Generally covariant geometric momentum and geometric potential for a Dirac fermion on a two-dimensional hypersurface

TL;DR

This work extends the notion of geometric momentum to a generally covariant form suitable for multi-component quantum states of Dirac fermions constrained to two-dimensional hypersurfaces. It derives an explicit momentum operator ${\bf p}=-i\hbar\left(\nabla_{\Sigma}+\frac{M\mathbf{n}}{2}\right)-{\bf A}$ with gauge term ${\bf A}=\hbar\mathbf{r}^{\mu}\Omega_{\mu}$ and introduces a $2\times2$ geometric potential $V_G$ constrained by dynamical quantization conditions. Through two examples, a pseudosphere and a helical surface, it is shown that $V_G=C I + C'\sigma_z$, a constant matrix that can be eliminated by an energy reference, implying no curvature-induced geometric potential in these cases. The framework provides a practical method to obtain both the generally covariant geometric momentum and the geometric potential for spinful particles on parametrically defined surfaces, with potential implications for curvature-induced gauge fields in constrained quantum systems.

Abstract

Geometric momentum is the appropriate momentum for a particle constrained to move on a curved surface, which depends on the extrinsic curvature and leads to observable effects, and curvature-induced quantum potentials appear for a nonrelativistic free particle on the surface. In the context of multi-component quantum states, the geometric momentum should be rewritten as a generally covariant geometric momentum, which contains an additional term defined as the gauge potential. For a Dirac fermion constrained on a two-dimensional hypersurface, we derive the generally covariant geometric momentum, and demonstrate that no curvature-induced geometric potentials arise on a pseudosphere or a helical surface. The dynamical quantization conditions are verified to be effective in dealing with constrained systems on hypersurfaces, enabling the derivation of both the generally convariant geometric momentum and the geometric potential for a spin particle constrained on parametrically defined surfaces.

Figures (2)

• Figure 1: A pseudosphere with parametric equation ${\bf{r}}(u,v)=\left(\alpha\cos{u}\cos{v},\alpha\cos{u}\sin{v},\alpha[{\rm{ln}}(\sec{u}+\tan{v})-\sin{u}]\right)$, where $u \in [0,\frac{\pi}{2})$, $v\in [0, 2\pi)$, and $\alpha=1$.
• Figure 2: A helical surface with parametric equation ${\bf{r}}\left(u,v\right)=(u\cos{v},u\sin{v},\beta v)$, where $u \in (-\infty,\infty)$, $v \in (-\infty,\infty)$ and $\beta=1$.