Dirac particles, spin and photons

Abstract: We describe relativistic particles with spin as points moving in phase space $X=T^* R^{1,3}\times C^2_L\times C^2_R$, where $T^* R^{1,3}=R^{1,3}\times R^{1,3}$ is the space of coordinates and momenta, and $C^2_L$ and $C^2_R$ are the spaces of representation of the Lorentz group of type $(\frac12 , 0)$ and $(0, \frac12)$. Passing from relativistic mechanics with a Lorentz-invariant Hamiltonian function $H$ on the phase space $X$ to quantum mechanics with a Hamiltonian operator $\hat H$, we introduce two complex conjugate line bundles $L_C^+$ and $L_C^-$ over $X$. Quantum particles are introduced as sections $\Psi_+$ of the bundle $L_C^+$ holomorphic along the space $C^2_L\times C^2_R$, and antiparticles are sections $\Psi_-^{}$ of the bundle $L_C^-$ antiholomorphic along the internal spin space $C^2_L\times C^2_R$. The wave functions $\Psi_\pm$ are characterized by conserved charges $q_{\sf{v}}=\pm 1$ associated with the structure group U(1)${\sf{v}}$ of the bundles $L_C^\pm$. Wave functions $\Psi\pm$ are governed by relativistic analogue of the Schr\"odinger equation. We show how fields with spin $s=0$ (Klein-Gordon), spin $s=\frac12$ (Dirac) and spin $s=1$ (Proca fields) arise from these equations in the zeroth, first, and second order expansions of the functions $\Psi_\pm^{}$ in the coordinates of the spin space $C^2_L\times C^2_R$. The Klein-Gordon, Dirac and Proca equations for these fields follow from the Schr\"odinger equation on the extended phase space $T^* R^{1,3}\times C^2_L\times C^2_R$. Using these results, we also introduce equations describing first quantized photons. We show that taking into account the charges $q_{\sf{v}}=\pm 1$ of the fields $\Psi_\pm$ changes the definitions of the inner products and currents, which eliminates negative energies and negative probabilities from relativistic quantum mechanics.

• The paper introduces a differential geometric formulation that unifies geometric quantization, spin, and relativistic field equations.
• It systematically derives the Klein-Gordon, Dirac, and Proca equations by constructing line bundles and leveraging symplectic structures.
• The framework resolves negative energy issues and extends to first-quantized photons using a twistor-inspired, geometric approach.

Geometric Quantization, Spin, and Relativistic Quantum Mechanics: A Differential Geometric Framework

This paper presents a comprehensive differential geometric formulation of relativistic quantum mechanics, with a focus on the geometric quantization of phase spaces for particles with spin, the structure of antiparticles, and the first-quantized description of photons. The work systematically develops a framework in which classical and quantum properties of spin, charge conjugation, and field equations (Klein-Gordon, Dirac, Proca, Maxwell) are unified as consequences of the geometry and topology of extended phase spaces and associated vector bundles.

Extended Phase Space and Spin as a Geometric Degree of Freedom

The central construction is the extension of the classical phase space for a relativistic particle to include internal spin degrees of freedom. The phase space is taken as

$$X = T^*\mathbb{R}^{1,3} \times \mathbb{C}^2_L \times \mathbb{C}^2_R,$$

where $T^*\mathbb{R}^{1,3}$ is the cotangent bundle of Minkowski space, and $\mathbb{C}^2_L$, $\mathbb{C}^2_R$ are the left- and right-handed spinor spaces, corresponding to the $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ representations of the Lorentz group.

The dynamics of spin is encoded as motion on orbits in these internal spaces, with the classical spin variable $z \in \mathbb{C}^2$ evolving under a $U(1)$ action. The quantization of this structure leads to a discrete spectrum for spin already at the classical level, as the orbits are associated with lens spaces $S^3/\mathbb{Z}_n$, where $n=2s$ and $s$ is the spin. This topological discretization is linked to the first Chern class of the associated line bundle, providing a geometric origin for the quantization of spin.

Charge Conjugation and Antiparticles as Geometric Operations

A key innovation is the identification of charge conjugation and the particle-antiparticle distinction with an antilinear involution on the phase space, corresponding to the reversal of the evolution parameter $\tau \mapsto -\tau$. This operation maps the complex structure of the internal spin space to its conjugate, and, at the quantum level, exchanges holomorphic and antiholomorphic sections of the relevant line bundles. The charges $q_\pm = \pm 1$ associated with this operation are shown to be essential for the correct definition of inner products and currents, ensuring the elimination of negative energy and negative probability solutions at the first-quantized level.

Geometric Quantization: Bundles, Polarizations, and Quantum Observables

Quantization is formulated as the construction of complex line bundles $L_{\mathbb{C}}^\pm$ over the extended phase space, with connections determined by the symplectic structure. Quantum states are polarized sections of these bundles: holomorphic for particles, antiholomorphic for antiparticles. The structure group $U(1)_{\text{vac}}$ is distinct from the gauge groups of the Standard Model and encodes the "vacuum" connection of geometric quantization.

Quantum observables are realized as covariant derivatives in these bundles, and the canonical commutation relations arise from the curvature of the connection. The Schrödinger equation is generalized to a relativistic context as an evolution equation in the parameter $\tau$, with the Hamiltonian operator constructed from the covariant Laplacian on the bundle.

Derivation of Relativistic Field Equations from the Geometric Framework

A central result is the derivation of the Klein-Gordon, Dirac, and Proca equations as consequences of the geometric quantization of the extended phase space. The wave function $\Psi_\pm$ is expanded in the spinor coordinates, and the field equations emerge as constraints on the coefficients in this expansion:

• The zeroth order yields the Klein-Gordon equation.
• The first order yields the Dirac equation, with the correct projection from the full spinor space to the physical subspace enforced by Lorentz-covariant constraints.
• The second order yields the Proca equation for massive vector bosons.

The Bargmann-Wigner formalism is naturally incorporated, with higher-order symmetric spinor fields corresponding to higher-spin representations.

First-Quantized Photons and the Massless Limit

The framework is extended to massless particles, with particular attention to the photon. The massless limit of the Proca field yields the Maxwell equations, and the first-quantized photon is described as a complexified light-like momentum, in line with Penrose's twistor-theoretic approach. The photon wave functions are identified with self-dual and anti-self-dual solutions of Maxwell's equations, corresponding to right- and left-handed polarizations.

Elimination of Non-Physical States and Positive-Definite Inner Products

A significant claim is that, by correctly accounting for the charges $q_\pm$ and the bundle structure, the framework eliminates negative energy and negative probability solutions at the first-quantized level, without recourse to second quantization or operator-valued fields. The inner product and current definitions are modified to ensure positivity and Lorentz invariance, with the charge conjugation operator acting as required.

Implications and Theoretical Significance

The paper provides a unified geometric and topological foundation for relativistic quantum mechanics, clarifying the status of spin, antiparticles, and field equations. The approach resolves longstanding issues with negative energy solutions in first-quantized relativistic wave equations and offers a natural setting for the inclusion of higher-spin fields and internal symmetries.

The explicit geometric realization of quantization, spin, and charge conjugation suggests new avenues for the study of quantum field theory, particularly in contexts where the standard operator formalism is inadequate or ambiguous. The identification of photons as complexified light-like momenta may have implications for the geometric understanding of gauge theories and twistor methods.

Conclusion

This work establishes a rigorous differential geometric framework for the description of relativistic particles with spin, antiparticles, and photons at the first-quantized level. By extending the phase space to include internal spinor degrees of freedom and employing geometric quantization, the paper derives the standard relativistic field equations as consequences of the underlying geometry. The approach resolves the issue of non-physical states in relativistic quantum mechanics and provides a coherent picture of spin, charge conjugation, and field quantization rooted in the topology and geometry of phase space and associated bundles. This framework has the potential to inform future developments in the mathematical foundations of quantum theory and the geometric analysis of field equations.