Quantum Field Theory in Low Dimensional Space Time

Matthew P. A. Fisher

R. Grobe

Gerald V. Dunne

Quantum field theory (QFT) is a rich and robust part of mathematical and theoretical physics, with a wide range of applications in statistical physics, condensed matter physics, particle physics and string theory. Many (if not most) of the applications in four space-time dimensions require some sort of systematic approximations. In contrast, quantum field theories in lower dimensions often display special properties which open the door, at least in principle, to their exact solution. In many cases, the methods used to obtain exact solutions are peculiar to low-dimensional space-times, but the exact solutions thus obtained can teach us a great deal about more realistic QFTs in higher dimensions. Thus, lower-dimensional QFTs provide excellent models for exploring deep theoretical and computational issues, in addition to their direct applications to physical processes in reduced dimensions, which include polymers, boundary fluctuations, spin chains, and quantum Hall systems. Such models are also vital testing grounds for fundamental theoretical ideas such as integrability, phase transitions, collective and non-perturbative quantum behaviour, and non-equilibrium statistical mechanics. We should also mention the effective dimensional reduction which occurs in the high temperature limit, or is induced by disorder. Low-dimensional quantum field theory thus sits at a unique intersection point of contemporary mathematics and physics, allowing many of these issues to be addressed rigorously and analytically. Surprising inter-relations continue to arise, from the ubiquitous nature of the Bethe ansatz and Calogero models, to the link between integrable spin chains and the anomalous dimensions of operators in gauge theories. Aspects of the latter connection have been known at least since the 1990s; they have recently received tremendous attention in connection with N =4 supersymmetric Yang--Mills theory (in four space-time dimensions) and the AdS/CFT correspondence. One unifying idea is the Bethe ansatz, whose 75th anniversary we celebrate this year, and whose wide-ranging impact in diverse branches of modern mathematical physics is made abundantly clear in this special issue. In addition, new ideas and techniques of significance to both mathematics and physics have emerged: the recent and spectacular progress on geometrical aspects of phase transitions made possible through the ideas of stochastic (Schramm) Loewner evolution, or SLE, is a particularly noteworthy example. All papers in this special issue are invited, refereed contributions from leading experts. Many are pedagogical reviews, capturing the current state of knowledge in a particular subject, putting the subject in context and outlining open challenges. Others are research papers, reflecting the rapid flux of new ideas and results in this rich and exciting field. Together, they represent a broad sample of the entire field. We would like to take this opportunity to thank all of the authors for their efforts in making this special issue possible, and Kazuhiro Hikami for his help in the early stages of this project. We hope it inspires many new developments!

Shailesh Chandrasekharan A

To study quantum field theories on a quantum computer, we must begin with Hamiltonians defined on a finite-dimensional Hilbert space and then take appropriate limits. This approach can be seen as a new type of regularization for quantum field theories, which we refer to as qubit regularization. A related finite-dimensional regularization, known as the D-theory approach, was proposed long ago as a general framework for all quantum field theories. In this framework, the dimensionality of the local Hilbert space at each spatial point can increase as needed through an additional flavor index. To reproduce asymptotically free QFTs, most studies assume that qubit-regularized theories require extending the local Hilbert space to infinity. However, contrary to this common belief, recent discoveries in (1+1) dimensions have revealed two examples where asymptotic freedom appears to emerge within a strictly finite-dimensional local Hilbert space through a novel renormalization group (RG) flow. These findings motivate further investigation into whether asymptotically free gauge theories could also emerge within a strictly finite-dimensional local Hilbert space. To support these explorations, we propose an orthonormal basis called the monomer-dimer-tensor-network (MDTN) basis and use it to construct new types of qubit-regularized lattice gauge theories.

Jonathan Bain

Pieter Naaijkens

Anyons, comprising a special type of particles that exhibit braid statistics,\nhave received considerable attention in the past decade, from physicists,\nmathematicians, and computer scientists. It turns out that so-called modular\ntensor categories naturally appear in the description of the key properties of\nanyons, which partly explains the interest of mathematicians in the subject.\nIn this thesis, we study the way such modular tensor categories emerge in two\nclasses of models, viz. quantum field theory in low-dimensional Minkowski\nspace-time, and quantum spin systems on an infinite spatial lattice. Although\nthe nature of these two classes of models is quite different, we show in this\nthesis that they may both be discussed in a similar (mathematically rigorous)\nframework. This approach is inspired by the work of Dpplicher, Haag and\nRoberts (DHR) in algebraic quantum field theory.\nThe DHR approach leads to braided tensor categories, which describe the charges\nin the theory (in our case, these are anyons). These categories need not be\nmodular. In this thesis we present a method to remove an obstruction for\nmodularity, in the case of a space-time of dimension three. We then study (a\nvariant of) Kitaev's toric code, regarded as a quantum spin model on an\ninfinite lattice. Using techniques inspired by the DHR programme, we show that\none can obtain a modular tensor category describing all relevant properties of\nthe anyons in this system. Finally, some generalisations to non-abelian models\nare discussed.