# Heuristic explanation of why quantum mechanics plus SR imply antiparticles

 P: 194 Quantum mechanics plus special relativity does not necessarily require antiparticles: although it naturally accommodates them. ----The straightforward generalization of quantum mechanics to include special relativity requires a modification of the Schrodinger equation. Modifying the Hamiltonian to reflect the energy-momentum relation, $E=\sqrt{m^2c^4+p^2c^2}$, for relativistic particles, $$\hat{H}=\sqrt{-\hbar^2c^2\nabla^2+m^2c^4}\,,$$ leads to a non-local theory due to the differential under the square-root. ----Modifying the entire time-dependent Schrodinger equation to reflect the squared energy-momentum relation $E^2=m^2c^4+p^2c^2$ gives $$-\hbar^2\frac{\partial^2}{\partial t^2}\psi(\mathbf{x},t)=-\hbar^2c^2\nabla^2+m^2c^4\psi(\mathbf{x},t)\,,$$ the Klein-Gordon equation. While this gives a local theory, it contains negative-energy states in its spectrum. This is a problem since perturbations can cause transitions indefinitely into lower states (hence, this theory is unstable). ----The modern view is to abandon any attempt to directly modify the Schrodinger equation, and instead, to quantize a continuous field, $\phi(\mathbf{x},t)$, using ordinary quantum mechanics. The fields that are quantized, however, have dispersion relations of the same form as the squared energy-momentum relation $E^2=m^2c^4+p^2c^2$, with the energy, $E$, identified as the frequency, $\omega_p$, of propagating plane waves. The resulting Schrodinger equation, has no negative energy solutions, and is local. There are, however, negative frequency solutions associated with the field's dispersion relations. In the case that a real-valued field is quantized, negative and positive frequency solutions are identified, and there are no antiparticles. In the case that a complex-valued field is quantized, negative and positive frequency solutions are the particle and anti-particle solutions, respectively.