Vacuum catastrophe: An elementary exposition of the cosmological constant problem

Quantum field theory predicts a very large energy density for the vacuum, and this density should have large gravitational effects. However these effects are not observed, and the discrepancy between theory and observation is an incredible 120 orders of magnitude. There is no generally accepted explanation for this discrepancy, although numerous papers have been written about it. As usually stated the problem requires a knowledge of quantum field theory and general relativity, topics not normally studied by undergraduates. We have tried to make the problem accessible to undergraduates by using only the simplest ideas of quantum theory, such as the uncertainty principle and the theory of the harmonic oscillator, and classical gravitational theory. We believe that such simplification is not only an amusing pedagogical exercise but clarifies how basic is the conflict between quantum theory and gravitational theory. We do not here discuss various proposed solutions to the problem, beyond the trivial and unsatisfactory one of assuming an ad hoc canceling term in the Hamiltonian or field equations.