So currently I'm reading the Birkhoff–Neumann (1936) quantum logic paper 1, and I'm basically stuck in the interpretation of section (13) (p. 833-834); for what I understand, they are trying to construct the elements of a projective field trough a definition of partially ordered inclusion of n-tuples of a field F.
In any case, they start by defining a "right-ratio" $[x_{1}:...:x_{n}]_{r}$ as n elements of F not all = 0, and additionally they define an equivalence relation with any other "right-ratio" iff there is a $z \in F$ with $\xi_{i} = x_{i}z$; a definition for a "left-ratio" $[y_{1}:...:y_{n}]_{l}$ with the equivalence relation expectedly reversed for non-commutative cases, another "left-ratio" being equal iff $z \in F$ with $\eta_{i} = zy_{i}$.
I get lost however when they start defining the relation with a projective geometry, it is stated: a n-1-dimensional projective geometry $P_{n-1}(F)$ is defined as having as "points" the right-ratios $[x_{1}:...:x_{n}]_{r}$, and that the "linear subspace" of $P_{m-1}(F)$ are those sets of points, which are defined by systems of equations:
$\alpha_{k1}x_{1}+ \cdots + \alpha_{kn}x_{n} = 0, k=1, \cdots, m$ ,and state "$m=1,2, \cdots ,$ the $\alpha_{ki}$ are fixed, but arbitrary elements of F"
After this they continue on and state: "The same considerations show, (n-2-dimensional) hyperplane in $P_{m-1}(F)$ correspond to $m=1$, not all $\alpha_{i} = 0$. Put $\alpha_{1i} = y_{i}$ then we have (*) $y_{1}x_{1}+\cdots+y_{n}x_{n}=0$, not all $y_{i}=0$." This supposedly proves a one-to-one correspondence between left-ratios and hyperplanes, and furthermore, the inclusión of "points" in the "hyperplanes" as the definition of "incidence".
So given all that, my doubts are the following:
- ¿Am I supposed to understand that each "right-ratio" is some kind of "linear dependance" given the equation and each "subspace m" part of the equivalence class defined at the beginning?¿Or what is the systems of equations trying to express?¿what does the change of the n-1 index to the m-1 index even imply?
- ¿why do "hyperplanes" only correspond to $m=1$?¿How does substituting the $\alpha_{1i}=y_{i}$ make the left-ratios hyperplanes?
- And returning to the field F ¿what is the intuition on the subset relation between the "right-ratios" and "left-ratios" that is implied by the "incidence" relation between hyperplanes and points?
I know those are a lot of questions, I'm not expecting a measured answered for each of them, more than anything, I'm looking for general intuition that can make sense of the series of definitions that are given in this part of the paper. Addtionally, any interpretation of the actual meaning of the n-tuples in respect to quantum mechanics entities would be appreciated, but I understand if it's out of the bounds of the question.
1: Birkhoff, Garrett; von Neumann, John “The logic of quantum mechanics.” Annals of Mathemathics (2nd series) 37 (1936), no. 4, 823–843. 2