Symmetry is one of the most powerful ideas in physics. Emmy Noether, the most important woman in the history of mathematics, determined that every continuous symmetry in a physical system results in a conserved quantity. This is called Noether’s Theorem. Every beginning physics student learns that any system’s total energy, momentum, and angular momentum are conserved. Noether’s Theorem explains why. Each law comes from one of our universe’s symmetries. Let’s take a look.
Conservation of Energy
If each conservation law stems from an underlying symmetry, what causes energy conservation? First of all, what does it mean for energy to be conserved? Suppose we take a box and count the total energy inside. If come back later and count again, we should get the same answer, provided nothing entered or left the box. So, for an isolated system, whether we turn the clock forward or backward, we should see the same energy. That’s energy conservation. If you pick any system and study its “equations of motions”, which are the equations which govern its behavior, you’ll find that the equations look the same at any time
t0 and at time
t0 + t1. Or in other words, the laws of physics are symmetric in time. Energy is conserved because the laws of physics (or the universe) are homogeneous in time, i.e., whether we turn the clock forward or backward the equations are the same.
Conservation of Momentum
Similar to the discussion for energy, if we compute the total momentum for the box now and the total momentum at a later time, then the total momentum should be unchanged. If we look at the laws of physics, like we did for energy, we’ll find that they don’t explicitly depend on position. The laws of physics are the same no matter where you are in the universe. If we are at position
x0 and then shift to position
x0 + x1, the laws of physics are unchanged. This symmetry is called the homogeneity of space and it causes momentum conservation.
Conservation of Angular Momentum
You’re probably catching on now. When the laws of physics don’t explicitly depend on a variable like time or space, we get a conservation law. So what about rotational symmetry? If we rotate a system through an angle, the equations of physics are the same. So, there is no preferred direction in space. This symmetry is called the isotropy of space and it causes the conservation of angular momentum.
The Underlying Mathematics
If you look at classical (or quantum) mechanics, you’ll encounter Noether’s theorem and see that there’s a conservation law between every pair of canonically conjugate quantities. Each member of a pair will be an “observable” quantity that we could measure. One member of the pair is what we usually think of as an independent variable, like time, and can produce transformations in the system (like translation in space or time). The other will be a physical quantity like energy. So, at this level of understanding, time and energy are canonically conjugate. The same is true for space and momentum and for angular position and angular momentum.
Every conservation law stems from some underlying continuous symmetry. Energy conservation is due to the homogeneity of time, momentum conservation to the homogeneity of space, and the isotropy of space produces angular momentum conservation.
To learn more about this, you can take a look at any of the following resources, listed in order of increasing sophistication:
Richard Feynman’s The Character of Physical Law; from Cornell’s Messenger Lectures
John Baez’ Classical Mechanics lecture notes (http://math.ucr.edu/home/baez/classical/)