This book is written with the graduate student in mind. I had in mind to write a text that would introduce my students to the basic ideas and concepts behind many-body physics. At the same time, I felt very strongly that I would like to share my excitement about this field, for without feeling the thrill of entering uncharted territory I do not think one has the motivation to learn and to make the passage from learning to research. Traditionally, as physicists we ask “what are the microscopic laws of nature?”, often proceeding with the brash certainty that, once revealed, these laws will have such profound beauty and symmetry that the properties of the universe at large will be self-evident. This basic philosophy can be traced from the earliest atomistic philosophy of Democritus to the most modern quests to unify quantum mechanics and gravity. The dreams and aspirations of many-body physics interwine the reductionist approach with a complementary philosophy: that of emergent phenomena . In this view, fundamentally new kinds of phenomena emerge within complex assemblies of particles which cannot be anticipated from an a priori knowledge of the microscopic laws of nature. Many-body physics aspires to synthesize, from the microscopic laws, new principles that govern the macroscopic realm, asking: What emergent principles and laws develop as we make the journey from the microscopic to the macroscopic? This is a comparatively modern and far less familiar scientific philosophy. Charles Darwin was perhaps the first to seek an understanding of emergent laws of nature. Following in his footsteps, Ludwig Boltzmann and James Clerk Maxwell were among the first physicists to appreciate the need to understand how emergent principles are linked to microscopic physics. From Boltzmann's biography [1], we learn that he was strongly influenced and inspired by Charles Darwin. In more modern times, a strong advocate of this philosophy has been Philip W. Anderson, who first introduced the phrase “emergent phenomenon” into physics. In an influential article entitled “More is different,” written in 1967 [2], he captured the philosophy of emergence, writing: The behavior of large and complex aggregations of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles.[…]
This book is written with the graduate student in mind. I had in mind to write a text that would introduce my students to the basic ideas and concepts behind many-body physics. At the same time, I felt very strongly that I would like to share my excitement about this field, for without feeling the thrill of entering uncharted territory I do not think one has the motivation to learn and to make the passage from learning to research. Traditionally, as physicists we ask “what are the microscopic laws of nature?”, often proceeding with the brash certainty that, once revealed, these laws will have such profound beauty and symmetry that the properties of the universe at large will be self-evident. This basic philosophy can be traced from the earliest atomistic philosophy of Democritus to the most modern quests to unify quantum mechanics and gravity. The dreams and aspirations of many-body physics interwine the reductionist approach with a complementary philosophy: that of emergent phenomena . In this view, fundamentally new kinds of phenomena emerge within complex assemblies of particles which cannot be anticipated from an a priori knowledge of the microscopic laws of nature. Many-body physics aspires to synthesize, from the microscopic laws, new principles that govern the macroscopic realm, asking: What emergent principles and laws develop as we make the journey from the microscopic to the macroscopic? This is a comparatively modern and far less familiar scientific philosophy. Charles Darwin was perhaps the first to seek an understanding of emergent laws of nature. Following in his footsteps, Ludwig Boltzmann and James Clerk Maxwell were among the first physicists to appreciate the need to understand how emergent principles are linked to microscopic physics. From Boltzmann's biography [1], we learn that he was strongly influenced and inspired by Charles Darwin. In more modern times, a strong advocate of this philosophy has been Philip W. Anderson, who first introduced the phrase “emergent phenomenon” into physics. In an influential article entitled “More is different,” written in 1967 [2], he captured the philosophy of emergence, writing: The behavior of large and complex aggregations of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles.[…]
The study of many-body physics has provided a scientic playground of surprise and continuing revolution over the past half century. The serendipitous discovery of new states and properties of matter, phenomena such as superfluidity, the Meissner, the Kondo and the fractional quantum hall eects, have driven the development of new conceptual frameworks for our understanding about collective behavior, the ramications of which have spread far beyond the connes of terres- trial condensed matter physics- to cosmology, nuclear and particle physics. Here I shall selectively review some of the developments in this eld, from the cold-war period, until the present day. I describe how, with the discovery of new classes of collective order, the unfolding puzzles of high temperature superconductivity and quantum criticality, the prospects for major conceptual discoveries remain as bright today as they were more than half a century ago. 1 Emergent Matter: a new Frontier Since the time of the Greeks, scholars have pondered over the principles that govern the universe on its tiniest and most vast scales. The icons that exemplify these frontiers are very well known - the swirling galaxy denoting the cosmos and the massive accelerators used to probe matter at successively smaller scales- from the atom down to the quark and beyond. These traditional frontiers of physics are largely concerned with reductionism: the notion that once we know the laws of nature that operate on the smallest possible scales, the mysteries of the universe will nally be revealed to us(1). Over the last century and a half, a period that stretches back to Darwin and Boltzmann- scientists have also become fascinated by another notion: the idea that to understand nature, one also needs to understand and study the princi- ples that govern collective behavior of vast assemblies of matter. For a wide range of purposes, we already know the microscopic laws that govern matter on the tiniest scales. For example, a gold atom can be completely understood with the Schrodinger equation and the laws of quantum mechanics established more than seventy years ago. Yet, a gold atom is spherical and featureless- quite unlike the lustrous malleable and conducting metal which human society so prizes. To under- stand how crystalline assemblies of gold atoms acquire the properties of metallic gold, we need new principles{ principles that describe the collective behavior of matter when humungous numbers of gold atoms congregate to form a metallic crystal. It is the search for these new principles that denes the frontiers of many- body physics in the realms of condensed matter physics and its closely related
It is difficult to venture into the world of the ultimately small, or even to talk about it, without a very good understanding of the laws of Nature that govern that world. The forces one finds there determine the way in which the tiny particles we wish to study move about, and also all their other properties. Whether, and how, we can actually observe them will also depend on these forces. And this is not easy, because the laws of Nature are complicated. More and more experts in this field seek refuge in a kind of mathematical gibberish no ‘normal’ person can follow, unless he or she belongs to the in-crowd. To really appreciate the rock-solid logic of the laws of physics, one actually cannot avoid math. Nonetheless, we physicists feel the need to share the joy of all our beautiful discoveries with whoever wants to listen. We are advised then to avoid all math. This then is what I shall reluctantly do. It is my intention to provide a narrative of twenty-five years of research on the very tiniest particles of matter. During those twenty-five years, I began to view Nature as an intelligence test to which humanity as a whole has been subjected, as a giant jigsaw puzzle given to us to play with. Time and again, we stumble upon new pieces, large or small, that fit beautifully with those we already have. I want to share with you the sense of triumph we feel at such moments.
The announcement in 2012 that the Higgs boson had been discovered was understood as a watershed moment for the Standard Model of particle physics. It was deemed a triumphant event in the reductionist quest that had begun centuries ago with the ancient Greek natural philosophers. Physicists basked in the satisfaction of explaining to the world that the ultimate cause of mass in our universe had been unveiled at CERN, Switzerland. The Standard Model of particle physics is now understood by many to have arrived at a satisfactory description of entities and interactions on the smallest physical scales: elementary quarks, leptons, and intermediary gauge bosons residing within a four-dimensional spacetime continuum. Throughout the historical journey of reductionist physics, mathematics has played an increasingly dominant role. Indeed, abstract mathematics has now become indispensable in guiding our discovery of the physical world. Elementary particles are endowed with abstract existence in accordance with their appearance in complicated equations. Heisenberg’s uncertainty principle, originally intended to estimate practical measurement uncertainties, now bequeaths a numerical fuzziness to the structure of reality. Particle physicists have borrowed effective mathematical tools originally invented and employed by condensed matter physicists to approximate the complex structures and dynamics of solids and liquids and bestowed on them the authority to define basic physical reality. The discovery of the Higgs boson was a result of these kinds of strategies, used by particle physicists to take the latest steps on the reductionist quest. This book offers a constructive critique of the modern orthodoxy into which all aspiring young physicists are now trained, that the ever-evolving mathematical models of modern physics are leading us toward a truer understanding of the real physical world. The authors propose that among modern physicists, physical realism has been largely replaced—in actual practice—by quasirealism, a problematic philosophical approach that interprets the statements of abstract, effective mathematical models as providing direct information about reality. History may judge that physics in the twentieth century, despite its seeming successes, involved a profound deviation from the historical reductionist voyage to fathom the mysteries of the physical universe.
The following news item is taken in part from the March 10, 2010 issue of Nature titled ''Exotic matter,'' by Dan Csontos.Crucially, the macroscopic properties of matter usually differ substantially from those of its microscopic constituents.In fact, numerous complex many body systems show emergent phenomena that are associated with the ''whole'' but that cannot be understood solely in terms of the fundamental laws that govern their microscopic ''parts.''This complexity is perhaps best encapsulated in Philip Anderson's ''more is different'' idea,... and is part of the challenge and the beauty of condensed-matter physics.In this Insight, we expose examples of matter for which ''more is different'' holds true in striking ways....
Liquids and solids, atoms and molecules, nuclei—all these clearly are interacting many-body systems. Even a nucleon may be regarded as a manyparticle system, not just because it is now known to consist of three quarks interacting via gluons, but because of the possibility in quantum field theory of virtual excitation of many particles from the vacuum. in the table page 56 we list some of the many-particle systems we encounter in the physical world at length scales that range from a few centimeters to a few fermis. Many of these systems exhibit phenomena—superconductivity in solids and fission in nuclei, for example—whose understanding does not follow immediately from knowledge of the constituents of the system and the interactions among the constituents, but requires new concepts and ideas. Many-body physics is the branch of theoretical physics that studies the new phenomena or “emergent properties” that arise from interactions among “elementary” constituents of a many-particle system and provides means and methods for carrying out precise calculations of such characteristic properties of these systems as may be compared with experimental results to verify hypotheses about the nature of the constituents and their interactions.
The book gives an introduction to the concepts and methods of many-body problems and quantum fields for graduate students and researchers. The formalism is developed in close conjunction with the description of a number of physical systems: cohesion and dielectric properties of the electron gas, superconductivity, superfluidity, nuclear matter and nucleon pairing, matter and radiation, interaction of fields by particle exchange and mass generation. Emphasis is put on analogies between the various systems rather than on advanced or specialized aspects, with the purpose of illustrating common ideas in different domains of physics. The exposition is self-contained and displays in a coherent way all details of derivations starting from a basic knowledge of quantum mechanics and classical electromagnetism.