For a free scalar boson field and for U(1) gauge theory finite volume (infrared) and other corrections to the energy-momentum dispersion in the lattice regularization are investigated calculating energy eigenstates from the fall off behavior of two-point correlation functions. For small lattices the squared dispersion energy defined by $E_{ \rm dis}^2=E_{ \vec{ k}}^2-E_0^2-4\sum_{ i=1}^{ d-1}\sin(k_i/2)^2$ is in both cases negative ($d$ is the Euclidean space-time dimension and $E_{ \vec{ k}}$ the energy of momentum $\vec{ k}$ eigenstates). Observation of $E_{ \rm dis}^2=0$ has been an accepted method to demonstrate the existence of a massless photon ($E_0=0$) in 4D lattice gauge theory, which we supplement here by a study of its finite size corrections. A surprise from the lattice regularization of the free field is that infrared corrections do { \it not} eliminate a difference between the groundstate energy $E_0$ and the mass parameter $M$ of the free scalar lattice action. Instead, the relation $E_0=\cosh^{ -1} (1+M^2/2)$ is derived independently of the spatial lattice size.
Categories: hep-lat physics.comp-ph