Annals of Physics

# Solving the quantum conundrum: Uniting quantum and classical physics

## New research suggests that the key to bridging the gap between quantum and classical mechanics may lie in an algebraic approach to an old idea.

Without a shadow of a doubt, the crowning achievement of early twentieth-century physics was the development of quantum mechanics. Perhaps for the first time since the development of Newtonian mechanics and classical mechanics, our view of the physics that govern the world around us was fundamentally changed. But, there is a fly in the ‘quantum ointment’ – the struggle to unite this theory to that of Newton’s physics, represented in particular by that most stubborn of stand-outs, quantum gravity.

A new paper in the *Annals of Physics**,* by Peter Morgan of the Physics Department at Yale University, Connecticut, USA, suggests an algebraic approach to bringing these disparate fields together. “If this can be called a unification, it doesn't immediately set us on track to construct a theory of quantum gravity,” says Morgan. “But a mathematically better understanding of the relationship between classical and quantum mechanics should clear the air for future attempts to solve quantum gravity.”

The approach towards unification put forward by Morgan isn't exactly new, but he certainly tackles it in a novel way. He expands upon the ideas of Bernard Osgood Koopman, a French-born American mathematician, who in 1931 put forward a description of classical physics that involves the mathematical concept of ‘Hilbert Space’ – a generalisation of traditional Euclidean space to an abstract vector space.

“Quantum theory has become a central idea in almost everyone’s thinking, but for 90 years there has been confusion about the relationship between classical and quantum ways of thinking,” says Morgan. “Koopman’s ideas, first put forward in 1931, can show us a way to end this confusion.”

Morgan explains that his algebraic approach to Koopman’s model, which he labels AlgKoopman, shows us why and how incompatible measurements – measurements that can’t be performed together – are entirely natural in classical mechanics. Thus, AlgKoopman applies algebraic quantum measurement theory to classical mechanics – enabling a unification of the disciplines.

This is done through the application of non-commutative operators, which are mathematical objects that are commonly used in quantum mechanics, to classical mechanics and thus devising a similar mathematical language for both. “I apply the methods of algebraic quantum mechanics and algebraic quantum field theory to classical mechanics,” Morgan says. “The result is a classical algebraic and Hilbert space structure that looks rather like the quantum algebraic and Hilbert space structure, with more similarities than differences.”

One of the key differences between classical mechanics and quantum mechanics revolves around what exactly a measurement is. “AlgKoopman discusses experiments in terms of fields and events instead of in terms of particles: we see events, not particles,” Morgan says. “I try to bring to descriptions of experiments an engineer’s sensibility not just about events but also about the signals in the apparatus that we have to examine to find the events. Signal analysis is a major theme.”

In particular, much of the mystery around this measurement issue involves the switch between wave behaviour and particle-like action in quantum systems. Physicists refer to this as the collapse of the wave function, but much confusion surrounds what actually causes this collapse, or whether indeed such a collapse occurs at all. “No Collapse interpretations are usually thought of as associated with the many-worlds interpretation of QM, but Bohr’s Copenhagen interpretation also eschews collapse,” Morgan says. “Another hook that Bohr gives us is his emphasis on classical modelling of experiments, which goes a whole lot further, indeed all the way, if we use all the tools that Koopman gives us.”

Much discussion of the unification of quantum and classic mechanics exists within the scientific literature, yet many of these ideas are incompatible with each other and, in some cases, outright contradictory. Morgan believes the AlgKoopman concept can alleviate this situation.

“Semi-classical methods are everywhere in the literature as ad-hoc ways to describe some parts of an experiment classically and other parts of the experiment in a ‘quantum-like’ way,” he says. “AlgKoopman will hopefully help clean up the mess.”

Additionally, the author believes that the approach described in his paper could deliver a new way of thinking about quantum computing with quantum fluctuations functioning as a resource rather than a hindrance. “If we can make ‘quantum’ seem more of an everyday thing, that might lead to more investment,” Morgan says. “On the basis of AlgKoopman, quantum computing is a form of analogue computing, in which probabilities and noise evolve as the analogue components of the system.”

**Article is free to read until March 2021**

Morgan. P.: “An algebraic approach to Koopman classical mechanics,” *Annals of Physics* (2020)