SIXTH MEETING

Hey everybody! I know this isn't much of a heads up, but if you're available this Saturday, come on down to our SIXTH QUANTUM MEETUP. (We could also meet Sunday instead, if people are interested in that--let me know!) Ideally, we'll be discussing a number of topics. First: the polarization of a photon. I have some real life linear and circular polarizing filters that we can play around with and put in sequences (as far as I know, this is the cheapest "quantum" experiment money can buy!). We'll discuss how the "ellipse" traced out by a propagating light wave corresponds (mathematically and physically, in a beautiful way) to a qubit! It turns out we can apply much of what we know about spin systems and Stern-Gerlach apparatuses to photons and polarizing filters: the sphere, as usual, comes to our rescue. I have some 3D graphics to demonstrate. Then, time allowing, I'd like to discuss some foundational issues in mathematics. I want to give a sense of how quantum mechanics arises in a natural and organic way from the concept of "number" itself, stretching back to the Ancient Babylonians and earlier. We'll see how due to the simple demand of consistency, the concept of number evolved from something you can count like a pebble and put in a pile, to something you can measure out with a string, or enclose with a fence, to something which can convert one currency to another, or sound against another to create harmony or dissonance in accordance with its primes. But then numbers like the square root of 2 were discovered to be irrational and -1 to have no square root at all? We need to move to 2D: to be consistent with algebra, numbers turn out to have to be right triangles, aka points on a plane. Multiplication now corresponds to a stretch by a hypotenuse and a rotation by an angle. Furthermore: if you fold up the 2D plane into a sphere, closing it at the north pole by adding an extra "point at infinity," we might say a number is: a point on the sphere aka a qubit. (At this juncture, we'll return to the polarization ellipse and consider it anew.) But is there a next step? We'll begin to explore how the amazing answer to that is just: more points on the sphere!--thanks to the fundamental theorem of algebra. (To wit: there's a progression from: natural numbers -> integers -> rationals -> complex numbers aka monomials aka qubits -> polynomials. Then miraculously, polynomials turn out to be vectors and matrices in disguise--and we have linear algebra! (Bonus: Continuing the story, since all finite groups have linear representations, this takes us pretty far up the hierarchy of abstraction, all within the sphere. Also, lots of infinite groups have linear representations: like the symmetry group of the sphere itself, for example. Then, there are the nonlinear infinite groups...) Anyways, hope to see you there!