Err… I mean happy 12th anniversary, Lindsay! ❤ On a beautiful summer day in August 2008, Lindsay and I eloped at the municipal building overlooking the majestic Detroit river. We rented Pineapple Express in the Windsor Casino Hotel room and drove back to the University of Waterloo in my 15 year old Hyundai Scoupe, which was called the “scope” since the ‘u’ had gone missing for several years. Sigh… young love. Millennials just don’t understand romance these days…
See, we went back to the University of Waterloo that day because I was just finishing up my Masters degree. But fast forward two weeks earlier and I had another, this time professional, milestone. My very first academic paper was published. It has been at least 10 years since I have looked at it, or even thought about the topic I was researching back then. I thought, when I looked back on it now after 12 years of intense research and academic experience, I would cringe when reading it. Much to my surprise, though, it is actually not bad. In fact, it is actually still pretty good! So, let me tell you about it.
First, the ugly academic title, Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations (arXiv link). Though, it’s actually quite descriptive if you know the jargon. Let’s unpack it.
Frame: A frame is mathematical object used in linear algebra. It it a generalisation of a basis, which is the minimal set of objects that let’s you build any other object. A simple example is the Cardinal directions, north and east. From some place, you can get to any other place by saying how far you need to travel north and east. All four Cardinal directions are more than a basis — they are a frame.
Representation of quantum mechanics: quantum mechanics is an abstract mathematical theory. It’s like a sphere. A basketball is a representation of a sphere, as is a ping-pong ball. And we can think of the two as the “same” in the sense that they are both representations of a sphere. In quantum mechanics, we have several concrete sets of tools that are representations of the abstract rules of the theory. The most famous are Schrodinger's wave mechanics, called the “Schrodinger representation”, and Heisenberg’s matrix mechanics, called the “Heisenberg representation”. But there is another representation that has been around since the early days of quantum mechanics.
Quasi-probability: in 1932 Eugene Wigner proposed a way to solve quantum mechanical calculations in the same way one would have done in classical physics with uncertainty. The rules were simple equations that showed how probabilities of particle positions and velocities should be updated and changed. And the quantum calculation fit perfectly into this framework, except for one small hiccup — the probabilities kept going negative. In fact, it would seem that the more “quantum” the phenomenon being calculated, the more these “negative probabilities” showed up. Since “negative probability” is an oxymoron, a new name was given to this representation: quasi-probability.
So, the title says “this paper is about a new kind of representation we call frame representations and how we can use them to answer questions about these negative probabilities that show up in quasi-probability representations”. Either way it’s a mouthful.
The reason I still like this paper when going back to it is because we were able to distil a lot of research into a simple framework. We showed how any representation whatsoever uniquely identified a frame, and vice-versa — they are in perfect correspondence. These representations can be nasty and full of terrible notation and mathematical gymnastics that are hard to follow. But we united them all by showing they only differ essentially by the frame that defines them.
Among the myriad of representations out there, many were variations on Wigner’s original proposal. There are dozens of so-called quasi-probability representations out there. One reason why people keep trying for more is the search to make sense of these “negative probabilities”, or what is now called negativity. In some representation they would show up often, and in another, not at all. The big open question was: is negativity necessary? To be fair, everyone knew it to be true, since there would be no difference between quantum and classical physics otherwise — but nobody had yet proven it. It was folklore.
Using our frame representation formalism, we were able to prove it quite easily. Of course, in the paper, the grand finale comes at the end. So, there are lots of definitions and things you need to know first to understand how it works. However, I’ll show you here because I think it beautiful how succinct it is.
Ahh… it is a thing of beauty, I must say!
I can say at the time I felt about it the same way I feel about my work now: meh. When it’s done, it’s done. The fun is doing it. And when I am done with a a project, I don’t usually feel proud because I want to move on to the next project, which I hope will be even better!
I actually rather enjoyed this exercise, though, because for the first time in long while, I actually feel proud. And that’s a nice feeling to have circa 2020.
If you are a student writing your first paper, know this. This paper had dozens of versions, including one called “long version” that was over 30 pages (the final published version is 10 pages). I know how frustrating it can be nearing the end, when you just want it done, and you are thinking no one is even going to read this. Well, one person will read it. It might be 12 years from now, but believe me, you’ll want them to be proud when they do.