Abstract
A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation.
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Notes
According to the Felix Klein Erlangen program, the affine geometry deals with intrinsic geometric properties that remain unchanged under affine transformations (affinities). These preserve collinearity transformations e.g. sending parallels into parallels, and ratios of distances along parallel lines. The Weyl geometry is considered a kind of affine geometry preserving angles. Cfr: H. Coxeter, Introduction to Geometry (Wiley, New York 1969).
The configuration space of the top described by the Lagrangian L is the principal fiber bundle whose base is the Minkowski space-time \({\mathcal{M}}_{4}\) and whose fiber is SO(3,1), conceived as a proper Lorentz frame manifold. The dynamical invariance group is the whole Poincaré group of the inhomogeneous Lorentz transformations.
More specifically, \(A_{\alpha}=\xi^{r}_{\alpha}(\theta)A_{r}(x)\) (α,r=1,…,6), where \(A_{r}(x)=-\frac{\kappa}{2}\{\boldsymbol{H}(x), \boldsymbol{E}(x)\}\) and \(\xi^{r}_{\alpha}(\theta)\) are the Killing vectors of the Lorentz group SO(3,1).
The action of D i over a tensor field F of Weyl type w(F) is given by \(D_{i} F = \nabla^{(\varGamma)}_{i} F -2w(F)\phi_{i}F\), where ϕ i is the Weyl potential and \(\nabla^{(\varGamma)}_{i}\) is the covariant derivative built up from the Weyl connections \(\varGamma^{i}_{jk}\) given by Eq. (4). The Weyl type of D i F is the same as of F. Because w(S)=0, we have D i S=∂S/∂q i.
The two matrices are related by [D (u,v)(Λ)]†=[D (v,u)(Λ)]−1.
The spinors \(\psi^{\sigma '}_{\sigma}(x)\) and \(\psi^{\dot{\sigma}'}_{\dot{\sigma}}(x)\) are invariant with respect to their lower indices, which are related to the spin component along the top axis.
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We thank dott. Paolo Aniello for useful suggestions.
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Santamato, E., De Martini, F. Derivation of the Dirac Equation by Conformal Differential Geometry. Found Phys 43, 631–641 (2013). https://doi.org/10.1007/s10701-013-9703-y
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DOI: https://doi.org/10.1007/s10701-013-9703-y