I'd disagree with the title of this a bit. Rather than describing it as "probability to allow minus signs", I'd describe it as "probability in L^2 instead of L^1". The author actually discusses this a bit towards the end.
It makes sense to ask what is the distance between two continuous probability distributions. It's given by:
\int | p1(x) - p2(x) | dx
L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors".
Quantum mechanics changes this to:
\int |p1(x) - p2(x)|^2 dx
Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc.
So I'd argue that QM is probability in L^2.
It makes sense to ask what is the distance between two continuous probability distributions. It's given by:
\int | p1(x) - p2(x) | dx
L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors".
Quantum mechanics changes this to:
\int |p1(x) - p2(x)|^2 dx
Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc.
So I'd argue that QM is probability in L^2.