1 What Is Effective Field Theory?

The laws of nature have a hierarchical structure, stratified by the resolution scale of our observations. When Isaac Newton published in 1687 his gravitational law explaining Kepler’s laws of planetary motion, he did not need to know the details of the planets’ inner structure. His theory gives an accurate description of the dynamics of the solar system while treating the planets as point-like objects.

The phenomenological understanding of the basic material properties of solids, liquids and gases culminated in the mid-ninetenth century through the work of Clausius, Gibbs, Kelvin, Maxwell and others. Much about these different phases of matter and the transitions between them can be, and was, learned without insight into the molecular structure of matter. It was only about a half-century later that quantum mechanics provided an adequate framework for the description of atoms and molecules.

Quantum mechanics itself is extremely successful in explaining the chemical properties of different substances and mechanical properties of solids, liquids and gases. In fact, virtually all natural phenomena observed at macroscopic scales boil down either to gravity or to Maxwell’s electromagnetism combined with quantum mechanics. Up to rare exceptions, for instance radioactivity, one does not need to understand the structure of atomic nuclei. Nuclear physics itself only advanced later, in the 1930s. And again, much about nuclear structure and reactions was understood before the discovery in the 1960s that individual nucleons are composed of yet smaller particles, the quarks.

The smallest currently known constituents of matter are quarks, bound in atomic nuclei, and leptons, of which the electron is but one example. How do we know that the chain of successive divisions into smaller and smaller constituents ends here? We do not. Perhaps, one day, even smaller building blocks of nature will be discovered. Nevertheless, we do have a successful theory that is able to account for experimental observations at the current resolution frontier: the Standard Model of particle physics. This theory does not depend on the details of as yet undiscovered microscopic physics. Our ignorance of such microscopic structure currently out of reach is subsumed into the values of the input parameters of the Standard Model.

This is the essence of effective field theory (EFT). Going back to where we started, the information about the detailed structure of planets, unknown to Newton, could be subsumed into the values of the planets’ masses. Likewise, the quantum-mechanical nature of molecular interactions only affects macroscopic thermodynamics through material constants such as specific heats or elastic moduli. All that quantum mechanics in turn needs to know about atomic nuclei are their masses and electric charges. Finally, the quark structure of nucleons only manifests itself through binding forces between nucleons that hold the nuclei together. These nuclear forces were studied thoroughly already in the mid-1930s by Fermi, Yukawa and others, three decades before the existence of quarks was conceived by Gell-Mann and Zweig.

Generally speaking, EFT is a framework for a quantitative description of nature at certain level of resolution. Whatever physics might exist at shorter, unresolved length scales can be taken into account through the values of the parameters of the EFT. In the absence of further information about the microscopic physics, these parameters have to be determined by experiment. This line of reasoning has implicitly accompanied the evolution of physics from its early stages. In the second half of the twentieth century, it was however developed into a full-fledged quantitative formalism with tremendous impact, from concrete physical applications to the philosophy of science. One of the founders of modern EFT, Steven Weinberg, offers a nice account of the early history of EFT in [1]. The array of successful applications of the EFT program has grown so long in the last decades that I cannot even list them all here. Instead, I point the reader to the literature. References to introductory-level expositions of EFT with applications are given below in Sect. 1.3.1.

2 Broken Symmetry Zoo

Most macroscopic phenomena observed in nature are dominated by the collective dynamics of the fundamental constituents of matter. In hindsight, this is the reason why nineteenth-century physics was so successful in describing nature. Consider, for instance, a single-component fluid in thermodynamic equilibrium. All the thermodynamic properties of the fluid are determined by the values of two independent observables such as temperature and pressure. If we now disturb the fluid locally, the interactions among the fluid molecules will cause the perturbation to propagate. This propagation manifests macroscopically as a sound wave, possibly accompanied by a background flow. Such small perturbations from thermodynamic equilibrium are described by hydrodynamics, which is an early example of an EFT of the modern type. The microscopic input required here is the equation of state of the fluid along with a set of transport coefficients. Other than that, hydrodynamics is based only on a set of local conservation laws including energy, momentum and whatever other conserved charge the fluid might carry. We know from Noether’s theorem that local conservation laws are a consequence of continuous symmetries. One can therefore say that the dynamics of fluids near equilibrium is largely dictated by symmetry. That is typical of spontaneous symmetry breaking (SSB).

Let me illustrate this on another example. Ferromagnets are materials which, below certain temperature called the Curie temperature, exhibit spontaneous alignment of magnetic moments of their atoms. A solid ferromagnet can be pictured as a lattice, each of whose nodes carries a single spin degree of freedom, see Fig. 1.1. The mutual interactions between the spins may be perfectly isotropic, that is invariant under spatial rotations. Yet, the ferromagnetic state below the Curie temperature obviously possesses a preferred direction of alignment of the spins. This direction is a priori arbitrary and may be set for instance by boundary conditions, not by the internal dynamics of the ferromagnet itself. The emergence of an equilibrium state that breaks the intrinsic symmetry of the system is a hallmark of SSB.

Fig. 1.1
Two schematic diagrams of two different arrangements of ferromagnet under 2 conditions. A, When T C is greater than T, the ferromagnets are arranged systematically. B, When T is greater than T C, the ferromagnets are arranged randomly.

Schematic picture of the ordered state of a ferromagnet below the Curie temperature \(T_{\mathrm {C}}\) and of the disordered state above the Curie temperature \(T_{\mathrm {C}}\)

Full size image

A local perturbation of the ferromagnetic state will be propagated by spin–spin interactions and manifest itself macroscopically as a spin wave. The existence of wave-like excitations which dominate the physics at long distances is another general feature of SSB; these are called Nambu–Goldstone (NG) bosons. Just like for sound waves in fluids, the dynamics of ferromagnetic spin waves is controlled by symmetry. A quantitatively accurate EFT of spin waves can be constructed solely based on the underlying spacetime translation invariance, spatial rotation invariance, and the fundamental commutation relations of angular momentum (spin).

The above two examples hint at a particular type of EFT that governs the long-distance physics of systems with SSB. The relevant degrees of freedom of this EFT are the NG bosons whereas the details of the EFT are dictated by symmetry. It should therefore not come as a surprise that many of the techniques developed in this book rely on the theory of Lie groups, and to a lesser extent on differential geometry. The precise mathematical structure of the EFT depends on the kind of symmetry in question. It is common to use the term internal symmetry for symmetries that act directly on the dynamical degrees of freedom of a given physical system. SSB of an internal symmetry lies for instance behind the phenomena of (anti)ferromagnetism or superfluidity. On the other hand, a spacetime symmetry is a geometric property of space and time. It affects the dynamical degrees of freedom indirectly as fields living in the physical spacetime. SSB of spacetime symmetries is more subtle yet ubiquitous and relevant for all phases of matter. This is the reason why the form of hydrodynamics is largely fixed by the local conservation laws of energy and momentum. A different pattern of SSB gives rise to the elasticity theory of solids. There are also examples of phenomena that exhibit a combination of SSB of internal and spacetime symmetry. These include for instance vortex lattices in rotating superfluids or type-II superconductors, anisotropic Cooper pairing in p-wave superconductors, or the helical order in chiral magnets.

3 Structure of This Book

The subject of this book is of central importance to quantum field and many-body theory. As such, it connects several branches of physics including, but not limited to, high-energy physics, condensed-matter physics, astrophysics and cosmology. With the explicit aim to cater to these different communities, I have tried my best to make the text relevant and comprehensible to audiences with different backgrounds. It is up to the reader to judge to what extent this effort has been successful.

The book aims primarily at graduate students of theoretical physics regardless of their concrete specialization. The text should in principle be accessible to anyone who has taken a first course on quantum field theory. With this target group in mind, I have interspersed the text with two didactic elements. On the one hand, there are numbered examples, graphically separated from the main text, that mostly serve to illustrate newly introduced theoretical concepts or arguments. Occasional more advanced examples mention interesting applications that would disrupt the line of discussion if included in the text. On the other hand, gray blocks such as this one point to important facts or subtleties that might otherwise be missed. They are meant to encourage the reader to take a critical look at the presented material.

The mathematics background required to benefit from this book likewise corresponds to that of a typical student of theoretical physics. It consists mostly of linear algebra, advanced calculus including complex calculus and the calculus of variations, and the theory of partial differential equations. The only piece of background I assume that might go beyond the curriculum of some graduate programs is certain familiarity with group theory and its applications to physics. The level and extent as covered by Georgi [2] is fully sufficient; Chaps. 14 and 15 of [3] are a good start that will get the reader very far. Some more advanced parts of the book rely on basic background in differential geometry. Everything that is needed in this regard (and presumably more) is covered in a self-contained manner in Appendix A.

In order to make the book potentially useful also for more experienced researchers, the content is composed of several layers. In the following Chaps. 2 and 3, I introduce the concepts of SSB, NG bosons and the EFT description thereof through the case study of a simple toy model. This is intended for an uninitiated reader. Anybody else can proceed directly to Part II. Chapters 5 and 6 of this part give a thorough introduction to the physics of SSB. This serves as a supplement to advanced courses on field theory and may thus be suitable for graduate students or junior researchers. An expert reader can skip this part as well.

Parts III and IV constitute the core of this book that will bring the reader close to the research frontier. Part III is dedicated to spontaneously broken internal symmetries. Here the EFT methodology is by now well-settled, and is developed in detail in Chaps. 7 and 8. Chapters 9 and 10 work out some concrete applications of the general formalism. In contrast to internal symmetries, spontaneously broken spacetime symmetries remain a rather active area of research with a number of open questions left to answer. The choice and organization of material in Part IV is therefore necessarily more subjective. I however try to draw on the analogy with broken internal symmetries as much as possible. This largely fixes the content of Chap. 12, which details the necessary modifications to the techniques of Chap. 7. The following Chaps. 13 and 14 then develop the EFT for spontaneously broken spacetime symmetry using numerous examples, roughly sorted by increasing complexity.

In spite of the rather large extent of the book, I was forced to make compromises regarding the choice of material. Sadly, some very exciting aspects of SSB had to be left out altogether. Covering them at the same level of detail as in the rest of the book would have required a substantial amount of additional background. Some of these topics are mentioned at least briefly in Part V.

Let me conclude this introduction with a remark on the chosen style of the book. In writing the text, I have not tried to give a review of all existing research literature on the subject. My intention was instead to create a unified narrative that would give the reader the tools necessary to critically assess existing results and approaches, and to launch a research program of their own. In line with this philosophy, I have deliberately adopted a very restrictive policy regarding bibliography. The purpose of references in this book is primarily to aid the reader. Generally, I thus only include references to point the reader to specified supplementary information, or occasionally to fill in a gap in the argumentation. Only when I explicitly borrow an idea or example from a single identifiable source, do I credit this with a citation. I therefore apologize to all those who might be missing their name in the reference list.

3.1 Further Reading

This book revolves around the application of EFT to physical systems with SSB. Most of the graduate-level physics background that is needed to follow the advanced parts of the book is covered in Part II. This however does not mean that the text is comprehensive. Here is a (biased) list of suggestions for readers who wish to learn more details than what this book can cover.

A monograph fully devoted to EFT was missing on the market for years. This gap has now been filled by the new opus by Burgess [4]. A reader seeking a more concise introduction to EFT will find a variety of excellent resources with slightly different balance of topics included. Thus, for instance, [5] and [6] cover a spectrum of applications to particle phenomenology and nuclear physics. Reference [7] includes a discussion of physics beyond the Standard Model and of the application of EFT to fluid dynamics, which is hard to find in a student-friendly form elsewhere. Cohen’s lectures [8] are a comprehensive source on EFT with a very useful list of further references including annotations. There are also great texts that mostly focus on EFT for NG bosons. This is true especially of [9, 10] and to some extent of [11].

SSB is rarely a subject of a dedicated monograph. An exception is the classic by Strocchi [12], which will satisfy a mathematically oriented reader desiring a rigorous treatment of SSB. The comprehensive early review [13] remains a valuable source of insight, including details that seem to have been forgotten by practitioners. The lecture notes [14] give a modern introduction to SSB that covers recent developments. They omit a discussion of EFT for NG bosons though, and are thus nicely complemented by [15] which focuses largely on the latter. Finally, the specific topic of classification of NG bosons is addressed in detail by [16].