2026-10-22 · arXiv Quantum Physics
arXiv:2610.29791v1 Announce Type: cross
Abstract: We show that the concept of particles and antiparticles is related to the Weyl-Heisenberg group $W\!H_3$. Its generators are the position operators $\hat x^a$, the momentum operators $\hat p_a$, and the central element $Q_{\sf{v}}$, which defines a subgroup U(1)$_{\sf v}$ of $W\!H_3$. We show that the wave functions have a charge relative to the group U(1)$_{\sf{v}}$ equal to either $q_+=1/\hbar$ (particle $\psi_+$) or $q_-=-1/\hbar$ (antiparticle $\psi_-$). We call these eigenvalues of the operator $Q_{\sf{v}}$ quantum charges: $Q_{\sf{v}}\psi_\pm^{}=q_\pm^{}\psi_\pm^{}$. The wave functions $\psi_+$ and $\psi_-$ belong to the Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$, on which unitarily inequivalent representations $\pi_\hbar$ and $\pi_{-\hbar}$ of the Weyl-Heisenberg group $W\!H_3$ are defined. These representations are related to each other by the antilinear charge conjugation operator $\hat C$, 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)$_{\sf v}$-invariance in these theories. Restoring this invariance separates particle and antiparticles and eliminates all paradoxes at the level of first-quantized relativistic theories.
原文链接：https://arxiv.org/abs/2610.29791
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