To my regret, despite that I frequently invoke her name (she co-starred with Albert in the Special Relativity series), her work in mathematics is pretty far above my head, and I’m simply not qualified to write about it. I can say that her work connects mathematical symmetry with physical conservation laws. She also made significant contributions to abstract algebra.
Among physicists, Emmy Noether is most notable for her (first) theorem, which connects symmetry with physical conservation laws. For instance, conservation of energy and the conservation of angular momentum both derive from Noether’s theorem.
To quote Wikipedia, that theorem is:
[E]very differentiable symmetry of the action of a physical system has a corresponding conservation law. […] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.
If you look at the rest of the Wiki page, you’ll see that it gets seriously mathematical very quickly. (Starting with integrals and partial derivatives, both of which are beyond my current grasp.)
That’s about all I can say about this aspect of Noether’s work. Calculus is still mostly a mystery to me!
I can keep up a little better when it comes to algebra (at least the more pedestrian forms of it). Noether’s other body of work, the one most notable with mathematicians, is a little more accessible to me.
That said, I’ve only just begun to scratch at the surface of this. Suffice to say it’s a large area of mathematics, just parts of which are deep enough for a lifetime of study (let alone trying to take it all in).
The part of this where I have dipped my toe is called Group Theory.
This, as one might guess, is the study of groups (and operations performed on members of the group).
Obviously, we need to start with what a “group” is (mathematically speaking).
We already have a general sense of what a “group” is — a bunch of things we somehow identify as belonging together.
A mathematical group is a specific distillation of that idea. Crucially, it introduces the idea of an operation that can be applied to elements (members) of the group.
Doing so, applying the operator, results in a new element which must qualify as a member of the group. The operation is said to be closed, or the group is said to be closed under the operator.
For example, the integers form a closed group under multiplication.
The operator (multiplication) can be applied to any two elements of the group (of integers), and the result is always another integer.
¶ Another group requirement is an identity element. There must be some element that, when used in the operation with a second element, always results in (a copy of) that second element.
In the integer multiplication group, the number 1 is the identity element. Any number multiplied by 1 is that number. (Under integer addition, 0 is the identity element.)
¶ A group must also have an inverse element. Essentially, this is an element that can generate the identity.
Under integer multiplication, 1/x is the inverse element for x, because multiplying them returns 1, the multiplicative identity.
Under integer addition, -x is the inverse element for x, because adding them returns 0, the additive identity.
What’s really important (and interesting) here is that group theory applies to far more than just integers (or even the reals, which are their own group).
Group theory generalizes — abstracts — a behavior seen in things ranging from quantum particles to Rubik’s Cubes.
This is because we can treat states of a Rubik Cube as elements of a group. For instance, the pristine state, the “solved” state, is one element.
The various twists of the cube are the operator. Twisting the cube gives a new orientation, a new state — an element of the group.
This means group theory connects with permutation mathematics, since we can view states of the Cube as permutations and the operator as a function returning a new permutation.
I’ve been exploring group theory as it applies to something even simpler than twisting a Rubik’s Cube — I’ve been considering the symmetrical rotations of plain old squares, cubes, and tesseracts.
By “symmetrical” I mean rotations of 90° (or multiples of 90°) — rotations that would leave an unmarked object looking the same before and after.
We do mark, or label, the object so we can keep track of the rotation, but on some level, the rotations are symmetrical.
Consider figure 1. It shows four rotations of a square, which we’ll label I, II, III, and IV. They show rotations of, respectively, 0°, 90°, 180°, and 270°, although each represents a 90° rotation from the previous position.
Our group is the four rotations, our elements, and the operator is a 90° rotation, which we can apply successively to generate all four. Note how applying the operator to element IV returns us to element I.
What’s crucial here is that the order of the corner labels (a, b, c, d) does not change. It always winds clockwise around the square.
In contrast, consider figure 2.
It shows two other (separate) types of operation, that — for the moment — I’ll call flips.
The top example flips the I element along its horizontal (X) axis. The bottom example flips the same I element along its vertical (Y) axis.
In contrast to the (proper) rotation shown above, these operations reverse the order of the corners! Now they wind counter-clockwise.
We might also think of a flip as a mirror or reflection operation because of how it reverses the order.
Note that either the red or blue square can be rotated in 90° steps to produce four elements (of which the red and blue squares are two). All four of such elements have reversed corner order.
What’s really happening here is that what I keep calling the “proper” rotation turns the square on what would be its Z-axis if the square were described in three dimensions. (In two dimensions, it’s just the center point.)
A proper rotation does not disturb the orientation of the parts. (It’s proper!)
What I called a “flip” is, in fact, a rotation through the third dimension. The first one (red) along the X-axis and the second (blue) along the Y-axis.
But this requires rotating the 2D square through an additional dimension, in this case the Z-axis. (In these diagrams the Z-axis would stick into and out of the screen.)
Since squares don’t have a third dimension (they’re so square), this is called an “improper” rotation, a key effect of which is that it reverses or mirrors parts of the whole.
It turns out there’s kind of a neat mathematics that unifies all this into a wonderfully elegant package that easily and naturally extends to objects in three, four, or more, dimensions.
I’ll get into that (matrix rotation) in future posts!
If I ever hope to grasp quantum theory, group theory is a prerequisite. (One of many!)
And, if nothing else, in my dotage, it might help keep my mind from turning to complete mush. “Use it or lose it!” (As they say.)
Or not. In any event:
Happy Birthday to Emmy Noether!
Your legacy shines bright! Your work is a foundation of modern physics, and you are revered by mathematicians (and plain math lovers) everywhere. Ya done good!
Stay symmetrical, my friends!