Search Results

I don't know what you mean by "composition as in $\pi_3(G)$" unless either $M$ is a $3$-sphere or $G$ is, in addition, assumed to be simply connected. If the latter, then $G$ is $2$-connected, and so …

What we mean when we say that two quantities have different units is that if we change something about how we measure quantities, the two quantities will behave differently. For example, if one quanti …

Dolgachev has some lecture notes for an introduction to physics course he taught to math graduate students. Certainly it presumes mathematical maturity.

Mark Levi's The Mathematical Mechanic is a book of examples of how physical reasoning can be used to solve mathematical problems; another couple of examples is in this blog post at Concrete Nonsense. …

In statistical mechanics, the Boltzmann distribution gives the probability of a system being in state $i$ as $$\displaystyle \frac{e^{- \beta E_i}}{\sum_i e^{-\beta E_i}}$$ where $E_i$ is the energy …

Your first requirement suggests to me that you want to think of an infinite-dimensional vector space as an ind-object, namely the filtered colimit of its finite-dimensional subspaces. If so, one forma …

Let me work with arbitrary affine schemes $X = \text{Spec } k$. I believe there is a reasonable notion of a spin structure on a quadratic module $(V, q)$ over $k$, by which I mean a pair consisting of …

You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the classi …

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn q …

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a $3$-d …

Such a thing doesn't exist. The symmetric square of the cotangent bundle of a real $n$-dimensional manifold has dimension $n + {n+1 \choose 2}$, which is in particular odd whenever $n \equiv 2 \bmod 4 …

Well, unlike the determinant, the eigenvalues of an integer matrix aren't integers, so I don't know how much to expect here as far as a direct combinatorial interpretation of any kind. However, there …

Quantum mechanics is not just noncommutative probability; a commutative $C^{\ast}$-algebra alone corresponds via Gelfand duality to some (locally) compact Hausdorff space $X$, which is not equipped wi …

Let's agree that whatever "supersymmetry" means it has something to do with working in the symmetric monoidal category of super vector spaces (e.g. we might want to consider Lie algebras or commutativ …