Equal doesn't mean idential. You have an apple in one hand. Since 1 = 1/2 + 1/4 + ... + 1/2^n + ... you therefore have an infinite collection of apple pie

Equal doesn't mean idential. You have an apple in one hand. Since 1 = 1/2 + 1/4 + ... + 1/2^n + ... you therefore have an infinite collection of apple pieces in your hand? Whether that's a useful description, depends on context. Virtual particles are sort of like this decomposition. I think that a lot of confusion arises because of the innocuous seeming equal sign. Explicitly replace the equals by approprate (algebraic) maps, and at least for me, the relationship becomes a lot clearer. Maybe more familiar to HN, maybe it helps to do "typed" math? E.g. counting numbers have a different type than equivalence classes of Cauchy sequences. From a set theory perspective, it makes sense to ask the question whether the number 3 (as a set) equals the collection of triangles in R^3 (as a set). Of course, the two aren't equal, but it seems weird to even ask this question. Maybe it should be just as incongruous to say that some vacuum state equals it's decomposition into virtual particles. The two things are just differently-typed. You need some computation to convert between them.