechanics to replace standard quantum mechanics it will have to address more than the information capacity of qubits. In practice, given how entrenched the standard theory is, it will probably have to make a breakthrough that the standard theory has so far been unable to make. A theory of quantum gravity would be one such breakthrough and would suggest that we have been thinking about physics all wrong for over a century.