Quantum computational path summation for relativistic quantum mechanics and a time dilation relation for a Dirac Hamiltonian generator on a qubit array

Abstract

Dirac particle dynamics is encoded as a unitary path summation rule and implemented on a qubit array, where the qubit array represents both spacetime and the fermions contained therein. The unitary path summation rule gives a quantum algorithm to model a many-body system of Dirac particles in a gauge field with Lorentz invariance down to the grid scale (Planck scale)--the lattice-based model neither suffers the Fermi-sign problem nor breaks Lorentz invariance. Yet, for the Dirac Hamiltonian to generate the unitary evolution of the 4-spinor field at the Planck scale, there is time dilation between the shortest observable time near a single space point and that time measured at long-wavelength scales. We find gravitational time dilation where the model space around each point (with an even number of qubits) is curved like the space around a Schwarzschild black hole.