Search Results

This chapter argues that gauge symmetry is a purely formal feature of our theoretical representations with no empirical consequences. It distinguishes theoretical symmetries of a theory from empirical symmetries of situations to which a theory may be applied. A theoretical symmetry is purely formal if it implies no corresponding empirical symmetry. The chapter responds to several objections based on phenomena whose explanation apparently requires that gauge symmetry have empirical consequences. It argues that gauge symmetry is not observable and criticizes ‘the gauge argument’ from gauge symmetry to the existence of fundamental interactions. ‘Ghost’ fields are not real fields: despite its name, ‘spontaneous symmetry breaking’ does not involve any breaking of ‘local’ gauge symmetry; the theta-vacuum is not a superposition of distinct states inter-related by ‘large’ gauge transformations; anomalies in empirically successful theories do not involve breaking of any empirical gauge symmetry.

This chapter introduces the basic concepts and language that will be needed later on: order, symmetry, invariance, broken symmetry, Hamiltonian, condensed matter, order parameter, ground state, and several thermodynamic terms. It also presents the necessary concepts from thermodynamics and statistical mechanics that will be needed later. It boils down the latter to its most elemental and essential ingredient: that of temperature as controlling the relative probabilities of configurations of different energies. For much of statistical mechanics, all else is commentary. This is sufficient to present an intuitive understanding of why and how matter organizes itself into different phases as temperature varies, and leads to the all-important concept of a phase transition.

Symmetries play a fundamental role in modern physics. This chapter introduces this concept by discussing the meaning of symmetries in mathematics and in physics, explaining the significance of Lie groups. With the help of some analogies, the principle of gauge symmetry is illustrated. This principle is the basis for the construction of the Standard Model and determines the structure of the fundamental forces. However, the masses of the W and Z particles are in conflict with the gauge principle. The solution to this problem requires the introduction of new concepts such as spontaneous symmetry breaking and the Higgs mechanism. The chapter also describes the experimental strategies in the search for the Higgs boson, the significance of the possible discovery of this particle, and the remaining open problems.

This chapter and the next explore a response to a natural objection to the conclusion of Chapter 12. The natural objection is, based on models of steaming tea cups that idealize them as infinite in volume, the conclusion cuts no interpretive ice. The strategy underlying the response is to find in the physics of honestly infinite systems, like quantum fields, explanatory structures sharing the features of Chapter 12's model of phase structure on which that chapter's argument relied. Broken symmetry in quantum field theory could be such a structure. This chapter offers a bridge to that topic, in the form of an account of broken symmetry in quantum statistical mechanics, an account which, the chapter argues, shares the features which stymie rigid interpretation.

Chapter 12 argued that quantum statistical mechanics puts unitarily inequivalent representations to use in ways no rigid interpretation can make sense of. Two features of working QFTs which promise a quantum field theoretic realization of Chapter 12's argument are Goldstone bosons and the Higgs mechanism. This chapter explains why they're promising by presenting them as instance of broken symmetry. Then it tempers the promise by admitting that the working QFTs in which these features occur are less mathematically explicit than they need to be to persuasively realize the argument of Chapter 12. The chapter closes by extracting from this very circumstance a non-conclusive reason to lend the argument of Chapter 12 interpretive weight. The reason is that our best theories of physics are still under construction, and their successors could share with the models presented in Chapter 12 the features on which the argument of Chapter 12 hinged.

This chapter introduces a quantum field theory for interacting boson systems. It develops a mean-field theory to study the superfluid phase. A path integral formulation is then developed to re-derive the superfuid phase, which results in a low energy effective non-linear sigma model. A renormalization group approach is introduced to study the zero temperature quantum phase transition between superfluid and Mott insulator phase, and finite temperature phase transition between superfluid and normal phase. The physics and the importance of symmetry breaking in phase transitions and in protecting gapless excitations are discussed. The phenomenon of superfluidity and superconductivity is also discussed, where the coupling to U(1) gauge field is introduced.

This chapter introduces mean field theory, both as a general class of models and in its specific incarnation in spin glasses, the Sherrington–Kirkpatrick model. This is undoubtedly the most theoretically studied spin glass model by far, and the best understood. For the nonphysicist the going may get a little heavy in this chapter once replica symmetry breaking is introduced, with its attendant features of many states, non-self-averaging, and ultrametricity—but an attempt is made to define and explain what all of these things mean and why replica symmetry breaking represents such a radical departure from more conventional and familiar modes of symmetry breaking. While this is a central part of the story of spin glasses proper, the nonphysicist who wants to skip the technical details can safely omit certain sections in the chapter and continue on without losing the essential thread of the discussion that follows.

Statistical-mechanical systems often involve discrete elementary degrees of freedom such as spins in the Ising model. Field theories, on the other hand, have continuous fields, defined over the whole space-time or part of it, as fundamental degrees of freedom. These two seemingly different descriptions of physical phenomena can be related close to the critical point. The present chapter summarizes how the description by continuous fields emerges from discrete degrees of freedom in a more systematic manner than in previous chapters. The phenomenological Landau-Ginzburg approach, based on the concept of order parameter, is expanded to generate effective field theories. The important roles of symmetry and topology are also elucidated in some detail. Also shown are some of the important consequences of having a broken-symmetry phase, such as long-range order, the emergence of Nambu-Goldstone modes when the symmetry involved is continuous, and topological defects.

According to the principle of emergence, the properties of material are mainly determined by how the atoms are organized in the material. Such organization is formally called order. The vast range of materials is a result of the rich variety of orders that atoms can have. For a long time, it has been believed that all orders are described as symmetry breaking. A comprehensive theory for phases and phase transitions is developed here based on the symmetry breaking picture. The existence of FQH states (and superconducting states) indicates that there are new states of matter that cannot be described as symmetry breaking. Completely new theory is needed to describe those new states of matter. This chapter outlines the theory of topological order and theories of quantum order for the new states of matter, such as FQH states. Many new concepts and new language, such as topology-dependent degeneracy, fractional statistics, edge states, etc, are introduced to describe new states of matter.

The effective metric and effective gauge fields are simulated in superfluids by the inhomogeneity of the superfluid vacuum. In superfluids, many inhomogeneous configurations of the vacuum are stable and thus can be experimentally investigated in detail, since they are protected by r-space topology. In particular, the effect of the chiral anomaly has been verified using such topologically stable objects as vortex-skyrmions in ³He-A and quantized vortices in ³He-B. Other topological objects can produce non-trivial effective metrics. In addition, many topological defects have almost direct analogs in some relativistic quantum field theory. Topological defects are results of spontaneously broken symmetry. This chapter discusses the spontaneous symmetry breaking both in ³He-A and ³He-B, which is responsible for topologically stable objects in these phases, and analogous ‘superfluid’ phases in high-energy physics, such as chiral and color superfluidity in quantum chromodynamics (QCD).

A motivation for the development of atomically resolved spectroscopic imaging STM (SI-STM) has been to study the broken symmetries in the electronic structure of cuprate high temperature superconductors. Both the d-wave superconducting (dSC) and the pseudogap (PG) phases of underdoped cuprates exhibit two distinct classes of electronic states when studied using SI-STM. The class consists of the dispersive Bogoliubov quasiparticles of a homogeneous d-wave superconductor. These signature are detected below a lower energy scale |E| = D₀ and only upon a momentum space (k-space) arc which terminates near the lines connecting k = ±(p/a₀, 0) to k = ±(0̣,p/a₀). This ‘nodal’ arc shrinks continuously with decreasing hole density. In both the dSC and PG phases, the only broken symmetries detected in the |E|≤D₀ states are those of a d-wave superconductor. The second class of states occurs at energies near the pseudogap energy scale |E|~̣D₁ can be associated with the ‘antinodal’ states near k = ±(p/a₀,0) and k = ±(0̣,p/a₀). These states break the expected 90º-rotational (C₄) symmetry of electronic structure within CuO₂ unit cells, at least down to 180º-rotational (C₂) symmetry (nematic) but in a spatially disordered fashion. This intra-unit-cell C₄ symmetry breaking coexists at |E|~̣D₁ with incommensurate conductance modulations locally breaking both rotational and translational symmetries (smectic). Empirically, the characteristic wavevector Q of the latter is determined by the k-space points where Bogoliubov quasiparticle interference terminates and therefore evolves continuously with doping. The properties of these two classes of |E|~̣D₁ states are indistinguishable in the dSC and PG phases. To explain these two regimes of k-space that are distinguished by the symmetries of their electronic states and their energy scales |E|~D₁ and |E|≤D₀, and to understand their relationship to the electronic phase diagram and the mechanism of high-T_c superconductivity, represent key challenges for cuprate studies.

These lectures provide an introduction to the low-energy dynamics of Nambu–Goldstone fields, which associated with some spontaneous (or dynamical) symmetry breaking, using the powerful methods of effective field theory. The generic symmetry properties of these massless modes are described in detail and two very relevant phenomenological applications are worked out: chiral perturbation theory, the low-energy effective theory of QCD, and the (non-linear) electroweak effective theory. The similarities and differences between these two effective theories are emphasized, and their current status is reviewed. Special attention is given to the short-distance dynamical information encoded in the low-energy couplings of the effective Lagrangians. The successful methods developed in QCD could help us to uncover fingerprints of new physics scales from future measurements of the electroweak effective theory couplings.

A quantitative change can lead to a qualitative change. The system with many degrees of freedom can demonstrate qualitatively new phenomena. This introductory chapter summarizes the principle of emergence, which states that the properties of material are mainly determined by the organizations (or the orders) of atoms in the material. The different organizations of microscopic degrees of freedom not only give rise to traditional symmetry breaking orders (such as crystal, magnets, superfluids, etc), they also give rise to new topological/quantum orders (such as fractional quantum Hall states, superconductors, etc). Those different orders lead to the rich properties of materials and phenomena that we see every day. The collective excitations in the new topologically ordered states can even be gauge bosons and fermions that satisfy Coulomb's law. So the principle of emergence in many-body systems may explain the origin of light and electrons, as well as the beauty in laws of physics.

This chapter examines the role of global symmetries in local quantum field theories. It shows that exact global symmetries are rare — indeed, if we take gravitational effects into account, probably non-existent. Nevertheless, approximate global symmetries play an enormously important role in modern field theory. The appearance of massless Goldstone particles once an exact global symmetry is spontaneously broken is of enormous importance in modern field theory, and proof of the Goldstone theorem embodying this phenomenon is given. Dynamical aspects of spontaneous symmetry-breaking (SSB) are examined, and it is shown that the essence of SSB resides in the energetics of the theory in the infrared (i.e., at long distances).

This chapter gives a general path integral formulation of the quantum anomaly for chiral symmetry. The quantum anomaly is identified as the Jacobian arising from the symmetry transformation of path integral variables. The correct Ward-Takahashi identities with an anomalous term are thus defined from the beginning. To realize this idea, it is essential to have a suitable regulator which controls divergences by preserving as much symmetry as possible. In the path integral formulation, the gauge invariant mode cut-off is shown to be useful. As topics related to quantum anomalies, a brief account of instanton solutions and the Atiyah-Singer index theorem is given. As applications of these notions, it is explained that the (naive) unitary transformation to the interaction picture does not exist. When the relevant symmetry is spoiled by the quantum anomaly and instanton effects, the Nambu-Goldstone particles associated with spontaneous symmetry breaking is generally absent.

This chapter describes the experimental and theoretical studies of a spontaneous time-translation symmetry breaking transition in a non-equilibrium cold atom system. Such studies prepare two vibrating atomic clouds by periodically modulating the laser intensity in a magneto-optical trap (MOT). For a comparatively small total number of trapped atoms the clouds have equal populations, a consequence of fluctuation-induced inter-cloud transitions. As the total number of atoms is increased, spontaneous breaking of the discrete time-translation symmetry is observed with respect to the modulation period. The measured critical exponents and the frequency dispersion of the response show an ideal mean-field transition behaviour. The symmetry breaking is due to the modification of fluctuational dynamics by the interatomic interaction. The chapter provides a theoretical model based on the activation energy and master equation which explains the experimental results well. The enhancement of the fluctuation which cannot be described by the mean-field model, is also discussed.

The symmetry breaking pattern in A-phase of ³He is similar to that in electroweak phase transition. This chapter discusses the topology of singular topological defects in ³He-A and in analogous phases in high-energy physics. There are two types of hedgehog in ³He-A — one is analogous to ‘t Hooft–Polyakov magnetic monopole, while another one to Dirac monopole terminating Dirac string. Among the linear defects there is half-quantum vortex, which is analog of Alice string. A particle that moves around an Alice string continuously flips its charge, or parity, or enters the ‘shadow’ world. The pure mass vortex can continuously transform to disclination which is analog of antigravitating string. Singular doubly quantized vortex, which is analog of electroweak Z-string, is topologically unstable and may decay into non-singular texture. The fractional vorticity and fractional flux in cuprate d-wave superconductors and chiral superconductors are also discussed.

This chapter finally deals with the concept of spin glasses. The intention is not to provide anything approaching a thorough history of the subject. The field today is broad, with threads and subthreads extending in a multitude of different directions. Rather, the chapter focuses on a relatively narrow part of the overall subject. It discusses some of the history of their discovery, their basic properties and experimental phenomenology, and some of the mysteries surrounding them. It introduces some of the basic theoretical constructs that underlie much of the discussion in later chapters. Topics covered include dilute magnetic alloys and the Kondo effect, nonequilibrium and dynamical behavior, mechanisms underlying spin glass behavior, the Edwards–Anderson Hamiltonian, frustration, dimensionality and phase transitions, broken symmetry and the Edwards–Anderson Order Parameter, and energy landscapes and metastability.

Chrisoula Andreou says procrastination qua imprudent delay is modeled by Warren Quinn’s self-torturer, who supposedly has intransitive preferences that rank each indulgence in something that delays his global goals over working toward those goals and who finds it vague where best to stop indulging. His pair-wise choices to indulge result in his failing the goals, which he then regrets. This chapter argues, contra the money-pump argument, that it is not irrational to have or choose from intransitive preferences; so the agent’s delays are not imprudent, not instances of procrastination. Moreover, the self-torturer case is intelligible only if there is no vagueness and if the agent’s preferences are transitive. But then he would delay only from ordinary weakness of will. And when it is vague where best to stop indulging, rational agents would use symmetry-breaking techniques; so, again, any procrastination would be explained by standard weakness of will, not vagueness.