Concept Deep Dive: Noether's Theory, Symmetries and Operators

It's relatively easy to describe what Noether's theory says in abstract, less so where it comes from - and to a lesser extent where it goes. That's what we're here for today. Starting from the abstract description, we have gain symmetries for every conserved Noether current and every symmetry nets us a conserved quantity. There are of course some didactic holes in this application of Noether's theorem, i.e. that the connections here are a little bit wishy-washy. Matching the conserved quantity to the symmetry for example is really more of a separate labour of love, and intuitively I would like this to be a straight path, even if we have to do a little algebra to get there.

From the algebra front, we get a collection of groups, which may or may not display a set of special properties. In our case, we're explicitly looking for symmetries. We then introduce group symmetries.

A physical problem mapped onto spacetime will have some geometry in any solution space. These symmetries construct a group (let's call it "T") over that space. Let's take a look at the group axioms to convince ourselves of that first:

Any symmetric operation lands the geometry of the problem back where it started. We call this invariance under transformation. Of course the transformations here are just elements of the group. However this also means that all combinations of transformations in the group T are in T. We remind ourselves of the group axioms:

For all a, b, c in G: (ab)c = a(bc)

For all a, b in G: ab, ba are also in G

G has an unique identity element e

Every element g in G has an inverse element g* with gg* = e = g*g

As well as the definition of cyclic groups and their generators. The generator of a cyclic group is a subset of the group, out of which all other elements of the group can be constructed. Now, naively this is a somewhat redundant concept to introduce at this point, but in actuality, one has to make a distinction on which the geometries the group transformations are invariant over specifically. Take the group of 90° rotations around the Cartesian 3 dimensional axes.

In the pictured example, two rotations are done successively on a rotationally non-invariant shape. We see that the final position is not the same, so rotational groups are not abelian. Of course this is true for rotations not exactly by 90°.

Transformations are really only interested in the outcome, the path doesn't matter to us. We notice that this group is at the very most of degree 24 (6 "dice directions" with 4 orientations each). This automatically qualifies this group as finite. In fact, all discrete symmetry groups are finite. This may be interesting to know for now, but should not invite anybody to identify all elements of a discrete symmetry group. It's always advised to stick to the generators when describing cyclic groups.

We notice that such problems only crop up in non-Abelian groups, because we will have to make a distinction between the order of operations. Finding an abelian group consisting of invariant transformation thus is the easiest case to handle.

Though it's easy to visualize transformations in space, there are a number of other symmetries. The spatial symmetries are the most varied and interesting ones, but in general, physics identifies Parity, Charge, and Time. Everything else falls under gauge transformation, which are a whole different can of worms and we'll leave as an anecdote for now. Charge and time are very one-dimensional (at least for the most part) and so their symmetries don't even necessarily need to be visualized to be grasped, however they have to be dealt with a little differently than spatial (or space parity) symmetry. At this point, we want to not only look at the geometry of a problem, but rather at the geometry of its solution.

Often, there are Ansatzes that dictate the solution of a problem while not being remotely close to the solution. In theoretical physics, this is usually the Lagrangian or the Lagrangian Density. Take charge parity for example. One can always flip the charge, but this won't do anything with the geometry of the starting position. It's not useful then to apply the transformation at this point, but the solution (i.e. the equations of motion) are a direct consequence of the Lagrangian. Then it's sensible to apply the transformation to the Lagrangian and verify that the resulting equations of motion don't change before or after the charge symmetry transformation.

Time symmetry is a little easier, since we only ever really ask for homogeneity in time, i.e. we don't want to make distinctions depending where we start the physical process in time. In small enough sections of space-time that is assumed to be true (Einstein's Equivalence Theorem), and I can't be asked to find an example where it isn't for myself for the complexity this entails. Instead I want to take another look at the generators, and how to get them into a readable form.

Spatial symmetries are quite varied, considering that those are the only ones that can be symmetrical along several spatial dimensions. Like time symmetry, we can usually assume translation symmetry in space. For the sake of readability, I'll pretend we did a coordinate transformation before, so that the translation happens along one of the axes. Let's first assume that the symmetry we're looking at is discrete. Then, we would require a function f(x) with this symmetry to exhibit the behavior f(x) = f(x + dx) where dx is the fixed interval for the symmetry. An example of such a function would be the sine or cosine function. However, this is not usually the case that we're looking for. We are now introduced to the problem of continuous symmetries. Of course one can attempt to let dx tend to 0, but this alone won't yield a lot of useful results in itself. The background of these symmetries are interesting. Of course the group underlying this symmetry can't be finite, because ostensibly the coordinate axis is infinite, and doesn't loop back in on itself, and the number of translation on this axis is then infinite. In fact, considering we're looking to map a group bijectively onto a continuous group (the real numbers in this case), the generating group can't be cyclic and must be continuous as well. The smoothness that is required for all the operations to be combined with one another as group operation dictates, identifies the underlying group as a Lie group. We can even narrow this down further, but for this I will need to make a little jump.

In some previous posts, we've seen the significance of commutators for quantization. We won't be able to avoid dipping our toes into quantum mechanics for this topic entirely, but I want to put that off until a little later and just remind ourselves that those commutators come from something called the Poisson-Brackets, a binary operations that helps arrive at Hamiltonian equations of motion. They also satisfy Leibniz's law, so in sum, we're working with a Poisson algebra. In the Poisson algebra we can define a unique group of maps from an algebra of observables to itself, and the full time derivative is equal to the Poisson-Bracket of the element and the map. However, as the map is a group action, it retains group operations, so composing chains of these maps is unproblematic. Suppose we take an element of the Poisson algebra as the Hamiltonian. Then, the map would describe the time evolution of observables with the quantity as an argument that is only conserved, if it maps onto itself. If the same holds true when the element and the argument are reversed, we say that the conserved quantity generates symmetries of the Hamiltonian. We get this relation from the anti-symmetry of the Poisson-bracket, so the bracket of element and quantity should evaluate to zero both ways. They commute. This is the connection between observables (operators) and symmetries that we were looking for in Noether's theorem. This holds true up to quaternionic spaces, which I won't touch on here, but rather leave until algebraic topology is more present for me.

A short note on notation: In the realm of quantum mechanics, having something commute is always very nice, since it gives very concrete behavior in respect with the Hamiltonian. Because of the uncertainty principle leveraging the Fourier transformation, there already this precedent of exponential functions dictating behavior. Without getting too much into the definitions, all quantum mechanical objects exist in Hilbert space. The argument of the exponential functions usually includes a product of two commuting quantities. Having understood the symmetries as commuting with the Hamiltonian, symmetry operations turn out in the form of exp(k AB), where k is some factor with mass dimension 1 and [A, B] = 0. This gets us the generators of quantum mechanical operators.