In the Hamiltonian formulation, it is not a priori clear whether a symmetric configuration will keep its symmetry during evolution. In this paper, we give precise requirements of when this is the case and propose a symmetry restriction to the phase space of the symmetric variables. This can often ease computation, especially when transcending from the infinite dimensional phase space of a field theory to a possibly finite dimensional subspace. We will demonstrate this in the case of gravity. A prominent example is the restriction of full general relativity in its Hamiltonian formulation to the cosmological configurations of Robertson–Walker type. We will demonstrate our procedure in this setting and extend it to examples that appear useful in certain approaches to quantum gravity.

在哈密顿公式中，对称构型是否会在演化过程中保持其对称性并不清楚。在本文中，我们给出了这种情况下的精确要求，并提出了对对称变量相空间的对称限制。这通常可以简化计算，尤其是从场论的无限维相空间跨越到可能的有限维子空间时。我们将在重力的情况下证明这一点。一个突出的例子是将其哈密顿公式中的完全广义相对论限制为罗伯逊-沃克类型的宇宙构型。我们将在此设置中演示我们的程序，并将其扩展到在某些量子引力方法中似乎有用的示例。