Bibliography:

Pauli matrix.

This is a good and simple first example of Lie algebra to look into.

Take the group of all Translation in ${\mathbb{R}}^1$.

Let's see how the generator of this group is the derivative operator:

The way to think about this is:

So let's take the exponential map:and we notice that this is exactly the Taylor series of $f(x)$ around the identity element of the translation group, which is 0! Therefore, if $f(x)$ behaves nicely enough, within some radius of convergence around the origin we have for finite $x_0$:

This example shows clearly how the exponential map applied to a (differential) operator can generate finite (non-infinitesimal) Translation!

A law of physics is Galilean invariant if the same formula works both when you are standing still on land, or when you are on a boat moving at constant velocity.

For example, if we were describing the movement of a point particle, the exact same formulas that predict the evolution of $x_{land}(t)$ must also predict $x_{boat}(t)$, even though of course both of those $x(t)$ will have different values.

It would be extremely unsatisfactory if the formulas of the laws of physics did not obey Galilean invariance. Especially if you remember that Earth is travelling extremelly fast relative to the Sun. If there was no such invariance, that would mean for example that the laws of physics would be different in other planets that are moving at different speeds. That would be a strong sign that our laws of physics are not complete.

The consequence/cause of that is that you cannot know if you are moving at a constant speed or not.

Lorentz invariance generalizes Galilean invariance to also account for special relativity, in which a more complicated invariant that also takes into account different times observed in different inertial frames of reference is also taken into account. But the fundamental desire for the Lorentz invariance of the laws of physics remains the same.

Some sources distinguish "invariant" from "covariant" such that under some transformation (typically Lie group):TODO examples.

Bibliography:

Physics from Symmetry by Jakob Schwichtenberg (2015) page 66 shows one in terms of 4x4 complex matrices.

More importantly though, are the representations of the Lie algebra of the Lorentz group, which are generally also just also called "Representation of the Lorentz group" since you can reach the representation from the algebra via the exponential map.

Bibliography:

One of the representations of the Lorentz group that show up in the Representation theory of the Lorentz group.

TODO understand a bit more intuitively.

Two observers travel at fixed speed relative to each other. They synchronize origins at x=0 and t=0, and their spacial axes are perfectly aligned. This is a subset of the Lorentz group. TODO confirm it does not form a subgroup however.

Generalization of orthogonal group to preserve different bilinear forms. Important because the Lorentz group is $SO(1,3)$.

TODO who bought the Bitcoins? Is anyone else besides Jeremy Sturdivant

The original forum thread bitcointalk.org/index.php?topic=137.msg1195 suggests multiple purchases were made, until he had to withdrawl the offer. Perhaps an easier question is how many pizzas he got in the first place.

www.reddit.com/r/Bitcoin/comments/13on6px/comment/jl55025/?utm_source=reddit&utm_medium=web2x&context=3 mentions without source:One source is: bitcoinmagazine.com/culture/the-man-behind-bitcoin-pizza-day-is-more-than-a-meme-hes-a-mining-pioneer

Related thread from May 2023: bitcointalk.org/index.php?topic=5453728.msg62286606#msg62286606 "Did Laszlo Hanyecz exchange 40000 BTC for 8 pizzas, not 10000 BTC for 2 pizzas?" but their Googling is so bad no one had found the 100,000 quote before Ciro.

As per bitcoin.stackexchange.com/questions/113831/searching-the-blockchain-based-on-transaction-amount-and-or-date at blockchair.com/bitcoin/outputs?s=time(asc)&q=value(1000000000000),time(2010-05-18..2010-08-05) we can list all the transactions made between the offer and withdrawal dates for value exactly 10k. There are only about 20 of them, and including someone the 22nd of May, so it is extremely likely that this will contain the hits. No repeated recipients however, so it is hard to progress with more advanced analytics tools

Some of the transactions are:8 d1a429c05868f9be6cf312498b77f4e81c2d4db3268b007b6b80716fb56a35ad (29 May) is a common looking transaction with a single input from 1Bc7T7ygkKKvcburmEg14hJKBrLD7BXCkX and two outputs, one likely being the change to 1GH4dRUAagj67XVjr4TV6J9RFNmGYsLe7c and the other the actual value to 138eoqfNcEdeU9EG9CKfAxnYYz62uHRNrA.

The input chain is complex, but it does contain one block reward on the third level: 17PBFeDzks3LzBTyt6bAMATNhowrvx5kBw + 79 rewards 4th level at 045795627ca29ec72a94c23a65ee775ea1949d60b6fba0938b75e1cfe1e6643e.

Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.

First we can observe that the exact matrices are different. For example, taking the standard matrix of $O(2)$:and:both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector $x=(1,0)$, then $x\cdot x=1$. But after a rotation of 90 degrees, it becomes $x_2=(0,1)$, and now $x_2\cdot x_2=2$! Therefore, we have to search for an isomorphism between the two sets of matrices.

For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as: