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For a free scalar boson field and for U(1) gauge theory finite volume(infrared) and other corrections to the energy-momentum dispersion in thelattice regularization are investigated calculating energy eigenstates from thefall off behavior of two-point correlation functions. For small lattices thesquared dispersion energy defined by $E_{\rmdis}^2=E_{\vec{k}}^2-E_0^2-4\sum_{i=1}^{d-1}\sin(k_i/2)^2$ is in both casesnegative ($d$ is the Euclidean space-time dimension and $E_{\vec{k}}$ theenergy of momentum $\vec{k}$ eigenstates). Observation of $E_{\rm dis}^2=0$ hasbeen 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 ofits finite size corrections. A surprise from the lattice regularization of thefree field is that infrared corrections do {\it not} eliminate a differencebetween the groundstate energy $E_0$ and the mass parameter $M$ of the freescalar lattice action. Instead, the relation $E_0=\cosh^{-1} (1+M^2/2)$ isderived independently of the spatial lattice size.

On the energy momentum dispersion in the lattice regularization