Causality and analyticity place powerful, interlocking restrictions on the mathematical structure of the S-matrix, turning physical requirements about signal propagation into precise constraints on scattering amplitudes. Causality demands that effects cannot precede causes outside the light cone, which in quantum field theory is expressed by vanishing commutators for spacelike separations. Analyticity then follows as the statement that the S-matrix elements are analytic functions of complexified energy and momentum variables except at physical singularities. Steven Weinberg of University of Texas at Austin has emphasized how these axioms reduce freedom in constructing amplitudes and lead to model-independent relations.
Analyticity and dispersion relations
Analyticity organizes singularities in the S-matrix so that poles correspond to stable particles and resonances while branch cuts mark multiparticle production thresholds. This analytic structure allows the derivation of dispersion relations that relate the real part of an amplitude to an integral over its imaginary part across physical energies. The classic exposition by R. J. Eden, P. V. Landshoff, D. I. Olive, J. C. Polkinghorne of Cambridge University shows how dispersion relations enforce consistency between low-energy measurements and high-energy behaviour without committing to a particular dynamical model. Nuance arises because analyticity assumptions require control of asymptotic growth and the absence of unexpected singularities.
Causality, unitarity and bounds
Causality combined with unitarity which enforces probability conservation implies positivity constraints on the imaginary parts of amplitudes and leads to rigorous bounds on growth with energy. One important consequence is the Froissart bound which limits the rise of total cross sections at high energy. Crossing symmetry, a consequence of analyticity and Lorentz invariance, ties different reaction channels into a single analytic function and constrains possible extrapolations between processes. These mathematical constraints are not merely formal; they guide interpretation of accelerator experiments at facilities such as CERN and inform effective field theory constructions used in nuclear and condensed matter physics.
Taken together, causality and analyticity narrow the landscape of admissible S-matrices, turning physical intuition about signal propagation into quantitative tools for predicting and checking scattering phenomena. Where experimental data probe edges of analyticity assumptions, tensions can signal new physics or demand refined theoretical control.