Cognitive Dissonances: Maths

Showing posts with label Maths. Show all posts

Why is there a conserved quantity for every continuous global symmetry?

One of the most important theorems in physics is the theorem that states:

"To every differentiable global symmetry of a physical system there corresponds a conservation law"

This theorem is known as the first Noether's theorem, in honor of the great mathematician Emmy Noether, who proved it in 1915 in the context of classical mechanics (both relativistic and non-relativistic, but not quantum). By the way, Noether is part of the group of leading scientists and professors in her field who lost their jobs due to the intolerance of the Nazis when they came to power, as they immediately passed a law preventing Jews and Communists from working in universities and public institutions. This happened before the Holocaust and the Second World War. It is important to remember this so that it does not happen again.

This relationship between symmetries and conservation laws established by Noether is one of the most powerful ideas that human beings have ever had. Conservation laws are a very useful tool for finding out how the quantities of a physical system change over time. Knowing that there are physical quantities that do not change allows us to write equations where the unknowns are the quantities that do change. We can then use the quantities that do not change to find out how the other quantities change.

On the other hand, the symmetries of a physical system are related to its aesthetic aspect. For example, a sphere is beautiful because, no matter how you rotate it, it remains the same. Noether's theorem thus relates beauty to usefulness in physics in a certain way. Pragmatism and aesthetics go hand in hand.

However, for the physics student, it is not immediately evident that a continuous symmetry implies a conserved quantity. Apparently, they are two things that have nothing to do with each other. What is the reason for this relationship?

However, today we know that the world is not classical, but quantum, and that classical mechanics is nothing more than an approximation of the behavior of physical systems in a certain limit. Therefore, Noether's original proof does not serve us for the fundamental laws of nature. Does Noether's theorem still hold in quantum mechanics?

These are the two questions we are going to answer in this post.

Do all chlorine atoms have the same mass?

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 1, 2, 3, and so on. The method of induction requires two cases to be proved:

The first case, called the base case, proves that the property holds for the number 1.
The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1.

These two steps establish the property P(n) for every natural number n = 1, 2, 3, ...

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. (via GIPHY)

Do all chlorine atoms have the same mass?

First we establish a base case for one chlorine atom (n=1). The case with just one chlorine atom is trivial. If there was only one chlorine atom in the universe, then clearly all chlorine atoms in that universe would have the same mass.
We then prove that if in any random set of n chlorine atoms every atom had the same mass, then in any random set of n+1 chlorine atoms every atom must also have the same mass. First, exclude the last chlorine atom and look only at the first n chlorine atoms; all these have the same mass since n chlorine atoms always have the same mass. Likewise, exclude the first chlorine atom and look only at the last n chlorine atoms. These too, must also have the same mass. Therefore, the first chlorine atom in the group has the same mass as the chlorine atoms in the middle, who in turn have the same mass as the last chlorine atom. Hence the first chlorine atom, middle chlorine atoms, and last chlorine atom have all the same mass, and we have proven that: If n chlorine atoms have the same mass, then n+1 chlorine atoms will also have the same mass.

We already saw in the base case that the rule ("all chlorine atoms have the same mass") was valid for n=1. The inductive step showed that since the rule is valid for n=1, it must also be valid for n=2, which in turn implies that the rule is valid for n=3 and so on. Thus in any group of chlorine atoms, all chlorine atoms must have the same mass. We have proof that in any universe, no matter how many chlorine atoms exist, all chlorine atoms must have the same mass.

Nevertheless, it is an experimental fact that it is false that all the atoms of the same element have the same mass. For instance, chlorine's atomic mass of 35.5 a.m.u. is an average of the masses of the different isotopes of chlorine. This is calculated by working out the relative abundance of each isotope. For example, in any sample of Chlorine 25% will be Cl-37 and 75% Cl-35, so there are chlorine atoms with different (35 a.m.u. and 37 a.m.u.) masses.

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook, Twitter or Instagram to give your answers.

Why did the hunter wound the bear?

A hunter is located 100 meters due South from a bear. Then he goes East 100 meters. After that, he looks to the North and he shoots to the North, wounding the bear. Nevertheless, we know that the bear did not move!

You will be even more puzzled after realizing that the question is:

What color is the bear?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.