Dirac & Canonical Quantization

 

(Courtesy of Monica Dirac)

"I went back to Cambridge at the beginning of October 1925, and resumed my previous style of life, intense thinking about these problems during the week and relaxing on Sunday, going for a long walk in the country alone. The main purpose of these long walks was to have a rest so that I would start refreshed on the following Monday.

It was during one of the Sunday walks in October 1925, when I was thinking about this (uv − vu), in spite of my intention to relax, that I thought about Poisson brackets. I remembered something which I had read up previously, and from what I could remember, there seemed to be a close similarity between a Poisson bracket of two quantities and the commutator. The idea came in a flash, I suppose, and provided of course some excitement, and then came the reaction "No, this is probably wrong".

I did not remember very well the precise formula for a Poisson bracket, and only had some vague recollections. But there were exciting possibilities there, and I thought that I might be getting to some big idea. It was really a very disturbing situation, and it became imperative for me to brush up on my knowledge of Poisson brackets. Of course, I could not do that when I was right out in the countryside. I just had to hurry home and see what I could find about Poisson brackets. I looked through my lecture notes, the notes that I had taken at various lectures, and there was no reference there anywhere to Poisson brackets. The textbooks which I had at home were all too elementary to mention them.

There was nothing I could do, because it was Sunday evening then and the libraries were all closed. I just had to wait impatiently through that night without knowing whether this idea was really any good or not, but I still think that my confidence gradually grew during the course of the night. The next morning, I hurried along to one of the libraries as soon as it was open, and then I looked up Poisson brackets in Whittacker's Analytical Dynamics, and I found that they were just what I needed."

Dirac in his graduate days at St. John's College, recounting how he came up with an idea of a general map between a classical and a quantum system. This prescription for going between the classical (Poisson Brackets) and quantum theories (Commutation Relations) is known as canonical quantisation. (Excerpt from Prof. David Tong's notes on Classical Mecahnics)