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Recent progress in the treatment of Dirac fields in lattice gauge theory has allowed the chiral symmetry and associated anomaly on the lattice to be discussed in a manner similar to that in continuum theory. In particular, the index theorem on the lattice can be discussed. The analysis of the index theorem on the discrete lattice itself has certain subtle aspects, but lattice theory deals with completely regularized quantities, and thus some of the subtle aspects in continuum theory are now given a more rigorous basis. It is explained that all the results of chiral anomalies in continuum theory are reproduced in a suitable continuum limit of lattice gauge theory, providing a uniform and consistent treatment of both continuum and lattice theories.

This chapter gives an overview of methods and recent studies in the application of lattice gauge theory to physics beyond the standard model. It focuses on theories in which electroweak symmetry is broken spontaneously by new strong interactions. An introduction is given to some common features of such theories, described in terms of the chiral Lagrangian. Apparent difficulties in reconciling QCD-based models with precision experimental results provide the motivation for study of more general Yang-Mills theories. They are examined using the running coupling, which exhibits a perturbative infrared fixed point when enough light fermions are included, defining a "conformal window" in which confinement is lost and chiral symmetry is restored. Some analytic approaches to studying the conformal window are discussed, but the lack of an adequate method in the presence of a strongly-coupled fixed point leads to the consideration of non-perturbative lattice methods. Direct determination of the running coupling and its evolution is discussed, as well as the use of thermal phase transitions to distinguish confining from infrared-conformal theories. Finally, the direct simulation of the spectrum and other properties for theories outside the conformal window is described.

This chapter focuses on aspects of the long-distance behaviour of field theory. In a theory with only massive fields, this behaviour is essentially trivial: clustering is exponentially rapid, with the inverse of the smallest mass providing a length scale over which spatially separated processes decouple. The situation for theories with massless fields is radically different. Here, we need to distinguish between two important cases: when the massless field interpolates for a physical particle; and when massless fields are present in the underlying Lagrangian dynamics but do not interpolate for physical particles. The former case corresponds to quantum electrodynamics, where we have an exactly massless photon (and photon field). Sections 19.1 and 19.2 explore the specific problems arising from the introduction of massless fields, charged particle states, and a well-defined S-matrix in this situation. The second case, where massless fields exist in the theory, but do not interpolate for physical particles, corresponds to quantum chromodynamics and the physical phenomenon of colour confinement. The massless gluon fields of this theory, as well as the massive quark fields, specify the Lagrangian dynamics of the theory but do not interpolate for finite-energy asymptotic states. Section 19.3 explores this extraordinary behaviour and introduces the basic concepts and techniques of lattice gauge theory. It also presents a model where the physical mechanism of confinement can be exhibited using the three-dimensional gauge theory.

As the spatial dimensionality $d$ decreases, fluctuations become larger and the stability of the low-temperature ordered state deteriorates. The dimensionality where long-range order disappears is known as lower critical dimension. For instance, the Ising model in one dimension does not display long-range order at finite temperatures, however in two dimensions Peierls argument explains why the same model has an ordered phase below a certain critical temperature. If the basic variables and symmetries are continuous as in the $XY$ and Heisenberg models, the (long-range) ordered state at any finite temperature disappears already in two dimensions. This is the result of Mermin-Wagner's theorem. The $XY$ model nevertheless undergoes an unusual phase transition without an onset of long-range order in two dimensions, which is known as the Kosterlitz-Thouless transition. Gauge or local symmetries cannot spontaneously be broken as elucidated by Elitzur's theorem when applied to lattice gauge theories.

Chapter 3 illustrates by a number of examples the essential role of functional integrals in physics. For example, the path integral representation of quantum mechanics explains why many basic equations of classical physics satisfy a variational principle, the relation between quantum field theory and the theory of critical phenomena in macroscopic phase transitions. Field integrals are essential for gauge theory quantization, leading to the introduction of Faddeev–Popov ghost fields and BRST symmetry. Lattice gauge theory, the discretized form of field integrals, makes non–perturbative calculations possible. These are at the basis of the calculation of penetration effects in quantum field theory (instanton calculus).

The main thread of chapter 6 prompts the need for quantum gravity (QG) and introduces the RBW approach to QG, unification in particle physics, dark matter, and dark energy. The details of RBW’s modified Regge calculus and modified lattice gauge theory approaches are conveyed conceptually in the main thread. The RBW fits of galactic rotation curves, galactic cluster mass profiles, the angular power spectrum of the cosmic microwave background, and the Union2.1 supernova data associated with dark matter and dark energy are in Foundational Physics for Chapter 6. In Philosophy of Physics for Chapter 6, RBW’s taxonomic location with respect to other discrete approaches to QG is detailed and it is argued that the search for QG is stymied by the dynamical paradigm across the board. Further, it is maintained that an adynamical global constraint as the basis for QG in the block universe provides a self-vindicating unification of physics.