作为Cosserat弹性特例的无质量Dirac方程
Massless Dirac equation as a special case of Cosserat elasticity
摘要：
我们提出了一个无质量中微子的数学模型。考虑三维欧氏空间中的弹性连续体，并假设该连续体的点不能经历位移，只能经历旋转。这个框架是所谓的Cosserat弹性理论的一个特例。旋转的点的连续描述附加到每个点的正交基，它给出了一个领域的正交基称为coframe。作为我们理论的动力学变量（未知数），我们选择一个余框架和一个密度。我们写下一个共形不变的势能，然后用标准的牛顿方法，通过减去动能，来加入时间。最后，我们重写了一个未知的旋量场的非线性变分问题。我们寻找准稳态解，即在时间上谐波振荡的解。我们证明了在准定常条件下，我们的模型等价于一对无质量的Dirac方程。证明的关键要素是观察到我们的拉格朗日允许因子分解。
英文摘要：
We suggest an alternative mathematical model for the massless neutrino. Consider an elastic continuum in 3-dimensional Euclidean space and assume that points of this continuum can experience no displacements, only rotations. This framework is a special case of the so-called Cosserat theory of elasticity. Rotations of points of the continuum are described by attaching to each point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose a coframe and a density. We write down a potential energy which is conformally invariant and then incorporate time in the standard Newtonian way, by subtracting kinetic energy. Finally, we rewrite the resulting nonlinear variational problem in terms of an unknown spinor field. We look for quasi-stationary solutions, i.e. solutions that harmonically oscillate in time. We prove that in the quasi-stationary setting our model is equivalent to a pair of massless Dirac equations. The crucial element of the proof is the observation that our Lagrangian admits a factorisation.
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