When I was an undergraduate at the University of Pennsylvania, I took a class in general relativity (GR). It was taught by the eccentric Professor Jeffrey Cohen, not to be confused with the equally eccentric Professor Michael Cohen.
Michael Cohen had instilled in me a deep appreciation for truly understanding physics. Just as Michael Cohen felt that he would never attain the depth of understanding commanded by his adviser, THE Richard Feynman , I too feel that I will never approach the physical intuition of Michael Cohen. It is fortunate that the singularities that we call great physicists are born with abilities far superior to their contemporaries.
After more than three decades have passed, I recall little from my undergraduate flirtation with GR. However, some of the mathematical formalism of differential forms has taunted me for much of my career. I recall Jeffrey Cohen mentioning a paper on the topic of the properties of a black hole in some complex geometry that took forty pages of derivations in an article that appeared in The Physical Review. Using the trickery of differential forms, he was able to solve the problem in just a few steps.
The trick was to formulate the problem in a coordinate independent way, then to project the results into the coordinate system that reflected the symmetry of the problem. In contrast, the Physical Review paper used the inelegant brute-force approach of picking the coordinate system up front, and then by necessity painstakingly plodding through all the messy mathematics.
Given the complexity of the problems that we work on as a matter of daily routine in our research, I am always looking for simplifying tools. In teaching various classes, my intention is to sneak in a little bit of differential forms to wet the appetites of my acolytes and to teach my old brain some new tricks. Furthermore, the geometric interpretation of the mathematics adds a deeper layer of understanding.
In the upcoming spring semester, I am teaching graduate statistical mechanics for my first time. As usual, preparing for a new course if filled with grand excitement. You can imagine my elation when I realized that a homework assignment in the textbook could be done with ease using differential geometry. Since then, it has been difficult for me to think about anything else.
The problem is a simple one that normally requires a bit of math. The student is to show that the 6N-dimensional volume element in phase space for an N-particle system is invariant under a canonical transformation. To put this into simple English, the problem seeks to show that a transformation of coordinates does not change the nature of the results. Be reformulating the problem so that the volume element is represented as a wedge product of what are called one-forms, the volume element is shown to be the same when the so-called Poisson bracket yields unity -- the requirement of a canonical transformation. Thus, the problem is solved without the need for messy mathematics.
This realization makes me feel like a kid at Christmas. Ironically, tonight is Christmas Eve, the focal point of my family's celebration. My father has made what may be his last trip to Pullman from Philadelphia. He is 94 and still lives on his own, drives a car, and prepares meals for senior citizens at the Ukrainian Cultural Center in Fox Chase, Pennsylvania. Though still vigorous, his body betrays the telltale signs of wear and tear due to old age. Both of my children are home for the holidays, and all the fragrance from the traditional Ukrainian foods simmering on the stove and in the oven permeate the house. As I write this post, my wife is busily making last-minute preparations.
It is fortunate for me that my family values my passion for physics, and allows me to occasionally be a recluse. Just a few minutes ago, my wife called out a query about my whereabouts. I simply answered, "I am excited about something." Though she undoubtedly had some mundane duty for me to perform, she immediately signaled her understanding of my state of mind, and left me alone. I am truly fortunate to be living with someone who shares in my passions.
The intensity and meaningfulness of spirituality that I derive from physics far exceeds all others, including the times in my distant past when I had embraced religion. As my family turns in for the night, I continue to sit at my desk, full of excitement in my new-found understanding, and looking forward to sharing this understanding with my family and my students. It is a truly privileged life that allows me to rekindles the child-like wonder of Christmas on a daily basis.