Position eigenstates, symmetries, and the redundant hermiticity of free-particle Hamiltonians

The quantum state of a particle can be completely specified by a position at one instant of time. This implies a lack of information, hence a symmetry, as to where the particle will move. We here study the consequences for free particles of spin 0 and spin 1/2. On a cubic spatial lattice a hopping equation is derived, and the continuum limit taken. Spin 0 leads to the Schroedinger equation, and spin 1/2 to the Weyl equation. Both Hamiltonians are hermitian automatically, if time-reversal symmetry is assumed. Hopping amplitudes with a "slight" inhomogeneity lead to the Weyl equation in a metric-affine space-time.