Condensed matter physics is not axiomatic
When I was an undergraduate I loved taking courses in pure mathematics and physics. I never took the "Solid State Physics" course because the person who taught it was a hopeless teacher. In my honours year (final year) I wrote a thesis on "Gravitational Lenses" that involved proving some theorems in General Relativity. I wanted to do a PhD in mathematical physics, which is why I chose to take an offer from Princeton rather than Cornell. Nevertheless, I am very glad I ended up doing condensed matter.
I only just realised that there is a basic and fundamental thing about condensed matter that distinguishes it from all the physics courses I took and loved as an undergraduate. In simple and loose terms, CMP is not axiomatic. I don't meet this in a rigorous mathematical sense, but rather the following. Consider classical mechanics, thermodynamics, statistical mechanics, electromagnetism, special relativity, general relativity, and quantum mechanics. For all of them, particularly at the undergraduate level, you can write down just a few equations (or laws) and everything else follows. It can almost become an exercise in applied mathematics. At least that is how I viewed it. This is why undergraduate physics can be quite easy for nerds who are good at calculus and linear algebra.
On the positive side, this "axiomatic" character to these physics subjects is rather beautiful because of the simplicity of the fundamental laws/equation. Furthermore, in some cases, one can argue that one subject can be largely summarised in a single variational principle, such as the extrema of the action.
In contrast, CMP really does not have comparable "axioms" or laws. All it has are certain organising principles such as spontaneous symmetry breaking, emergence, quasi-particles, topological order, ...
Perhaps, I mean CMP is not reductionist or "fundamental", rather than not axiomatic.
This lack of an axiomatic character may be a second reason why CMP is hard for undergraduates, particularly for those like me who have found other physics subjects "easy".
What do you think?
As someone who just finished my master's degree (and had my first condensed matter class), I can definitely agree at how condensed matter can seem like a grab bag of techniques. Because of that, I was initially turned off by it, but I've come to appreciate this aspect of the subject.
ReplyDeleteWhat I find very hard is the rigorous need for approximations. It is necessary to find the relevant energy scales and the appropriate simplifications.
ReplyDeleteOften enough we find only afterwards that the approximation was good and much later why it was good. It often boils down to: we use this method, because it just happens to work surprisingly well.