Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families of subspaces. The complex Pythagorean theorems differ from the real ones in that measures are not squared, and this may have important implications for quantum theory. Projection factors of the complex line of a quantum state with the eigenspaces of an observable give the corresponding quantum probabilities. The complex Pythagorean theorem for lines corresponds to the condition of unit total probability, and may provide a way to solve the probability problem of Everettian quantum mechanics.

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投影因子与广义实数和复勾股定理

投影因子描述了Lebesgue测度在实际或复杂内积空间的子空间之间的正交投影中的收缩。它们与Grassmann和Clifford代数以及子空间之间的Grassmann角相连，并导致广义的勾股定理，从而关联实子空间或复子空间的子集的度量以及它们在子空间的某些族上的正交投影。复杂的毕达哥拉斯定理与真实定理不同，因为测度不是平方的，这可能对量子理论具有重要意义。量子态的复数线与可观察到的本征空间的投影因子给出相应的量子概率。线的复毕达哥拉斯定理对应于单位总概率的条件，