When someone asks you why it is −i here instead of i or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.
There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for E<0 is u:=p2+p02p2−p02n^+p2+p022p0p, where u is the position of the particle on 3-sphere (a 4-dimensional vector),
p is the momentum of the original particle in Kepler problem, n^ is a vector perpendicular to the 3-dimensional hyperplane where p lies, and p0:=−2mE.
When someone asks you why it is here instead of or the other way around, you can say that this is just a convention. My professor of quantum mechanics once asked the class similar a question, and I replied with this letter.
There is a canonical transform of the Kepler problem which is the same as the problem of motion of a free particle on 3-sphere. The explicit formula of the transform as well as some links about this topic is written in the article. The explicit formula for is where is the position of the particle on 3-sphere (a 4-dimensional vector),
is the momentum of the original particle in Kepler problem, is a vector perpendicular to the 3-dimensional hyperplane where lies, and .