# Scales IV: Long-distance structure of quantum field theory

# Scales IV: Long-distance structure of quantum field theory

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.

*Keywords:*
quantum field theory, long distance behaviour, massless field, quantum electrodynamics, quantum chromodynamics, colour confinement, lattice gauge theory, three-dimensional gauge theory

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