By Rivka Shapiro
While it may not seem like it, the relationship between mathematics and physics is inextricably intertwined. Evolutions in one field are supported by as well as contribute to those in the other. From simple quantitative observation, to new applications of Euler’s formula, to the marriage of derivative calculus and classical mechanics, there is no severing the deep connection between the fields. As a physicist, it is evident that mathematics is essential to understanding the complexity of our universe. In this article, I will explore one small portion of this relationship: Quantum Mechanics and Group Theory.
Quantum Mechanics, the study of subatomic particles and their properties, is a relatively new endeavor. The Bohr-Sommerfeld model was one of the first to begin tackling the organization of electrons within an atom. Their model described electron shells as circular orbits with specific quantized radii and energies. Soon after, Friederich Hund, a contemporary of Bohr and Sommerfeld, added a critical rule: electrons will always occupy empty orbitals before pairing up. The now-famous Exclusion Principle, proposed by Wolfgang Pauli in 1925, completed the Bohr-Sommerfeld model by theorizing that no two electrons may occupy the same quantum number in an orbital, necessitating the concept of “up” and “down” spins. This inclusion propelled the study of electron spin into the modern field of Quantum Electrodynamics.
Around half a century prior, mathematician Sophus Lie began to innovate in group theory, a topic in abstract algebra that considers the groups of numbers that define geometric spaces. Lie investigated all the possible changes and manipulations one could perform in a given space and characterized it in a basic form. For the particular groups Lie studied, the elements would follow a unique and particular set of rules, now called a “Lie Algebra.” Groups that follow Lie Algebras are termed Lie Groups, and all are differentiable under multiplication and inversion. An interesting type of Lie Group is the “Continuous Transformation” Lie Group, where each element transforms via an infinitesimal change under a continuous function.
Group Theory and Lie Groups are excellent tools for studying symmetry, which is useful for physicists since many of the laws and systems used to understand the physical world have symmetry. Quantum Mechanics is no exception. For example, an electron’s up and down spin states, though generally understood to be binary, can be added or superimposed to a combination of probabilities of either state. However, the total probability in the sum will always be the same, since the probability that the electron exists at all does not change. The conservation of probability is actually where the symmetry lies.
This is just one example of why the marriage of Quantum Physics and Lie Groups is so fortuitous. More broadly, the abstraction of physics that occurs when dealing with the quantum realm and the more abstract algebra of Lie groups not only go hand-in-hand but also support each other’s development.