New Lattice Formulation of the Continuum

摘要

It is well known that the Dirac equation on a discrete lattice in D dimension has 2D degenerate solutions. The usual method of removing these spurious solutions encounters difficulties with chiral symmetry when the lattice spacing l not equal 0, as demonstrated by the persistent problem of pion and kaon masses. On the other hand, we recall that in any crystal in nature, all the electrons do move in a lattice and satisfy the Dirac equation; yet there is not a single physical result that has ever been entangled with a spurious fermion solutions. Therefore it should not be difficult to eliminate these unphysical elements. On a discrete lattice, particles hop from point to point, whereas in a real crystal the lattice structure is embedded in a continuum and electrons move continuously from lattice cell to lattice cell. In a discrete system, the lattice function are defined only on individual points (or links, as in the case of gauge fields). However, in a crystal the electron state vector is represented by the Bloch wave functions which are continuous functions in tau(right arrow) and herein lies one of the essential differences.