The Diffractive Quantum Limits of Particle Colliders

Quantum Mechanics places limits on achievable transverse beam spot sizes of particle accelerators. We estimate this limit for a linear collider to be Δx ≥ \hstrok cf/Eδ₀ where f is the final focal length, E the beam energy, and δ₀ the intrinsic transverse Gaussian width of the electron wave-function. δ₀ is determined in the phase space damping rings, and for a linear focusing channel, is the Gaussian width of the transverse groundstate wave-function. A crude estimate in a circular damping ring yields δ ₀ ≈ √ \hbar c/eB} where B is the typical (wiggler) magnetic field strength in this system. For the NLC δ₀ 25nm, and Δ x ∼ { O} (0.06) nm, about two orders of magnitude smaller than the design goal. We can recover an estimate of the classical result when we include radiative relaxation effects. We also consider a self-replicating solution in a synchrotron and obtain Δ x>= √ {\hbar cf/E} ∼ { O} (1.0) nm, We discuss formulation of quantum beam optics relevant to these issues.