Noether’s Theorem: How Symmetry Shapes Physics
In 1915, mathematicians David Hilbert and Felix Klein came to Emmy Noether with a problem [1]. Einstein had published the field equations for his general relativity earlier that same year, and there seemed to be a worrying hole in the theory. Under certain circumstances, one of the most fundamental principles of physics was violated. Energy was not conserved.
Emmy Noether was born in 1882 to a wealthy Jewish family in Bavaria [1]. Her father, Max Noether, was a maths professor known for his studies in algebraic geometry [2], but it was languages which originally interested Emmy [1]. It was only after having qualified as a teacher of English and French that she began studying mathematics at university level, albeit unofficially as women were not allowed to matriculate at German universities at the time [1].
This rule was changed in 1904, and, by 1907, Noether had been awarded her doctorate. Her thesis was in the realm of abstract algebra, specifically on invariants — properties of functions or groups of functions which remain unaltered when the function is transformed [3] — and it was in this area of mathematics that she swiftly built her reputation, to the extent that she was invited to join the German Mathematical Society in 1909 [1].
It was as an expert in invariants that Hilbert and Klein came to Noether in 1915. If anyone could find a way to plug the gap in the theory, it would be her. They were not wrong. Noether’s resolution led to one of the most elegant and powerful results in theoretical physics.
A statement of Noether’s theorem is:
If the Lagrangian of a system has a continuous symmetry, then there exists an associated quantity which is conserved by the system, and vice versa.
Let’s unpack this statement. First, a conserved quantity is any property of a system which remains constant over time. For example, if I putt a golf ball, then the mass of the ball doesn’t change in the time between when I strike the ball and when it (hopefully) goes in the hole. Thus, the mass of the ball is a conserved quantity of the system.
Conversely, the speed of the ball changes over time, whether through friction with the grass or the contours of the green. The velocity of the ball is not a conserved quantity in this case.
Next is the idea of a continuous symmetry. Think back to primary school, and the idea that a square, say, is rotationally symmetric when rotated by an angle of 90°. This means that, when we rotate a square by 90°, the final state looks exactly like we hadn’t done anything at all. This, however, only holds for certain angles. If we were to instead rotate by 45°, the final state would be markedly different to the original orientation. So, we say a square has discrete rotational symmetry.
Now imagine performing the same process on a circle. This time, however, no matter what angle we rotate the circle by, the circle always looks exactly the same, even if that angle is a tiny fraction of a degree. This means the circle has continuous rotational symmetry.
Lastly, what is the Lagrangian of a system? To describe the Lagrangian, we must first understand another of the fundamental ideas in physics: The Principle of Least Action, or PLA as physicists affectionately know it. In effect, this states that the universe is lazy. Physical systems proceed in a manner which minimises the “effort” necessary for the evolution of the system from one state to another. We quantify this “effort” as what is known as the action of the system.
For example, when I hit my putt earlier, in broad terms the physical system evolved from the “ball at my feet” state to the “ball in the hole” state. There’s nothing stopping the ball from travelling the 10 feet from my feet to the hole via Timbuktu, except that the action of the system would be far larger than if the ball had followed the trajectory which we would expect. This latter path, the physical trajectory, is the trajectory which minimises the action. This is the PLA at work.
How does this relate to the Lagrangian? The Lagrangian describes how the energy of the system should change during a process to minimise the action of the system. By examining how it behaves over space and time in a set of differential equations known as the equations of motion, we can determine how the system evolves from one state to another according to the PLA.
So, back to Noether’s theorem. What does it mean for a Lagrangian to have a continuous symmetry? If, when transformed continuously along some coordinate, the Lagrangian for a system is unchanged, then the system is said to be continuously symmetric about that coordinate.
So, consider a classic physics exam question: the collision of two identical balls on the x-axis. Assuming no friction or air resistance, it can be easily shown that dynamics of the system depend only on the difference between the positions and velocities of the balls, and not on their absolute values.
If we translate both balls by the same arbitrary amount in the x direction, then the difference in position and velocity of the two balls is unaltered. Thus, the system must behave in the same way as if it hadn’t been moved at all. Since this behaviour is encoded in the Lagrangian, this means that it also cannot be altered by a translation in the x direction. This means that the Lagrangian for this system must be continuously symmetric in the x direction!
Noether tells us that this symmetry implies a conserved quantity, and in the case of this translational symmetry, the conserved quantity is momentum [4]. This is the origin of the law of conservation of momentum! The very tool we use to solve problems related to this scenario comes about as a consequence of the symmetries of the Lagrangian describing the system.
Similarly, if a Lagrangian for a system is rotationally symmetric, then angular momentum is conserved in the system [4]. The Lagrangian describing the gravitational force of planets is rotationally symmetric, so angular momentum is conserved in the orbits of planets, for example. Even electric charge is conserved due to a symmetry, this time the slightly more esoteric idea of local phase invariance of the wavefunction, the details of which deserve their own article.
A table showing some common symmetries and their associated conserved quantities. From [4].
This is fantastic, but how does it solve our problem in general relativity? Recall that Hilbert and Klein had realised that under certain circumstances, energy was not conserved by general relativity. We know that energy is usually conserved, so, using Noether’s theorem, what symmetry is invoked by conservation of energy? The answer is time. If a system is symmetric under time translations — if the Lagrangian of the system does not explicitly depend on time — then it must conserve energy [4]. And this, Noether realised, was the answer.
Time in general relativity is not an absolute quantity like it is in the Newtonian world, it flows and warps as spacetime is curved by the contents of the universe. Time translation symmetry only holds in general relativity under certain special circumstances, namely in the case of a flat or asymptotically flat spacetime, so, in general, energy need not be conserved!
So, through a combination of her expertise in abstract algebra and a fantastically analytical mind, Noether not only plugged a hole in one of the most important theories of the 20th century, but brought to light a truly fundamental idea in theoretical physics. Her theorem underlies a huge amount of the physics which we encounter daily, both in the classroom and in the world at large.
And yet, Noether is, for the large part, unknown. This is a woman who was described by Einstein himself as
“The most significant, creative, mathematical genius thus far produced since the higher education of women began.” [5]
Despite this, she never occupied a permanent faculty role. Hilbert was forced to advertise her lecture courses under his own name at Göttingen, such was the opposition to female academics in the university hierarchy at that time [1]. With the rise of the Nazis in 1933, she moved to the US, and was due to begin a position at Princeton alongside Einstein before her sudden death from cancer in 1935 [1]. We can only imagine what might have been discovered had fate allowed their genius to combine.
References
[1] J. O’Connor and E. Robertson, “Emmy Amalie Noether,” November 2014. [Online]. Available: https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Emmy/. [Accessed 3 June 2020].
[2] J. O’Connor and E. Robertson, “Max Noether,” December 2008. [Online]. Available: https://mathshistory.st-andrews.ac.uk/Biographies/Noether_Max/. [Accessed 3 June 2020].
[3] Encyclopedia of Mathematics, “Theory of Invariants,” [Online]. Available: https://encyclopediaofmath.org/wiki/Invariants,_theory_of. [Accessed 3 June 2020].
[4] D. Feng and G. Jin, Introduction to Condensed Matter Physics, vol. 1, Singapore: World Scientific Publishing, 2005, p. 18.
[5] D. Radford, On Emmy Noether and Her Algebraic Works, Governor’s State University, 2016.
