⇤Math diary
May 7, 2023:
Today I read a lot more about conformal groups, especially the conformal group of Minkowski space $\mathbb{R}^{1,1}$. This is truly infinite dimensional, and is isomorphic to $\text{diff}(\mathbb{R}) \times \text{diff}(\mathbb{R})$. Compactifying to $\mathbb{S}^{1,1}$, the minkowksi torus, we embed this in $\text{diff}(S^1) \times \text{diff}(S^1)$. This fact is why the diffeomorphism group of the circle plays such a big role in 2D conformal field theory.
I read about the lie algebra $\text{Lie}(\text{Diff}(S^1)) \cong \text{Vect}_{S^1}$, and its comlpexification, the Witt algebra. In quantum mechanics, we are interested in projective unitary represetnations of algebras, which are classified by possible central extentions. The Witt algebra has a unique nontrivial central extention, the Viasoro algebra. This is the fundamental object of study of 2D conformal field theory. All representations of 2D conformal field theory are built on verma modules of the Viasoro algebra, thought of like fock spaces. In a rather cursed fact, there is no lie group with lie algebra the Viasoro algebra, in stark contrast to finite dimenisional lie algebras.
Finally, I found another instance of the fact from yesterday, that field therory is geometry on the space of functions. If you start with a classical phase space $M$, you get a quantum theory by studying $L^2(M)$, functions on $M$. The space of quantum states $\mathbb{P}(L^2(M))$ is an infinite dimensional symplectic manifold. If we treat it as a new classical system, we can quantize again, and look at functions on the space of functions. This procedure is called Second quantization, and yeilds Fock spaces and quantum field theory. The operators which come up in a second quantized theory have some (to me) subtle functional analysis, called “reduced operators”, which also come up in the theory of loop groups. I think I should try to understand this example well, if I want to understand loop spaces.