Consequences of the eikonal formula

We analyze the implications of a model for high-energy hadron-hadron scattering in which the $S$ matrix is given by an eikonal formula and is therefore explicitly unitary. The eikonal $\chi$ is a Hermitian operator which represents the lowest-order amplitudes for all reactions, elastic and inelastic. (Consequently, the matrix elements of $S=e^{i\chi }$ are not simply related to those of $\chi$.) We derive a physical picture of high-energy scattering as a stochastic process in which quanta associated with harmonic oscillators at each point in a subspace of three-dimensional space are created or annihilated randomly. As the energy increases, the length of this subspace in the longitudinal direction expands and more harmonic oscillators become excited. The expectation values of ${\chi }^2,{\chi }^3,{\chi }^4,\dots$ become very large, but the $S$ matrix remains unitary. We derive a partial differential equation for a generating functional for the $S$ matrix, through which we show that the target particle becomes completely absorptive as the energy of the projectile goes to infinity.