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arXiv:2610.29791

Quantum Charge as a Particle/Antiparticle Label

Alexander D. Popov

math-phhep-thquant-phSubmitted Oct 21, 2026• Announced Oct 22, 2026• 0 citations
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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 $ψ_+$) or $q_-=-1/\hbar$ (antiparticle $ψ_-$). We call these eigenvalues of the operator $Q_{\sf{v}}$ quantum charges: $Q_{\sf{v}}ψ_\pm^{}=q_\pm^{}ψ_\pm^{}$. The wave functions $ψ_+$ and $ψ_-$ belong to the Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$, on which unitarily inequivalent representations $π_\hbar$ and $π_{-\hbar}$ 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 $ψ_\pm^{}$, 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.

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28 pages

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We show that the concept of particles and antiparticles is related to the Weyl-Heisenberg group W ⁣H3W\!H_3WH3​. Its generators are the position operators x^a\hat x^ax^a, the momentum operators p^a\hat p_ap^​a​, and the central element QvQ_{\sf{v}}Qv​, which defines a subgroup U(1)v_{\sf v}v​ of W ⁣H3W\!H_3WH3​. We show that the wave functions have a charge relative to the group U(1)v_{\sf{v}}v​ equal to either q+=1/ℏq_+=1/\hbarq+​=1/ℏ (particle ψ+ψ_+ψ+​) or q−=−1/ℏq_-=-1/\hbarq−​=−1/ℏ (antiparticle ψ−ψ_-ψ−​). We call these eigenvalues of the operator QvQ_{\sf{v}}Qv​ quantum charges: Qvψ±=q±ψ±Q_{\sf{v}}ψ_\pm^{}=q_\pm^{}ψ_\pm^{}Qv​ψ±​=q±​ψ±​. The wave functions ψ+ψ_+ψ+​ and ψ−ψ_-ψ−​ belong to the Hilbert spaces H+\mathcal{H}_+H+​ and H−\mathcal{H}_-H−​, on which unitarily inequivalent representations πℏπ_\hbarπℏ​ and π−ℏπ_{-\hbar}π−ℏ​ of the Weyl-Heisenberg group W ⁣H3W\!H_3WH3​ are defined. These representations are related to each other by the antilinear charge conjugation operator C^\hat CC^, which maps H+,q+\mathcal{H}_+, q_+H+​,q+​ to H−,q−\mathcal{H}_-, q_-H−​,q−​. Charges q±q_\pm^{}q±​ are intrinsic characteristics of the wave functions ψ±ψ_\pm^{}ψ±​, similar to spin and electric charge. The wave function possesses one of these charges, q+q_+q+​ or q−q_-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)v_{\sf v}v​-invariance in these theories. Restoring this invariance separates particle and antiparticles and eliminates all paradoxes at the level of first-quantized relativistic theories.
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Essentially, we solve the particle-field equations obtained in the matter distribution quantum mechanics. In the framework of this theory, a particle is described by a density amplitude function, we call wave function, as a packet of waves with the time dependent phases proportional to the relativistic Lagrangian. The mass quantization of such a particle, as a distribution of matter, is obtained from the equality of the spatial integral of the density with the mass as a characteristic of the relativistic dynamics of this matter. The particle dynamics, as a motion of a mass spatial distribution, is obtained from the equality of the wave propagation velocities with the distribution coordinate velocities. This description is a physical one, compared to the Schrödinger-Heisenberg description leading to unphysical results as a null velocity for the kinetic energy as a constant of motion, or Zitterbewegung. The description of a particle interaction with an electromagnetic field by additional Lagrangian terms, with a vector potential conjugated to coordinates, as in the Aharonov-Bohm effect, and an electric potential conjugated to time, leads to the Lorentz force, and the Maxwell equations. In the two conjugated spaces, of the coordinates and of the momentum, the dynamics of a quantum particle is described by propagation operators, depending on the coordinate-momentum product, applied to time dependent wave functions satisfying matter-field equations. Unlike the standard Schrödinger or Schrödinger-Dirac equations, depending only on the Hamiltonian operator, instead of the Hamiltonian these equations contain the Lagrangian, which, besides the Hamiltonian, include additional terms with an explicit dependence on velocity, as it is expected in a relativistic theory. We obtain the wave function of a particle-antiparticle system interacting with an electromagnetic field, as a wave packet of spinors propagating in space with finite dimensions, given by the finite definition domain of the density function, in the space of the matter-field momentum, as a constant of motion. We show that an electrically charged quantum particle, rotating in a central electric field, generates a vector potential with a magnetic field perpendicular to the rotational plane and an electric field in this plane. By a quantum transition, a photon is emitted in the particle rotational plane, with the magnetic field perpendicular to this plane, and the electric field contained in this plane. In this framework, the vector potential rotation is correlated to a particle momentum rotation, for the canonical matter-field momentum as a constant of motion, without any electromagnetic field radiated outside.
Relativistic Schroedinger Theory (RST), as a general gauge theory for the description of relativistic N-particle systems, is shown to be a mathematically consistent and physically reasonable framework for an arbitrary assemblage of positive and negative charges. The electromagnetic plus exchange interactions within the subset of {\it identical} particles are accounted for in a consistent way, whereas {\it different} particles can undergo only the electromagnetic interactions. The origin of this different interaction mechanism for the subsets of identical and non-identical particles is traced back to the fundamental conservation laws for charge and energy-momentum: in order that these conservation laws can hold also for different particles, the structure group U(N)\mathcal U(N)U(N) of the fibre bundles must be reduced to its maximal Abelian subgroup U(1)×U(1)×...×U(1)\mathcal U(1) \times \mathcal U(1) \times ... \times \mathcal U(1)U(1)×U(1)×...×U(1), which eliminates the exchange part of the bundle connection. The persisting Abelian gauge symmetry adopts the meaning of the proper gauge group for the electromagnetic interactions which apply to the identical and non-identical particles in the same way. Thus in RST there is an intrinsic dynamical foundation of the emergence of exchange effects for identical particles, whereas the conventional theory is invaded by the exchange phenomenon via a purely kinematical postulate, namely the antisymmetrization postulate for the wave functions due to Pauli's exclusion principle. As a concrete demonstration, a three-particle system is considered which consists of a positively charged particle of arbitrary rest mass and of two negatively charged particles of equal spin, mass and charge (e.g. electrons).
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Department ofScience Educationand Research Institute forBasicSciences, EwhaWomansUniversity, Seoul 120-750 Korea(Dated: August 15, 2011)We construct a relativistic quantum mechanics for a boson which sustains the positive definiteparticle probability for the relativistic boson. In this theory, we settle down the well-known 85 yearsold unsolved problem on the particle probability of the Klein-Gordon quantum mechanics. Thesolutions for a boson obtained from our theory possess a negative solution as well as a positive one,similar to those of the Dirac quantum mechanics for a fermion. We next study Lorentz transforma-tion, and a vacuum state with all fermion and boson negative energy levels filled and all fermion andboson positive energy levels empty, so that we can have unification of the bosonic sea and the Diracfermionic one. In this unified sea we investigate the bosonic and fermionic antiparticles associatedwith the unified vacuum on an equal footing, and we obtain a charge conjugation operator to studydecay modes of the particles and antiparticles.I. INTRODUTION