By Bartek Fecko, Fall 2024.
Perhaps no intellectual symbiosis has been as profound as that between physics and mathematics in recent centuries. Since the development of modern physics, the two fields have largely co-evolved: first, mathematicians develop the symbolic language for formulating physical laws. At the same time, physicists search for better mathematical concepts to tackle complex physical problems. For example, Newton developed calculus- alongside Leibniz- to describe his laws of motion. A more recent example is the introduction of the delta function by theoretical physicist Paul Dirac. Originally used to describe the quantum mechanical wave function of pointlike particles, it has become valuable in the mathematical theory of ordinary and partial differential equations. Thus, developments in both fields can influence each other, and in both directions. Unsurprisingly, there tends to be significant overlap between modern-day physics and mathematics departments [1]. However, physics possesses specific qualities that distinguish it from subfields of mathematics. One such quality, of course, is its experimental nature. Yet even in theoretical physics, it can be distinguished from applied mathematics in one critical sense. For physicists, mathematical accuracy, while necessary, is often insufficient. Physicists seek an accurate, elegant description of the physical world, in which the same phenomena can be explained by increasingly simple or intuitive mathematical theories that better communicate the universe’s structure than do relatively abstract ones. This creative, imaginative quality of physics is what distinguishes it as a discipline and manifests most clearly in those areas of physics where physicists deal most with complex mathematical abstractions that they may wish to simplify.
A case-and-point demonstration of this is a recent development in quantum field theory (QFT). In short, this describes all quantum mechanical phenomena- the physics of microscopic particles exhibiting probabilistic wave-like behavior- by incorporating special relativity and extending the notion of what physicists call a field [2]. Whereas fields like the electromagnetic and gravitational fields are usually used in classical physics as mathematical constructs that describe forces, in truth, fields are real physical entities that are more fundamental than particles themselves. “Fundamental” particles like photons and electrons are only “excitations” of their respective fields- photons, of the electromagnetic field, and electrons, of the electron field- which exist even if no particle- a vacuum- is present. Electromagnetic-electron field interactions then generate the electromagnetic force between photons and charged particles, in which the well-known behaviors of charged particles- for example, electron repulsion- can be understood from the transfer of photons between electrons. This is quantum electrodynamics, the “quantum field theory” of charged particles. Similarly, quantum chromodynamics describes interactions between quarks mediated by particles called gluons and explains the strong nuclear force responsible for the stability of all atomic nuclei. Explaining nature’s fundamental forces and other quantum phenomena through particle interactions is the cornerstone of quantum field theory.
Traditionally, physicists have approached quantum field theory through tools called Feynman diagrams. These diagrams track possible interaction outcomes by stringing together lines representing particles’ spacetime trajectories. Then, all possible trajectories are “integrated” via what are called path integrals to yield a “probability amplitude” that represents the likelihood of an interaction occurring in a particular manner [3]. By using quantum mechanical field equations- for example, the quantized Maxwell’s equations, which govern quantum electrodynamics- physicists can investigate the probabilities of various particle interactions. However, the Feynman method has encountered significant roadblocks. For one, applying it becomes mathematically cumbersome for complex particle interactions. Moreover, it requires a flat spacetime, and therefore cannot accommodate general relativity. Thus, physicists are in search of alternative formulations for both reasons of mathematical simplicity and physical applicability. More importantly, physicists have observed that the mathematical calculations for various particle interactions often simplify tremendously out of very complicated integrals, suggesting that more fundamental structures underlie particle collisions. This observation in particular has motivated the search for more fundamental formalisms of particle physics.
Figure 1. Feynman diagrams [6]. Particle interactions are performed by transferring a mediator particle represented as a wave (e.g. photons), and incoming and outgoing particle trajectories and identities are also shown. A rightward motion indicates forward motion in space, and an upward motion indicates forward motion in time.
A recent string of publications has demonstrated the utility of an alternative formalism called surfaceology. Rather than calculate probability amplitudes strictly from field equations, these equations and other information about particle collisions- particle trajectories- are encoded in the geometry of a three-dimensional surface whose volume equals the interaction’s probability amplitude [4]. Of course, for different quantum field theories, different governing equations lead to different three-dimensional structures. Though physicists have explored this for several years, it took a neat mathematical insight to achieve a real breakthrough: namely, that special polynomial functions can describe the curvatures of these surfaces. Then, the surface can be geometrically related to the Feynman formalism through the mathematical connection between these polynomials and the mathematics of Feynman path integrals. This “conversion” begins with the thickening of Feynman diagrams to create a surface, with incoming and outgoing streams (input and output particles) associated with the surface’s open junctions. Particle trajectories are converted into curves that traverse these junctions, and when described by special polynomials, can be used to calculate probability amplitudes with the aid of experimental data. An immediate advantage of this method is greater experimentation. Namely, one can manipulate the surface’s volume and analyze the structure to reveal interactions of various probabilities, including highly frequent and incompatible collisions, rather than having to handle the full details of the mathematical equations. Thus, surfaceology can describe complex collisions with many particles as well as exotic trajectories. For example, a particle being destroyed and later re-created in an interaction, while laborious to analyze mathematically, can be depicted as a cyclic curve on the surface. Furthermore, by abstractifying quantum field theory out of Feynman diagrams, which depend on flat spacetime geometry, surfaceology extends to regimes where quantum gravity s considered.
Figure 2. Diagramatic illustration of the surfaceology procedure. After all possible curves on the surface are drawn, the curves are converted into representations called “mountainscapes”, whose decomposition provides the mathematical information necessary to calculate the amplitude.
With surfaceology, these physicists successfully linked different, ostensibly unrelated quantum field theories by analyzing prohibited interactions. First, they investigated a perturbative quantum field theory called trace phi cubed theory. In short, it approximates particle interactions as small perturbations of a simpler “free field theory” in which the particles’ underlying quantum fields are non-interacting [5]. Here, particle trajectories in quantum chromodynamics were manipulated to yield flattened surfaces, corresponding to interactions with negligible probability amplitudes. Strikingly, these same particle collisions were outlawed when the physicists turned to the less understood quantum field theory of particles called pions. Moreover, these same zero-amplitude collisions were also observed in the better understood Yang-Mills Theory, a general class of quantum field theories encompassing quantum chromodynamics and the unification of quantum electrodynamics and the weak force. Thus, surfaceology has been demonstrated to link three seemingly independent quantum field theories under one unified umbrella.
Surfaceology has tremendous implications for quantum theory. The demonstrated linkage of seemingly disparate quantum field theories motivates the probing of a more fundamental, possibly geometric, reality that governs particle interactions, irrespective of vast differences in their particular spatiotemporal mathematics. The opening of a more rigorous geometric study of quantum field theory can also motivate physicists to finally unify quantum field theory with general relativity, which, as of yet, has no quantum mechanical description. Additionally, surfaceology does well in illustrating the nature of the entire physical enterprise. As previously explained, physicists seek not only mathematical accuracy but also an elegant simplicity, so as not to derail the creative function of physics by becoming too deeply submerged in complex mathematical formalisms, and develop theories that give the universe a kind of comprehensive structure. It is this structure and its inherent beauty that ultimately captivates the physicist. Surfaceology does precisely this by replacing a vast array of complicated mathematical equations with a comprehensive geometric picture of the same physical phenomena. With surfaceology now more thoroughly developed, it will be interesting to see in which direction physicists choose to take this new way of describing particle physics.
[1] Lamb, Evelyn. 2018. “The co-evolution of physics and math.” Symmetry Magazine, April 24. https://www.symmetrymagazine.org/article/the-coevolution-of-physics-and-math?language_content_entity=und
[2] Kuhlmann, Meinard. 2023. “Quantum Field Theory”, The Stanford Encyclopedia of Philosophy (Summer 2023 Edition), Edward N. Zalta & Uri Nodelman (eds.). https://plato.stanford.edu/entries/quantum-field-theory/#WhatQFT
[3] Brown, James Robert. 2018. “How Do Feynman Diagrams Work?” Perspectives on Science; 26 (4): 423–442. doi: https://doi.org/10.1162/posc_a_00281
[4] Wood, Charlie. 2024. “Physicists Reveal a Quantum Geometry That Exists Outside of Space And Time.” Quanta Magazine, September 25. https://www.quantamagazine.org/physicists-reveal-a-quantum-geometry-that-exists-outside-of-space-and-time-20240925/
[5] Arkani-Hamed, Nima, Qu Cao, Jin Dong, Carolina Figueiredo, and Song He. 2024. “Nonlinear Sigma model amplitudes to all loop orders are contained in the Tr (Φ 3) theory.” Physical Review D 110, no. 6: 065018.
[6] Effects of the CPT-even and Lorentz violation on the Bhabha scattering at finite temperature – Scientific Figure on ResearchGate. Available from: https://www.researchgate.net/figure/Feynman-diagrams-with-different-vertices-Each-diagram-is-composed-of-the-scattering_fig1_350087221