Woit’s Way

Andrew Jordan in Inference Review:

Peter Woit is a senior lecturer in the department of mathematics at Columbia University. Educated at Harvard and at Princeton, Woit is known for his book Not Even Wrong, and for his blog of the same name.1 Quantum Theory, Groups and Representations is based on a series of lectures that he gave at Columbia University.

And it is excellent.

An introduction to quantum mechanics very often follows a well-worn path. Wave functions are defined on a Hilbert space. Observables are represented as operators acting on quantum states. The evolution of a system is specified by Schrödinger’s equation. Analytic functions, Fourier transforms, eigenvalues, and matrices all play their accustomed roles. Matrices are particularly useful in quantum mechanics because, unlike the numbers, they do not necessarily commute under multiplication.

Is there a deeper structure beneath the quantum formalism? Yes, of course there is. It is a structure that appears when the symmetries of a quantum system are under analysis. In 1915, Emmy Noether demonstrated that differentiable symmetries give rise to conservation laws. Her work is a foundational document in quantum theory because it verifies the ancient insight that what is most important in any physical system is what remains the same in the system as the system is changing.

More here.