In a paper published today in the Proceedings of the National Academy of Sciences (PNAS), Emeritus Professor Tim Palmer FRS considers quantum mechanics (QM) from a different perspective. The paper questions whether there could be a simple tweak to the underlying mathematics of QM that could turn it from an incomprehensible to a comprehensible theory – and is there an experiment that could be performed to test such a tweak?
QM is our most successful theory of physics, having passed every experimental test thrown at it. And yet its meaning remains deeply mysterious, many would say incomprehensible. Professor Palmer thinks there is such a tweak and that it is testable.
The words 'quantum jump' and 'quantum leap' have become part of everyday language. As so, for many who are not familiar with QM at the technical level, it might seem that QM is a profoundly discontinuous theory, in comparison with the smoother classical physics it replaced. However, as Professor Palmer is keen to stress in his paper, this is completely wrong.
‘QM is more vitally dependent on the continuum of real numbers, indeed complex numbers, than is classical physics,’ explains Professor Palmer. ‘If one banishes the continuum from QM, the basic axioms – for example that quantum states are elements of a vector space called Hilbert Space - would be mathematically inconsistent. Banish the continuum and QM disintegrates into an incoherent mess.’
Professor Palmer is in strong agreement with David Hilbert himself, who claimed that the notion of infinity, and hence the infinitesimal, is nowhere to be found in reality. In Professor Palmer’s pithy words: ‘Nature abhors a continuum’. With this as motivation, he has developed a novel theory of quantum physics which he calls rational quantum mechanics (RaQM), where the continuum of complex Hilbert Space is excised.
In Professor Palmer’s theory, the famous Schrödinger equation is left unmodified, even during the measurement process. However, importantly, the quantum state is only defined in Hilbert-Space bases where the associated squared amplitudes and complex phases of the projected quantum state are rational numbers. Here, a basis is defined by the orthogonal eigenfunctions of a Hermitian operator known as an observable in QM. In RaQM, unlike QM, the quantum state is therefore only defined with respect to certain “rational” observables.
‘This could perhaps be thought of as a minor tweak to QM,’ explains Professor Palmer. ‘However, it has profound implications. The key idea to make this a mathematically coherent idea is to rethink what we mean by the imaginary number √(-1), which plays a pivotal role in QM, and yet is an undefined quantity introduced by axiom. In RaQM, by contrast, √(-1) is a constructively defined permutation/negation operator acting on strings of bits (applying it twice negates all the bits).
‘This reimagining of complex numbers (and so-called quaternions) allows a realistic interpretation of the quantum state in RaQM, compared with QM. In RaQM, cats are no longer simultaneously alive and dead.’
In a companion paper in review in Proceedings of the Royal Society, Professor Palmer shows that the conceptual problems that have dogged QM since its inception 100 years ago, vanish once the continuum is excised in this way. Most importantly perhaps, RaQM allows a new interpretation of the experimental violation of Bell’s inequality. The canonical interpretation of Bell’s Theorem is that either there is no such thing as objective reality or that the choice of experiment performed on the other side of the galaxy could instantaneously determine the outcome of an experiment performed here on Earth (as occurs in David Bohm’s 'pilot-wave' interpretation of QM),’ continues Professor Palmer. ‘Neither of these options is appealing.’
In RaQM, by contrast, Bell’s Theorem allows for something much less problematic: an objective reality where the laws of physics are not nonlocal but are instead holistic. One example of holism is Mach’s Principle – that inertia here is due to mass there. This idea was pioneered by Dennis Sciama – under whom, Professor Palmer studied for his DPhil in the 1970s.
Another example is the fractal geometry of a chaotic attractor – Professor Palmer is best known for his work on chaos theory and climate. He believes the universe as a whole may be a chaotic system evolving precisely on a fractal attractor in cosmological state space.
‘QM has met all the experimental challenges thrown at it,’ Professor Palmer states, ‘and so, in the paper, I propose an experiment that could be performed in a few years – if one is to believe the quantum tech roadmaps – for testing RaQM against QM.’
First, Professor Palmer proposes that the physical reason why Hilbert Space is discretised is gravity. This provides a quantitative measure of the granularity of Hilbert Space (very fine because gravity is so weak). Professor Palmer uses this measure of granularity to come up with his experimental test, drawing on ideas in quantum computing.
In QM the number of dimensions in Hilbert Space grows exponentially with the number of qubits (the building blocks of quantum computers). In RaQM, the information content in the quantum state only grows linearly with the number of qubits.
‘In RaQM, above a critical number of entangled qubits, there simply isn’t enough information in the quantum state to allocate even one bit of information to each dimension of Hilbert Space,’ explains Professor Palmer. ‘When this happens, quantum algorithms that utilise all of Hilbert Space will stop having a quantum advantage over classical algorithms. An example is Shor’s algorithm for factoring integers and hence decrypting RSA-encrypted messages.’
Professor Palmer predicts that Shor’s algorithm will start to fail in this way when a few hundred (error-corrected) qubits are entangled. This is bad news for those looking to develop quantum computers for practical applications. However, Professor Palmer views things more positively: ‘If quantum computers provide the experiments not only to find a successor theory to QM, but more importantly to find the theory which synthesises quantum and gravitational physics, that would surely be an extraordinarily good outcome for all the work that has been put into quantum computing over the years.’
Just as it was impossible for Einstein to have predicted the practical importance of general relativity for the development of the GPS, so it is impossible to imagine today what new commercial opportunities may accrue from a theory which synthesises quantum and gravitational physics.
Rational quantum mechanics: Testing quantum theory with quantum computers, T Palmer, Proceedings of the National Academy of Sciences, 16 March 2026