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R. E. Borcherds

Publications and source records attributed to R. E. Borcherds.

8 recordsLinked to original sources
arXiv2011

Renormalization and quantum field theory

The aim of this paper is to describe how to use regularization and renormalization to construct a perturbative quantum field theory from a Lagrangian. We first define renormalizations and Feynman measures, and show that although there need not exist a canonical Feynman measure, there is a canonical orbit of Feynman measures under renormalization. We then construct a perturbative quantum field theory from a Lagrangian and a Feynman measure, and show that it satisfies perturbative analogues of the Wightman axioms, extended to allow time-ordered composite operators over curved spacetimes.

math-ph↗
arXiv2002

Lectures on Quantum Field Theory

These are notes from a 15 week course aimed at graduate mathematicians. They provide an essentially self-contained introduction to some of the ideas and terminology of QFT.

math-ph↗
arXiv1999

Classification of positive definite lattices

In this paper we describe an algorithm for classifying orbits of vectors in Lorentzian lattices. The main point of this is that isomorphism classes of positive definite lattices in some genus often correspond to orbits of vectors in some Lorentzian lattice, so we can classify some positive definite lattices. As an application we give the classification of the 665 25-dimensional unimodular positive definite lattices and the 121 even 25 dimensional positive definite lattices of determinant 2. We also use this algorithm to show that there is a unique 26 dimensional unimodular positive definite lattice with no roots.

math.NT↗
arXiv1999

The Leech lattice and other lattices

This is an unpublished manuscript written in 1983-4. It contains several results about lattices (=integral quadratic forms) including the classification of the unimodular lattices in dimensions up to 25 and the construction of unimodular lattices with no roots in dimensions 26 and 27.

math.NT↗
arXiv1999

Reflection groups of Lorentzian lattices

The aim of this paper is to provide evidence for the following new principle: interesting reflection groups of Lorentzian lattices are controlled by certain modular forms with poles at cusps. We use this principle to explain many of the known examples of such reflection groups, and to find several new examples of reflection groups of Lorentzian lattices, including one whose fundamental domain has 960 faces.

math.GR↗
arXiv1998

What is moonshine?

This is an informal write up of my talk in Berlin. It gives some background to Goddard's talk (math.QA/9808136) about the moonshine conjectures.

math.QA↗