  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.
