Schrödinger, Navier and Stokes walk into a bar: anecdotes and insights about quantum physics

I have always been fascinated by the stories of people who pursue quantum physics. What drew them to it in the first place? What do they find most intriguing, or most unsettling? My own story begins in the land between physics and mathematics.

As a physics undergraduate and mathematics postgraduate, I walked uncomfortably among both physicists and mathematicians specialising in quantum mechanics. And although I learned a lot from both groups, they have each left me with unease for different reasons. Based on my experience, I would characterise physicists and mathematicians by their approaches to mathematical problems:

A physicist is the person who points at the intractable term in an equation and confidently announces “This one drops out” and so solves the problem.

A mathematician is the person who, in carefully solving one mathematical problem, finds a whole family of even harder problems and announces “You’re welcome!”

The approximations made by physicists have sometimes seemed wild to me. Wilder still was the realisation that the resulting solutions could be far more accurate and useful than seems reasonable. Eventually I came to realise that these approximations are part of what makes physics so successful: linearising a non-linear problem, cancelling infinities and using other techniques that might make a mathematician queasy but without which much of physics would not be possible. The key is physical intuition---the ability to approach a difficult problem through a deep understanding of the underlying physical system and a healthy disregard for mathematical rigour.

That, however, is seemingly not a skill taught over the course of my undergraduate degree, nor does it even seem to be acknowledged. Instead, mathematical approximations are demonstrated to solve problems, almost like magic tricks. For me personally, the acknowledgement of physical intuition as the motivation underpinning such approximations would have been hugely helpful and not left me feeling quite as uneasy at times.

A prime example is quantum mechanics, both in its non-relativistic and its relativistic flavours (albeit for different reasons).

Quantum mechanics is a linear theory. That makes the mathematical framework of quantum mechanics relatively straightforward, at least when compared to the notoriously difficult non-linear mathematics that physicists like to hand over to the mathematicians. As noted above, physicists favour solutions whereas mathematicians appreciate problems. Yet quantum mechanics combines that manageable mathematical framework with extraordinarily challenging physics, from entanglement to Schrödinger’s cat. This bothered me for a long time: how can all this strangeness be described in such simple mathematical terms?

There are many questions one can ask about linear quantum mechanics. For example, there is a question as to what extent this limits the predictive power of quantum mechanics. For me, there was a more fundamental question about the theory itself: what was the cost of the linearisation? Rather than contemplating what that means for what we can get out of the theory (i.e. limits of the predictive power), I was concerned about the price we had already paid---something everybody else apparently took it in their stride. Is quantum mechanics fundamentally limited in its scope and ambition due to its linearity?

Enter Messrs Navier and Stokes: It was some time into my PhD when I contemplated some conceptual similarities between the physics of quantum mechanics, as described by linear quantum theory, and the physics of fluids, as described by the Navier-Stokes equations. At a very high level:

• quantum mechanics is concerned with the evolution of quantum waves/states and with the detection of discrete quantum particles;

• fluid dynamics is about a continuous medium and discrete bubbles therein;

• so both share the continuous/discrete divide.

The big difference is that quantum mechanics is a linear theory, which is mathematically so straightforward that we teach it to every physics undergraduate. By contrast, the non-linear Navier-Stokes equations are so difficult that an entire academic CV could read simply “Navier-Stokes” and I would be duly impressed.

The non-linearity of fluid dynamics allows interactions within a continuous medium to produce localised structures such as bubbles. In standard quantum mechanics, by contrast, the continuously evolving quantum state and the discrete quantum particles ultimately detected are treated differently, with the bolted-on Born rule connecting the two. It thereby bridges the mathematical description of the quantum world and the “real” world.

The Born rule can initially look like an inelegant addition to the theory of quantum mechanics. I came eventually to see it as an elegant way of providing a linear theory of quantum mechanics. That is to say, by separating the quantum waves and the discrete quantum particles, which are rejoined by the postulated Born rule, quantum mechanics sidesteps the issues that make fluid dynamics non-linear. I found some comfort in this, as it suggests that the so-called measurement problem (the transition from the continuous quantum waves to the discrete quantum particles) is not an isolated oddity; it emerges from the structure of the theory itself and, in a sense, seems to be the price we pay for its linearity. That said, I am unsure how many students of quantum mechanics will appreciate it to the same extent, since for the most part we are taught how to “do” quantum mechanics, but are rarely prompted to stop and think about “what” the theory of quantum mechanics is, how it comes together, and where its natural boundaries reside.

Fast-forward many years, and that habit of looking towards the assumptions beneath a technical argument has stayed with me. It shapes how I approach technology in my work, particularly high-tech and quantum inventions: I focus not only on how a technical solution works, but also on what has been simplified, what has been left implicit, and where the inventor’s insight fundamentally lies. I once had the pleasure of working with an inventor who had linearised what is conventionally a non-linear system. The inventor did not describe the invention in those terms, and I suspect never would have, but viewing it through that lens provided me with a useful understanding of the invention. Perhaps that is one of the key lessons quantum mechanics has taught me, and one that remains with me today: to be comfortable with uncomfortable questions.