arXiv subjects

Theo Johnson-Freyd

Publications and source records attributed to Theo Johnson-Freyd.

40 records · Page 3Linked to original sources

arXiv2012

The fundamental pro-groupoid of an affine 2-scheme

A natural question in the theory of Tannakian categories is: What if you don't remember $\Forget$? Working over an arbitrary commutative ring $R$, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid $π_1(\spec(R))$, i.e.\ the separable absolute Galois group of $R$ when it is a field. This gives a new definition for étale $π_1(\spec(R))$ in terms of the category of $R$-modules rather than the category of étale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of $π_1$ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the étale fundamental group of a scheme preserves finite products but not all products.

math.CT↗

arXiv2010

The formal path integral and quantum mechanics

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.

math-ph↗

arXiv2010

Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics

We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on R^n, with magnetic and potential terms. In particular, for each classical path γconnecting points q_0 and q_1 in time t, we define a formal power series V_γ(t,q_0,q_1) in \hbar, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) satisfies Schrödinger's equation, and explain in what sense the t\to 0 limit approaches the δdistribution. As such, our construction gives explicitly the full \hbar\to 0 asymptotics of the fundamental solution to Schrödinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams.

math-ph↗

arXiv2010

On the coordinate (in)dependence of the formal path integral

When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences.

math-ph↗