Emmy Noether

Emmy Noether was born in 1882 in Germany to become one of the greatest mathematicians of all time and yet, to remain still nearly unknown by most of us. Below are the highlights of such an extraordinary person and life.

Double jeopardy

She faced the double handicap of being Jewish and a woman in early 1900s. She was born to become a mathematician. Her father was a math professor. She just loved maths and it is fair to say that it was her life’s main passion. However, as a woman she could not register for classes at the university. It is only through a loophole that she managed to attend classes and obtained a master’s degree (with honors!) without ever registering. She could not get a faculty position and took unpaid positions until 1922. Even after migrating to the US in 1931, she had periods of unemployment. By then, she was probably the world’s best mathematician, and she could not get a job at Princeton or anywhere else for that matter!!!

Incredible set of friends, in the thick of things!

She managed to be assistant lecturer at the university of Gottingen which at that time was  the Holy Grail of mathematics and physics, where all the heavy weights were working. Planck had just discovered the quantic nature of light, so it was an exciting time of discovery. During her tenure at Gottingen, from 1908 to 1931, her friends and mentors were giants of general relativity and quantum mechanics such as Felix Klein (group theory and automorphic functions), Hermann Weyl (gauge theory, classical groups), David Hilbert (physics’ mathematization, ), and Minkowski (geometry of relativistic spacetime). An incredible set of friends!!! Even Einstein was her mentee: he came to her for help in formulating the equations of general relativity, which she solved for him. When she died of complication of ovarian cyst surgery at the age of 53, the New York Times published a short, unremarkable obituary. Einstein had them correct it with a long personal statement emphasizing the importance of her life and scientific contribution. https://mathshistory.st-andrews.ac.uk/Obituaries/Noether_Emmy_Einstein/

Her legacy to the world: the Noether’s Theorem

In 1916, she formulated the Noether’s theorem. which states that “for every symmetry, there is a corresponding conservation law.” In other words, “the conservation laws of nature can be formulated from fundamental symmetry of a system”. Let’s try to understand what this means.

Conservation laws include conservation of mass (now conservation of mass and energy after Einstein’s Theory of Relativity), conservation of linear momentum, conservation of angular momentum, and conservation of electric charge.  There are also many approximate conservation laws, which apply to such quantities as mass, parity, lepton number, baryon number, strangeness, hypercharge, etc. These quantities are conserved in certain classes of physics processes, but not in all.

In physics, a symmetry of a physical system is a physical or mathematical feature of the system that is preserved (covariance) or remains unchanged (invariance)  under some transformation. Emmy was focused on continuous and smooth transformations over physical space. Such fundamental transformations or symmetries are rotations, spatial translations, reflections, and glide reflections. Examples of invariance include rotation of a sphere around any of its axes, rotation of circles around an axis perpendicular to their plane and passing through their center. Examples of covariance include mirror symmetry which transforms a system into a covariant form called its chiral with all dimensions unchanged. It is somewhat confusing that the term symmetries is often applied to the group of transformations that leaves a system invariant or covariant.

We also know one of the most fundamental principles of physics: the principle of least action, which was developed successively by Fermat, Maupertuis, Euler, Lagrange and Hamilton. Nature is essentially lazy and accomplishes things in the easiest way it can. Action is defined at any moment as the difference in kinetic (the energy of motion) and potential energy (the energy available to do work). So, any movement is characterized by a path of least action, that minimizes the aggregate of difference in kinetic energy and potential energy at every step. Think of kinetic energy as energy needed by a car to move, and potential energy as the energy contained in its fuel. In nature, cars naturally move efficiently, with the minimum of gas consumption. Nature loves laziness and actually equals it to perfection!

Emmy intuited that the laws of nature that we observe are mere consequences of applying the basic principle of least action to systems while effecting symmetries. The conservation of principle of least action under spatial translation yields Newton first and second laws of gravitation. Its conservation by time translation yields the law of conservation of energy (first law  of thermodynamics) which essentially states that energy can never be created nor destroyed: it can only change form. Conservation by rotation yields the conservation of angular momentum.

The laws of nature preserve the coherence of what we observe through invariance or covariance. Emmy Noether’s theorem demonstrates that these laws of nature are dictated by the geometry of space and time. Laws of motion result from space geometry and work, heat and mass from the geometry of time. space and time combine to be a thing, a “fabric”. The laws of nature are determined more by the fabric of space and time than by the objects that nature contains.

A continuing cornerstone of physics research  

Noether’s theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system. As an illustration, suppose that a physical theory is proposed which conserves a quantity X. A researcher can calculate the types of Lagrangians that conserve X through a continuous symmetry. Due to Noether’s theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.

Wow Emmy! You lived at a most exciting time for physics and a worst possible time for Jewish German woman.