On the energy momentum dispersion in the lattice regularization

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 $E_{\mathrm{d}\mathrm{i}\mathrm{s}}^2=0$ 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 $E_0={\cosh }^{-1}(1+M^2\mathrm{/}2)$ is derived independently of the spatial lattice size.