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Given a compact, simple Lie group $G$, and a compact, oriented three manifold $M$, we can consider the following smooth homotopy groups: $(C^\infty(M;G)/{\sim},\cdot )$: Here, $\sim$ is smooth …

Consider $\mathfrak U(\mathfrak H)$, the CAR algebra over a separable Hilbert space $\mathfrak H$. The states $E_{\mathfrak U}$ over this algebra are defined to be positive linear functionals of norm …

This is primarily a linear algebra question, but for motivation I want to state this question in its natural, global context. Whenever we have a non-relativistic quantum field theory (renormalized, o …

Let $\tilde{\mathcal H}$ be a Hilbert space, and let $L(\tilde{\mathcal H})$ denote the corresponding space of linear operators. By fixing a basis, we can, via Fourier transform, identify an important …

Consider the following chain $\{A_1,A_2,A_3,\cdots,A_{n}\}$ of orbit spaces of even-rank anti-symmetric tensors, where $$A_k:=\frac{\Lambda^{2k}(\mathbb{R}^{2n})}{e_{i_1}\wedge \cdots \wedge e_{i_{2k} …

In page 9 of the introductory chaper of Renormalization and Effective Field Theory (the introductory chapter is available free here), Kevin Costello defines a propagator $P$ for the Laplace operator $ …

There is an elegant formulation of the Bogolyubov transformation in terms of Clifford algebras. Note that a quadratic Hamiltonian (noted by a hat), is a hermitian element of the representation of a Cl …

Letting the standard Clifford algebra of dimension $2k$ be denoted by $Cl_{2k}$, let's denote the corresponding complex Clifford algebra via $$\mathbb{C}l_{2k}\equiv Cl_{2k}\otimes_{\mathbb{R}}\mathbb …