Noether's Theorem and the Symmetries of Reality (Script) (Patreon)

Conservation laws are  among the most important tools in physics. They feel as fundamental as you can get. And yet they’re wrong - or at least they’re only right sometimes.  These laws are consequences of a much deeper, more fundamental principle: Noether’s theorem.

Conservation laws are the cheat codes of physics. They make it possible to solve physics problems that would otherwise be painfully difficult, or even impossible. More than cheat codes – conservation laws are close to the source code – they emerge from profound and simple truths about the basis of reality. They emerge from the fundamental symmetries of nature. The connection between conservation laws and symmetry is encapsulated in Noether’s theorem. But before we dive into this extremely elegant idea, let’s talk about the seeming paradox that inspired it, and the genius who discovered it.

When it was published in 1915, Einstein’s general theory of relativity opened as many new questions as it answered. Among them is the fact that energy is not always conserved in general relativity. The simplest example of this is the case of cosmological redshift. As the universe expands, light traveling through that expanding space is stretched out – its wavelength increases, and so the energy of each photon drops. Where does the energy from redshifted photons go? In 1915 the expansion of the universe hadn’t yet been discovered, but the failure of energy conservation was still clear from the math of general relativity.

Two of the greats of the era - David Hilbert and Felix Klein- sought the help of a young mathematician, Emmy Noether, to understanding this seeming paradox. She

discovered why the law of conservation of energy broke down in general relativity. The law was NOT fundamental after all. She realised that ALL conservation laws arise from a more fundamental relationship, which we call Noether’s theorem.

A simple expression of Noether’s theorem is this: For every continuous symmetry of the universe, there exists a conserved quantity. Let’s unpack this. First, what do we mean by symmetry? Actually, first, what don’t we mean. We say a face is symmetric if it looks the same under a mirror reflection. Snowflakes are symmetric under 60 degree rotation, playing cards under 90 degree rotation. But these are what we call discrete symmetries – single flips around one axis, or rotation by specific amounts. Noether’s theorem applies to continuous symmetries. Something is continuously symmetric if it stays the same for any size shift in a given coordinate.

For example a long road in the middle of nowhere is continuously symmetric under spatial translation in the direction of the road. A perfect sphere is continuously symmetric under rotational translations. In both cases, the environment stays the same for shifts along the symmetric coordinate. In the case of Noether’s theorem, when we say the environment stays the same, we mean the equations that give the laws of motion for the system. For example,  moving along a perfectly flat road, the downward force of gravity stays constant. We have symmetry to spatial translation, and Noether’s theorem tells us there’s a corresponding conserved quantity – that quantity is momentum. If two cars collide on that road, the sum of their combined momentum stays the same.

But what if the road is hilly? Momentum doesn’t appear to be conserved – it can be lost or gained to the gravitational field. This is because the direction of the gravitational field changes with respect to the road – it’s not symmetric to translations along the road. On the other hand, the gravitational field across the whole stretch of road doesn’t change from one in point in time to the next. The system is symmetric to time translations. It doesn’t matter when the collision happens, the results are the same. Noether’s theorem reveals that this time-translation symmetry gives energy conservation.

And the last classic example – if the factors driving the laws of motion are symmetric under rotation, for example in the spherically symmetric gravitational field experienced by a satellite orbiting the Earth, then Noether’s theorem predicts another conserved quantity – angular momentum.

By revealing the underlying source of the conservation laws, Noether’s theorem handily explains when and why they are broken. That includes the apparent breaking of conservation of energy in general relativity. See, Einstein’s description of gravity reveals the dimensions of space and time to be dynamic and changeable. If the very nature of space can change over time then continuous time symmetry is broken. That’s the case with the expanding universe: energy CAN be lost in the case of cosmological redshift, and it can be created from nowhere in the case of dark energy.

The law conservation of energy is fundamental in Newtonian mechanics, in which space and time are unvarying and eternal. But in Einstein’s universe, energy conservation is only valid as a special case; it only applies for parts of the universe where we can approximate space as unchanging over time. However it’s possible to use Noether’s theorem to come up with an analogous quantity that IS conserved. That quantity is the rather esoteric Landau-Lifshitz pseudotensor. It “saves energy conservation” by incorporating the entire universe’s gravitational potential energy to offset the seeming gains or losses to redshift and dark energy. This quantity is, however, controversial in its application and interpretation. It’s also worth an entire episode, so I’ll leave it alone. . . for now.

Despite it’s profound implications, the math behind Noether’s theorem is surprisingly straightforward. It falls like magic out of another deep law of the universe: the principle of least action, which states that the universe will always “choose” the path between two states that minimizes the change in the action. This is a rather abstract quantity that measures the effort involved in moving between two states over some time interval. It’s a generalization of Fermat’s principle, which states that light will always take the path between two points that minimizes the travel time. The principle of least action extends Fermat’s principle to any object moving on any path, or indeed any system – quantum mechanical to cosmological – evolving between two states.

The principle of least action can  be used to derive of laws of motion from the equations of classical mechanics to Feynman’s path integral formulation of quantum mechanics. In a sense the principle is axiomatic – it’s a founding assumption behind these derivations, and it’s also the founding assumption behind Noether’s theorem. However, as a founding assumption that is as close to fundamental as we get.

Noether’s theorem allows us to figure out the true conserved quantities for any system that is evolving according to the principle of least action, as long as we can identify that system’s symmetries. This is useful in cosmology, but it’s also useful in quantum physics. The general nature of the theorem means we can apply it not just for symmetries in the dimensions of space and time, but also more abstract stuff. For example, another conserved quantity in physics is electric charge. Noether’s theorem tells us that there should be a corresponding symmetry of nature. And there is. That symmetry is the phase of the quantum field. You can rotate the complex phase of an oscillation in a quantum field by any amount and the observable properties of that field - like its particles - don’t change. This symmetry leads to the conservation of electric charge and electric current. 

This quantum symmetry is simplest of a large number of symmetries exhibited by quantum fields: the so-called gauge symmetries. They predict a rich family of conserved charges that govern the interactions of the particles of the standard model. For example, the color charge of quantum chromodynamics describes the strong interaction between quarks and gluons. The entire standard model of particle physics is what we call a gauge theory – it’s founded on the fundamental symmetries of quantum fields. It’s going to take a few episodes to explain what exactly what these symmetries really are, and how they lead to the family of particles and interactions that make up our universe. And we’ll jump into that before too long.

Emmy Noether was one of the greatest mathematicians of golden era of modern physics, but she gained little recognition in her time. Even 11know she’s only known to the more eager students of math and physics. During her life she was repeatedly refused any paid academic position until late in her career. Hermann Weyl, also a giant in the mathematical foundation of quantum mechanics, said in her  memorial address that “I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as mathematician in many respects”. Einstein called her a genius. Her contributions to mathematics, particularly abstract algebra, redefined entire fields. Let’s be thankful that she took a moment to offer a little of her genius to physics, taking us a step closer to understanding the fundamental workings of the universe through the continuous symmetries of Space Time.