欧氏量子引力的谱几何方法
Towards Spectral Geometric Methods for Euclidean Quantum Gravity
摘要：
广义相对论与量子理论的统一还需要将广义相对论和量子理论这两种完全不同的数学语言结合起来，即，微分几何和泛函分析的基本原理。在这方面特别感兴趣的是谱几何领域，它研究黎曼流形的形状在多大程度上可以用定义在流形上的微分算子的谱来描述。光谱几何是困难的，因为它是高度非线性的，但线性化的光谱几何，即，从小的光谱变化确定小的形状变化的任务更加容易处理，并且可以迭代以近似整个问题。在这里，我们推广这种方法，允许，特别是，不相等的有限数量的形状和光谱自由度。这使我们能够研究形状自由度在特征值中的编码情况。我们将这种策略数值应用到一类平面域，发现如果使用足够的特征值，则从小的光谱变化重建小的形状变化是可能的。虽然isospectral非等距形状是已知的存在，我们发现的证据表明，一般形状isospectral非等距形状，如果存在的话，是非常罕见的。
英文摘要：
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis respectively. Of particular interest in this regard is the field of spectral geometry, which studies to which extent the shape of a Riemannian manifold is describable in terms of the spectra of differential operators defined on the manifold. Spectral geometry is hard because it is highly nonlinear, but linearized spectral geometry, i.e., the task to determine small shape changes from small spectral changes, is much more tractable, and may be iterated to approximate the full problem. Here, we generalize this approach, allowing, in particular, non-equal finite numbers of shape and spectral degrees of freedom. This allows us to study how well the shape degrees of freedom are encoded in the eigenvalues. We apply this strategy numerically to a class of planar domains and find that the reconstruction of small shape changes from small spectral changes is possible if enough eigenvalues are used. While isospectral non-isometric shapes are known to exist, we find evidence that generically shaped isospectral non-isometric shapes, if existing, are exceedingly rare.
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