March 2, 2020
Why is there just 1 thing that acts like 1?
2 nice things about 1:
and
There are number systems with many left units, or many right units. But if there's a left unit and a right unit, there's just one.
For any set you can define a funny multiplication like this: $$ ab = b $$ Then every element is a left unit! This multiplication is even associative.
Similarly, if you define
$$ab = a$$ then every element is a right unit.
But suppose you have a multiplication with a left unit, say \(L\). Then there can be at most one right unit! For suppose you have two right units, say \(R\) and \(R'\). Then $$ R = LR = L = LR' = R' $$ Similarly, if you have a multiplication with a right unit \(R\), there can be at most one left unit. For suppose you have two left units, say \(L\) and \(L'\). Then $$L = LR = R = L'R = L'$$ So if your multiplication has both a left unit \(L\) and a right unit \(R\), they are both unique.
Furthermore, they are equal! Why? We've already seen why: $$L = LR = R $$ In the battle of left and right units, both win... so they must be equal!
So: if you have a binary operation with a left unit and a right unit, they are both unique — and they're equal.
If we call the binary operation 'multiplication', then it makes sense to call this unique left and right unit '1'.
After all, there's just one.
March 7, 2020
The drastic measures taken by China to contain coronavirus seem to be working there, if their data can be trusted.
However, outside China the disease is spreading rapidly.
Why is there more matter than antimatter? Nobody knows, but the Standard Model does allow a process where 9 quarks and 3 leptons all annihilate each other! This 'sphaleron' process can also turn 9 antiquarks into 3 leptons. Is that where the antimatter went?
The sphaleron process is nonperturbative, so you can't understand it using the usual Feynman diagrams in the Standard Model. The minimum energy required to trigger it is about 9 TeV, but it's hard to get enough particles to collide to make it happen!
The early universe was very hot. As it cooled there was probably an electroweak phase transition (EWPT): bubbles formed, in which the electromagnetic and weak forces became different. Sphalerons might form at the bubble walls, preferentially destroying antimatter.
Alas, the best calculations people have done so far do not predict a 'first-order' electroweak phase transition &mdash a phase transition involving latent heat, like freezing water. So, the bubble walls would not be discontinuous enough to create lots of sphalerons.
So, physicists are looking for earlier phase transitions to explain 'baryogenesis': the creation of more baryons (protons, neutrons, etc.) than antibaryons.
Here's an excellent introduction:
Meanwhile, a dramatic announcement from U. C. Riverside! Three days ago classes were going to be done electronically for the first week of the next quarter. Now that's changed:
Since we got there after 7:30 on a Saturday night it's perhaps not surprising that the hoarders had taken all the paper towels and bleach and chicken. We managed to get hand lotion and peanut butter and something else. I forget to look for vinegar. The clerk said that they have a special 'seniors hour' from 6:00 to 7:00 to let people over 65 get the first crack at the goods. So, Lisa may try that.
At the ribs joint there was one woman eating dinner at the bar — maybe a friend of the management? The waitress took a little while to bring out our order. I took off my mask because it made me uncomfortable and there were just this one woman and two employees there.
At home I tried to take the ribs and onion rings and sweet potato fries out of the packaging without letting the packaging contaminate our house too much, and I wondered how much this was even possible. Of course I think there's a low probability that the packaging had COVID-19 viruses on it, since only a small number of cases have been detected in Riverside so far, but I figure I should get in the habit of being very careful, since things will only get worse. The whole idea of eating food that someone else has prepared becomes a lot less attractive to me under these cicumstances.