Quantum Charge as a Particle/Antiparticle Label

We show that the concept of particles and antiparticles is related to the Weyl-Heisenberg group $W\text{\hspace{0.17em}⁣}{H}_3$. Its generators are the position operators ${\widehat{x}}^a$, the momentum operators ${\widehat{p}}_a$, and the central element $Q_{\mathsf{v}}$, which defines a subgroup U(1)${}_{\mathsf{v}}$ of $W\text{\hspace{0.17em}⁣}{H}_3$. We show that the wave functions have a charge relative to the group U(1)${}_{\mathsf{v}}$ equal to either $q_{+}=1\mathrm{/}\mathrm{\hslash }$ (particle ${\psi }_{+}$) or $q_{-}=-1\mathrm{/}\mathrm{\hslash }$ (antiparticle ${\psi }_{-}$). We call these eigenvalues of the operator $Q_{\mathsf{v}}$ quantum charges: $Q_{\mathsf{v}}{\psi }_{\pm }^{}=q_{\pm }^{}{\psi }_{\pm }^{}$. The wave functions ${\psi }_{+}$ and ${\psi }_{-}$ belong to the Hilbert spaces ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$, on which unitarily inequivalent representations ${\pi }_{\mathrm{\hslash }}$ and ${\pi }_{-\mathrm{\hslash }}$ of the Weyl-Heisenberg group $W\text{\hspace{0.17em}⁣}{H}_3$ are defined. These representations are related to each other by the antilinear charge conjugation operator $\widehat{C}$, which maps ${\mathcal{H}}_{+},q_{+}$ to ${\mathcal{H}}_{-},q_{-}$. Charges $q_{\pm }^{}$ are intrinsic characteristics of the wave functions ${\psi }_{\pm }^{}$, similar to spin and electric charge. The wave function possesses one of these charges, $q_{+}$ or $q_{-}$, prior to the specification of any equations. We show that negative probabilities/energies and the Klein paradox in Klein-Gordon and Dirac theories are consequences of the violation of U(1)${}_{\mathsf{v}}$-invariance in these theories. Restoring this invariance separates particle and antiparticles and eliminates all paradoxes at the level of first-quantized relativistic theories.