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-dis,(k) over right arrow(2) = E-(k) over right arrow(2) - 4 Sigma(d=1)(i=1)sin(k(i)/2)(2) is in both cases negative (d is the Euclidean space-time dimension and E-(k) over right arrow the energy of momentum (k) over right arrow eigenstates). Observation of E-dis,(k) over right arrow(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 not eliminate a difference between the ground state energy E-0 and the mass parameter M of the free scalar lattice action. In stead, the relation E-0 = cosh(-1)(1+M-2/2) holds in dependently of the spatial lattice size.