I am presumably here to give you my perspective on quantum field theory from the point of view of a condensed matter theorist. I must begin with a disclaimer, a warning that I may not be really representing anyone but myself, since I find myself today working in condensed matter after following a fairly tortuous route. I must begin with a few words on this, not only since it will allow you to decide who else, if any, I represent, but also because my past encounters with field theory will parallel that of many others from my generation.

I started life in electrical engineering, as Professor Schweber said in the introduction. As an electrical engineering student, the only field theorists I knew about were Bethe and Schwinger, who had done some basic work on wave guides. When I graduated, I switched to physics and was soon working with Geoff Chew at Berkeley on particle physics. In those days (sixties and early seventies), the community was split into two camps, the field theorists and the S-matricists, and Geoff was the high priest of the latter camp. The split arose because, unlike in QED, where everyone agreed that the electron and the proton go into the Lagrangian, get coupled to the photon and out comes the hydrogen atom as a composite object, with strong interactions the situation seemed more murky.

The profound puzzles posed by quantum critical metals with Planckian dissipation and long-range entanglement, as observed in cuprates and heavy-fermion systems, cry out for a novel point of view. Holography can provide this new perspective. This book will propose that its concrete manifestation in terms of the AdS/CFT correspondence gives qualitatively new insights into these puzzles. The reason is that holography has to be understood above all as a “weak–strong” duality between two different descriptions of the same physics. In this regard it is qualitatively similar to the Kramers–Wannier or Abelian–Higgs duality we reviewed in chapter 2, but it takes the notion to a new level: it relates quantum field theories to a dual description that includes the gravitational force. For an extremely strongly coupled field theory, the weakly coupled theory is now Einstein's theory of general relativity. Vice versa, a strongly interacting gravitational theory has an equivalent description as a weakly coupled quantum field theory.

General relativity inherently contains the notion of a dynamically fluctuating space-time. The remarkable way in which this emerges in holography is by incorporating the renormalisation-group structure of the quantum field theory into the dualisation. As we previewed in the introduction, the renormalisation-group scale becomes part of the geometrical edifice as an additional space dimension in the gravitational theory.

It is still baffling that a quantitative duality relation can exist between two theories in different space-time dimensions. This paradox is resolved, however, by the holographic principle of quantum gravity. This lesson from black-hole physics insists that gravitational systems are less dense in information than conventional quantum field theories in a flat non-dynamical space-time, to the degree that the former can be encoded in a “holographic screen” with one dimension less. The dynamics of this “screen” can be thought of as the dynamics of the dual field theory.

In this chapter we will first provide a brief account of the conceptual and historical background of the holographic principle and in particular its manifestation within string theory (section 4.1). This is where the origin of the AdS/CFT correspondence lies. Fortunately one does not need all this material to understand holographic duality practically. In the remainder of this chapter we approach holography from a constructive angle instead, as was first put together in the excellent review [5].

The term ‘magnetic field’ was introduced by Faraday in 1845, and subsequently adopted by Thomson and Maxwell, whose usage clearly echoed Faraday's. Thomson first used the expression ‘field of feree’ in a letter to Faraday in 1849, following their discussion of the nature of magnetism; and Maxwell first referred to a ‘magnetic field’ in a letter to Thomson in 1854, in the context of a discussion of Faraday's ideas. Maxwell gave the term ‘field’ its first clear definition, in consonance with previous usage, in his paper ‘A dynamical theory of the electromagnetic field’ (1865); there he stated, ‘The theory I propose may therefore be called a theory of the Electromagnetic Field, because it has to do with the space in the neighbourhood of the electric or magnetic bodies’. The concept of a field was to be contrasted with an action-at-a-distance theory of electric action; that is, the mediation of forces by the agency of the contiguous elements of the field existing in the space between separated electrified bodies was to be distinguished from the action of forces operating directly between electrified bodies across finite distances of space.

The inclusive breadth of Maxwell's definition of the field makes it apparent that the physical status of the field was not defined uniquely. In a field theory the forces between bodies were mediated by some property of the ambient space or field.

If this book ever makes it to a second edition, this is most likely to be the chapter that will have to be most thoroughly rewritten. Is there a need in condensed matter physics for a theory that goes beyond the paradigm which we sketched in rough outline in the previous chapter? If so, would the lessons of AdS/CMT which are found in the later chapters be of any relevance for this purpose?

At present the fog of war is still obscuring the battlefield. This war started some thirty years ago with the discovery of high-Tc superconductivity. Before this event, there was a sense that insofar as metals and superconductors are involved the fundamentals could be understood in terms of the “fifties paradigm” of the previous chapter. In the frenzy that followed the high-Tc discovery, experiments showed that strange things were happening. The reaction of the mainstream was to try to tamper with the established paradigm, to accommodate the anomalies. However, Philip W. Anderson, who was very influential back then, took the lead in insisting that new physics is at work in the copper oxide electron systems [86]. This in turn had a great impact on the research agenda. During the subsequent thirty years the field diversified to other materials, while the repertoire of experimental methods employed to study the electrons in solids greatly expanded. Literally millions of papers were written on the subject. But some of the most basic questions formulated in the late 1980s are still awaiting a definitive answer. It is just impossible to do justice to this large and confusing literature in the present context (see e.g. [87]). We will therefore present here a small selection of subjects, which is intended to form a minimal background for the holographist to communicate with the condensed matter community.

Back in the late 1980s the great puzzle was why the superconducting transition temperature could be as high as 150 K, given that the conventional phonon mechanism runs out of steam at 40 K or so. It was also realised early on that the electron systems in cuprates are characterised by unusually strong inter-electron repulsions. An aspect that is well understood in these systems is the microscopic physics.

This chapter covers applications of quantum computing in the area of condensed matter physics. We discuss algorithms for simulating the Fermi-Hubbard model, which is used to study high-temperature superconductivity and other physical phenomena. We also discuss algorithms for simulating spin models such as the Ising model and Heisenberg model. Finally, we cover algorithms for simulating the Sachdev-Ye-Kitaev (SYK) model of strongly interacting fermions, which is used to model quantum chaos and has connections to black holes.