A Probabilist’s Perspective on Fundamental Physics
Benjamin Skuse
The primary purpose of the Heidelberg Laureate Forum is to bring together “some of the brightest minds in mathematics and computer science.” This is not simply a show of solidarity or a recognition of overlapping histories, it’s an opportunity to exchange and combine ideas from different fields and subfields in the hope of delivering new, perhaps unexpected, insights into up-to-now intractable problems.
No attendee of the Forum embodies this ethos better than Wendelin Werner. A Franco-German mathematician (whose older brother happens to be an outstanding computer scientist), his 2006 Fields Medal was awarded “for his contributions to the development of stochastic Loewner evolution, the geometry of two-dimensional Brownian motion, and conformal field theory,” work largely conducted in collaboration with fellow mathematicians Greg Lawler and Oded Schramm.
This ground-breaking research offered new mathematical tools in statistical physics for understanding and rigorously proving fundamental physical concepts, including what happens when physical systems reach critical temperature, like when a kettle boils, or how percolation works, like when water seeps through porous rock.
Almost two decades on from his Fields Medal – incidentally, the first to be awarded to a probabilist – Werner continues to straddle, and chip away at, the border between mathematics and physics, as he described in his Lecture at the 12th Heidelberg Laureate Forum.
Unanswered Question from Childhood
Werner’s fascination with the points where mathematics and physics intersect started from an early age. To begin his talk, he recalled a discussion in high school between him and his physics teacher regarding Newton’s universal law of gravitation. This law describes gravity as a force between two masses, attracting them in proportion to the product of their masses and inversely as the square of the distance between them.
“I raised my hand and said […] ‘Who’s measuring the distance between the Earth and the Sun in order to decide what the force is going to be – surely something has to travel back and forth.’” After a lengthy discussion digging deeper and deeper into the law, his teacher finally said: “Well, if you continue always asking why and how, you should rather be a mathematician than a physicist.”
This propensity to dig deeper and deeper in the hope of discovering more fundamental truths continues to drive Werner’s work to this day. He still remains unsatisfied with many physics-based explanations of fundamental real-world phenomena, and in particular gravity. “In some sense, the rule in physics is to find some formalism that produces equations or numbers that you can then confront to experimental data to measure something, and you see that this formalism fits with the real-world data,” he says. “But it will never tell you exactly how does it actually work.” For Werner, Newton’s law and Einstein’s theory of general relativity may predict what gravity does accurately, but they don’t fully explain the underlying mechanism.
Introducing Loops
Werner has attacked the problem of deeply understanding gravity on and off throughout his career. He often does so in a roundabout way, treating it like a mathematical puzzle, and picturing how mathematical objects could be incarnated in the world around us to form what we perceive as gravity.
The first mathematical object he introduced in his talk was the harmonic function. A harmonic function \( u(x,y) \) is a twice continuously differentiable function that also satisfies the Laplace equation:
\[ \nabla^{2}u = u_{xx} + u_{yy} = 0. \]
Such a function has the property that its value at a point is the same as its average value on any sphere around that point. If we look at similar properties in the real world, we come across gravitational potential, or equivalently Newton potential, \( 1/r \), where \( r \) is the distance between the centre of mass of the source object and the point in space where the potential is being measured. That it is harmonic is the only way to mathematically explain Newton potential’s properties, such as why we can treat a planet’s mass as if it were concentrated at a single central point.
The second mathematical object Werner introduced is called a Brownian loop, which he described as “the simplest possible, most natural, random structure that you can try to define around you without making any arbitrary choices.” Brownian loops are a nod to Brownian motion, first discovered by 19th Century botanist Robert Brown, who observed the jagged erratic motion of pollen immersed in water – one of the first tantalising experimental glimpses of atoms and molecules, and a crucial link between physical phenomena and the abstract world of probability theory and randomness. Regular Brownian motion requires a starting point, but Brownian loops do not, making them even simpler random structures that have only one fixed property: they are isotropic, i.e. their behaviour is uniform in all directions.
Imagining these Brownian loops appearing completely randomly in space and time, Werner conjured a reality in which there are infinitely many small independent loops everywhere at all scales. If you have a collection of these loops where they intermingle, called a ‘loop soup,’ it turns out that whether two loops are intersecting or whether a single loop is just overlapping itself, it doesn’t matter; the properties of the loop soup remain the same. Essentially, when they intermingle, each loop loses its individual identity. And they reestablish their identity when you extract an individual loop from the loop soup.
“In physics, bosons are particles that have this property that they’re indistinguishable when you see a collection of all these particles, but you can recover the individual particles,” said Werner. “This is a bit reminiscent to that … maybe there is something here, this exchangeability property, which is very elementary.”
Connecting It All Together
How do loops relate to Werner’s enduring desire to understand gravity at a fundamental level? He asked the audience to imagine two points \( x \) and \( y \) in three-dimensional space. “What is the probability that \( x \) and \( y \) are connected by a chain of loops?,” he said. “This probability is 1 over the distance – it’s our friend, the Newton potential.”
He next asked them to picture a single Brownian excursion going from \( x \) to \( y \), which can be thought of like a jagged telephone wire connecting the two points, with an unconditioned collection (i.e. not a chain) of loops on top. In simple terms, Werner showed recently that these two setups are exactly equivalent. This correspondence not only provides a useful tool to study loop-soup clusters, it very gently hints at relations to physics concepts including quantum field theory.
“When you mix randomness with continuum structures, some subtle things can appear that might have escaped from the radar of other types of approaches to this problem,” said Werner. But he was quick to point out that the work he presented should not be seen as having a direct relation to physics. “Disclaimer: I’m not doing any physics,” he underlined. “This is just what a mathematician is doing and dreaming about, playing around with fun math structures …; it might be inspired by physics, but in a way that physicists would not recognize.”
Nor did he feel that Brownian motion and loop soups are necessarily going to be any more satisfactory for others than conventional physics concepts such as particles and fields. However, he did hope that by presenting a probabilist’s unorthodox perspective on physics concepts, it might inspire young scientists to do the same.
“Some 10 years ago, I remember seeing a child looking out of the window of a train and trying to zoom in on what they were watching outside,” Werner recalled. “That child’s experience of what reality is, is very different from mine.” Because of this, the child on the train is unlikely to independently discover loop soups, but they do have the potential to unlock a completely fresh perspective – be it from probability theory, computer science, or some other discipline – to attack intransigent problems in science.
The post A Probabilist’s Perspective on Fundamental Physics originally appeared on the HLFF SciLogs blog.