Negative Energy States in Quantum Theory

Abstract: We analyze the Lagrangian density and canonical stress-energy tensor for the Dirac equation, where the Dirac bispinor has been recast as a multivector set of fields. For the massless Dirac field, the sign of the energy density is determined by the relative phase of uncoupled even- and odd-grade field components. These components become coupled in the massive Dirac equation, and the sign of the energy is determined by their spatial parity. The corresponding stress-energy tensors for the second-order equations also admit negative energy states, with the sign of the energy density again dependent on field parity. We apply the same multivector approach to electromagnetism, constructing new Lagrangian and energy densities in which the vector potential and the electromagnetic field are treated as independent field degrees of freedom.