After a long hiatus, I welcome you all to a fourth Brooklyn Quantum Meetup! We'll be discussing the interconnections between algebra, perspectival geometry, astronomy, and quantum physics. We'll be trying to imagine the night sky from the perspective of an electron. Fixing an observer here and now choses a slice of Minkowski space: all the intersecting worldlines can represented as points on the observer's celestial sphere. Lorentz/Mobius transformations on this celestial sphere represent motion through spacetime: the stars seen from a different point of view. Coincidentally, n-dimensional quantum states can also be represented as points on a 2-sphere, n-1 points specifically. This is called Majorana's stellar representation of the quantum state. Empirically, if you orient your Stern-Gerlach apparatus in the direction of a Majorana star, and send your state through the apparatus, one of the resulting spin states has 0 probability. Thus we can interpret applying Lorentz/Mobius transformations to the Majorana's stars on the sphere as correctly giving us the relativistic transformations of the quantum spin state. How does it work? To each quantum state is associated a complex polynomial, whose complex roots are stereographically projected from the plane onto the sphere, where "poles" of the function (where it goes to infinity in any direction in the plane) are all mapped to the north pole of the sphere. It works because a) the fundamental theorem of algebra: any polynomial equation of degree n has n roots in the complex numbers, and so any equation can be factored into 2d points b) the identification of all poles out in any 2d direction with a single "point at infinity" which is joined to the complex plane. Picture wrapping up the plane into a sphere, you have to add a navel, a bellybutton, where the surface comes together at the north pole. That's the point at infinity. Think of the point at infinity like the point where the railroad tracks meet in the distance. It's a perspectival point, and we've created a perspectival space where poles and roots can be represented on equal footing, and in which we can formulate quantum mechanics and relativity both.
We'll get to play around with some python code I've written that'll help us visualize and interact with the "3d geometry" of quantum spin. We can project it up on the wall!
In other news, I've started a blog that if you've enjoyed this meetup, you'll probably enjoy as well. You can find it at http://timespaceinterchange.com.