  We prove that Nelson's massless scalar field model is infrared divergent in
three dimensions. In particular, the Nelson Hamiltonian and the Hamiltonian
obtained from Euclidean quantization are not unitarily equivalent. In contrast,
for dimensions higher than three the Nelson Hamiltonian has a unique ground
state in Fock space and the two Hamiltonians are unitarily equivalent. We also
show that the Euclidean Hamiltonian has no spectral gap.
