For any admitted students reading this article: Welcome to Stevens! As you begin your college journey (hopefully as a Duck!) I thought I would write this week’s column in a style more like “vintage” submissions from For Math’s Sake’s early days (now almost three years ago). When I started writing this column, I tended to pick a mathematical concept that I found interesting and try to explain it to a general audience while also arguing about the concept’s real-world relevance.
The concept I have in mind this week is path integration in quantum physics — something that may evoke excitement, but just as well can induce indifference at best and horror at worst. I hope to get past the jargon and give an introduction of what this term means, and how it’s really helped physics (and mathematics) advance a great deal in the past century.
First, a quick introduction/refresher on integration. By this I mean evaluating integrals of functions, a process in calculus that one can visualize in a simplest sense as determining the “area” or “volume” under a function. One in turn can visualize a function by drawing a curve on a graph — the space lying between that curve and the axis of the function’s input variable between two points, is equal to the integral of that function.
To understand path integration, one has to become more abstract with what the axis represents. When evaluating a path integral, you are no longer simply calculating the area under a curve. Instead, you are studying how the term inside the integral—now called a “functional,” which takes functions rather than numbers as inputs and outputs a number—changes when you input different functions. The sum of the functional’s outputs over the range of different functions you input gives us a path integral. So, to try to extend the analogy above, now the “curve” is a set of numbers output by the functional, and points on the “axis” under the curve represent specific functions, like y = 2x or y = 1/x, that you input to the functional.
But what does any of this have to do with quantum physics? Due to the inherent uncertainty of quantum particles, it is impossible for us to say exactly how a quantum particle (like an atom, electron, or photon) moves from point A to point B. So, we instead imagine this particle taking all feasible paths from point A to point B, and calculate how much energy is exerted from taking each path. So now, instead of a mathematical abstraction, we have a physical intuition for the path integral: We go along the axis of functions describing a path from A to B, and evaluate a functional that outputs the energy a particle must exert to travel each path.
One can also see how this now gives a statistical interpretation of quantum physics — this path integration can also be viewed as an averaging procedure, giving us intuition for what a particle “on average” might be doing, if it were to repeat its travel from A to B many, many times. These path integrals, therefore, are crucial for describing quantum experiments and have been highly successful in doing so. One of the most precise predictions ever made is that of something called the anomalous magnetic dipole moment of an electron, whose experimental measurement aligns with the path integral calculation for the first 10 significant figures.
But there’s a lot that remains a mystery. How exactly does one choose which functions correspond to feasible paths, and how do we know that this definition of integration makes sense in general? One set of open problems involves physical theories that describe the interactions within an atomic nucleus — insights into path integrals in this setting may help solve a problem with a $1 million reward! This more general type of integral also has some relation to mathematical finance, a field that can also yield big returns.
Money aside, these path integrals are a fascinating area of mathematics, combining advanced theory with applications to describe the most fundamental interactions of our world. And they may not be quite like the integrals you learn about in introductory calculus courses, but they wouldn’t be possible otherwise. So if you feel skeptical about whether integral problem sheets are useful, remember that mastery of these can put you well on the way to shedding light on some of the great mysteries of the universe!
