# Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity

# Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity

Chapter 5 discussed two possible approaches to constructing a relativistically invariant theory of particle scattering. The first attempt — a frontal assault in which one directly writes down for each scattering sector (i.e., with a specified number of incoming and outgoing particles) a manifestly Lorentz-invariant interaction operator containing momentum-dependent Lorentz scalar amplitudes — led to disaster. The resultant theory led to particle interactions which could not be confined to finite regions of space-time. The second attempt, in which the interaction Hamiltonian is written as a spatial integral of a local, Lorentz (ultra-)scalar field, accomplishes the primary goal of producing a Lorentz-invariant set of scattering amplitudes, but its compliance with the clustering principle remains uncertain. This chapter puts this latter requirement into a precise mathematical framework, called second quantization, so that the process of identifying clustering relativistic scattering theories can be simplified and even to some degree automated.

*Keywords:*
relativistic quantum field theory, particle scattering, Hamiltonians, Lorentz transformations, clustering principle, second quantization

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