David W. Dreisigmeyer, Richard Clawson, R. Eykholt, Peter M. Young
We examine the dynamic and geometric phases of the electron in quantum mechanics using Hestenes' spacetime algebra formalism. First the standard dynamic phase formula is translated into the spacetime algebra. We then define new formulas for the dynamic and geometric phases that can be used in Hestenes' formalism.
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In contrast to formulations of the Dirac theory by Hestenes and by the current author, the formulation recently presented by W. P. Joyce [J. Phys. A: Math. Gen. 34 (2001) 1991--2005] is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a…
Defining a spin connection is necessary for formulating Dirac's bispinor equation in a curved space-time. Hestenes has shown that a bispinor field is equivalent to an orthonormal tetrad of vector fields together with a complex scalar field. In this paper, we show that using Hestenes' tetrad for the spin connection in a Riemannian space-time leads to a Yang-Mills formulation of the Dirac Lagrangian in which the bispinor field is mapped to a set of Yang-Mills gauge potentials and a complex scalar field. This result was previously proved for a Minkowski space-time using Fierz identities. As an ap…
Using the language of the Geometric Algebra, we recast the massive Dirac bispinor as a set of Lorentz scalar, vector, bivector, pseudovector, and pseudoscalar fields that obey a generalized form of Maxwell's equations of electromagnetism. This field-based formulation requires careful distinction between geometric and non-geometric implementations of the imaginary unit scalar in the Dirac algebra. This distinction, which is obscured in conventional treatments, allows us to find alternative constructions of the field bilinears and a more natural interpretation of the discrete C, P, and T transfo…
In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum densit…
Spin of elementary particles is the only kinematic degree of freedom not having classical corre- spondence. It arises when seeking for the finite-dimensional representations of the Lorentz group, which is the only symmetry group of relativistic quantum field theory acting on multiple-component quantum fields non-unitarily. We study linear transformations, acting on the space of spatial and proper-time velocities rather than on coordinates. While ensuring the relativistic in- variance, they avoid these two exceptions: they describe the spin degree of freedom of a pointlike particle yet at a cla…
We study the robustness of geometric phase in the presence of parametric noise. For that purpose we consider a simple case study, namely a semiclassical particle which moves adiabatically along a closed loop in a static magnetic field acquiring the Dirac phase. Parametric noise comes from the interaction with a classical environment which adds a Brownian component to the path followed by the particle. After defining a gauge invariant Dirac phase, we discuss the first and second moments of the distribution of the Dirac phase angle coming from the noisy trajectory.
Geometric phase in the wave function is important with regard to quantum non-locality and adiabatic evolution. We study the confinement of a particle by three-dimensional isotropically moving walls, of relevance to experimental trapping techniques, via a proposed approach that explains the physical origin of the geometric Dirac phase induced in the wave function. This phase depends only on the relative rate of change of the spatial scale factor. The approach also yields the class of external potentials that replicate adiabatic evolution in finite time. As illustrative examples, we consider uni…
Helical Current of Propagating Dirac Electrons and Geometric Coupling to Chiral Environments
研究发现狄拉克电子在传播过程中具有螺旋电流特性，通过几何耦合实现手性依赖的自旋选择性，无需自旋轨道耦合项。
We analyze the geometric phase in the neutrino oscillation phenomenon, which follows the pion decay π+ --> μ+ + ν_μ. Its value πis consistent with the present-day global analysis of the Standard Model neutrino oscillation parameters, accounting for the nonzero value of θ_13. The impact of the charge-parity (CP) violating phase δ, the neutrino's nature, and the new physics is discussed.
Note on Dirac monopole theory and Berry geometric phase
本工作通过将狄拉克单极子理论拓展到参数空间，利用双模哈密顿量模型可视化参数空间中的狄拉克弦，揭示其与贝里几何相位的内在联系，为相关研究提供统一框架。
For a free particle that non-relativistically moves on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states, but the experimental results in topological insulators, whenever curved or not, indicate no evidence of such a potential, implying that there does not exist such a quantum potential for the relativistic particles, constrained on the surface or not. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-in…
Operator ordering as an emergent geometric background in Dirac systems with spatially varying mass
研究狄拉克算符在空间变化质量下的算子排列对谱的影响，发现相对论性狄拉克算符有唯一一致的排列方式，产生对数梯度项并导致谱量化条件的变形。
Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface
本文推导二维超曲面上 Dirac 费米子的广义协变几何动量，证明伪球面和螺旋面不存在曲率诱导几何势，并验证动力学量子化条件的有效性。
We discuss a geometric approach to confining a Dirac neutral particle with a permanent magnetic dipole moment interacting with external fields to a hard-wall confining potential in the Minkowski spacetime through noninertial effects. We discuss the behaviour of external fields induced by noninertial effects, and a case where relativistic bound states can be achieved in analogous way to having a Dirac particle confined to a quantum dot. We show that this confinement of a Dirac neutral particle analogous to a quantum dot arises from noninertial effects that give rise to the geometry of the manif…
By starting from the modified Maxwell theory coupled to gravity, the arising of geometric quantum phases in the relativistic and nonrelativistic quantum dynamics of a Dirac neutral particle from the effects of the violation of the Lorentz symmetry in the cosmic string spacetime is investigated. It is shown that the Dirac equation can be written in terms of an effective metric and a relativistic geometric phase stems from the topology of the cosmic string spacetime and the Lorentz symmetry breaking effects. Further, the nonrelativistic limit of the Dirac equation is discussed and it is shown th…
A fundamental problem regarding the Dirac quantization of a free particle on an $N-1$ curved hypersurface embedded in $N$($\geq 2$) flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. We resolve this problem by means of previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free se…
For a particle that is constrained on an ($N-1$)-dimensional ($N\geq2$) curved surface, the Cartesian components of its momentum in $N$-dimensional flat space is believed to offer a proper form of momentum for the particle on the surface, which is called the geometric momentum as it depends on the mean curvature. Once the momentum is made general covariance, the spin connection part can be interpreted as a gauge potential. The present study consists in two parts, the first is a discussion of the general framework for the general covariant geometric momentum. The second is devoted to a study of…
In "Part I: Vector Analysis of Spinors", the author studied the geometry of two component spinors as points on the Riemann sphere in the geometric algebra of three dimensional Euclidean space. Here, these ideas are generalized to apply to four component Dirac spinors on the complex Riemann sphere in the complexified geometric algebra of spacetime, which includes Lorentz transformations. The development of generalized Pauli matrices eliminate the need for the traditional Dirac gamma matrices. We give the discrete probability distribution of measuring a spin 1/2 particle in an arbitrary spin sta…
Based on the observation that Cacic [10]'s characterization of almost commutative spectral triples as Clifford module bundles can be pushed to endomorphim algebras of Dirac bundles, with the geometric Dirac operator related to the Dirac operator of the spectral triple by a perturbation, the question of complete positivity of the heat semigroups generated by connection laplacian and Dirac and Kostant's cubic Dirac laplacians is approached using spin geometry and C *-Dirichlet forms. The geometric heat semigroups for on endomorphosm algebras of spinor bundles are shown to be quantum dynamical se…
Information geometry is an emergent branch of probability theory that consists of assigning a Riemannian differential geometry structure to the space of probability distributions. We present an information geometric investigation of gases following the Fermi-Dirac and the Bose-Einstein quantum statistics. For each quantum gas, we study the information geometry of the curved statistical manifolds associated with the grand canonical ensemble. The Fisher-Rao information metric and the scalar curvature are computed for both fermionic and bosonic models of non-interacting particles. In particular, …
Dirac monopole magnets in non-Hermitian systems
本文理论上证明非厄米微扰可将点状狄拉克单极子转化为扩展分布，揭示了微小能量差与复几何相移间的定量关系，并提出利用耗散两分量Bose-Einstein凝聚体孤子动力学直接测量该拓扑特征的方案。
Dirac fermions on wires confined to the graphene Moebius strip
本文研究石墨烯莫比乌斯带上受限电子的狄拉克方程精确解，揭示弯曲几何势导致的几何相位、宇称破缺引起的电子态内外侧集中以及由非平凡拓扑造成的能级半整数倍规律。
We derive a semiclassical time evolution kernel and a trace formula for the Dirac equation. The classical trajectories that enter the expressions are determined by the dynamics of relativistic point particles. We carefully investigate the transport of the spin degrees of freedom along the trajectories which can be understood geometrically as parallel transport in a vector bundle with SU(2) holonomy. Furthermore, we give an interpretation in terms of a classical spin vector that is transported along the trajectories and whose dynamics, dictated by the equation of Thomas precession, gives rise t…
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