Related papers: The Cotlar-Stein Lemma, Grothendiecks Inequality a…
The aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding…
Methods from scattering theory are introduced to analyze random Schroedinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz-Krein spectral shift function, which is related to the…
Feynman path integrals are now a standard tool in quantum physics and their use in differential geometry leads to new mathematical insights. A logical treatment of quantum phenomena seems to require a sustained mathematical analysis of path…
The Feynman Path Integral is extended in order to capture all solutions of a quantum field theory. This is done via a choice of appropriate integration cycles, parametrized by M in SL(2,C), i.e., the space of allowed integration cycles is…
This is mainly a brief review of some key achievements in a `hot'' area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying {\em integrable} quantum field theory models with {\em…
The Feynman quantum-classical isomorphism between classical statistical mechanics in 3+1 dimensions and quantum statistical mechanics in 3 dimensions is used to connect classical polymer self-consistent field theory with quantum…
We review completely monotone (CM) and Stieltjes functions, which are classes of functions obeying an infinite hierarchy of positivity constraints. While these are classical concepts in analysis, such properties have recently been shown to…
The scattering matrix (S-matrix), relating the initial and final states of a physical system undergoing a scattering process, is a fundamental object in quantum mechanics and quantum field theory. The study of factorised S-matrices, in…
In this paper we will demonstrate the use of Feynman Diagrams for one dimensional scattering in quantum mechanics. We will evaluate the S-Matrix explicitly for the Dirac delta and finite wall potentials by summing the full series of Feynman…
Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for…
The field of scattering amplitudes plays a central role in elementary-particle physics. This includes various problems of broader interest for collider physics, gravitational physics, and fundamental principles underlying quantum field…
Well defined quantum field theory (QFT) for the electroweak force including quantum electrodynamics (QED) and the weak force is obtained by considering natural unitary representations of a group $K\subset U(2,2)$, where $K$ is locally…
The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct…
This note is an introduction to methods of construction for Hilbert space realizations of relativistic quantum physics. The realizations satisfy a revision to Wightman's functional analytic axioms and exhibit interaction in physical…
The general boundary formulation of quantum field theory is applied to a massive scalar field in two dimensional Rindler space. The field is quantized according to both the Schr\"odinger-Feynman quantization prescription and the holomorphic…
We study path integrals in the Trotter-type form for the Schr\"odinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by…
These notes provide an introduction to recent work by Kevin Costello in which integrable lattice models of classical statistical mechanics in two dimensions are understood in terms of quantum gauge theory in four dimensions. This…
In this paper we will analyse some interesting structures that occur in scalar quantum field theory. We will quantize this theory using path integrals. We will analyse the Bogomolny Bound for scalar quantum field theory in two dimensions.…
Krein space quantization and the ambient space formalism have been successfully applied to address challenges in quantum geometry (e.g., quantum gravity) and the axiomatic formulation of quantum Yang-Mills theory, including phenomena such…
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time…