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Qiaochu_Yuan comments on Causal Universes - Less Wrong
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Several mathematicians I know (and, I would guess, a sizable population of physicists as well) regard Feynman sums-over-histories as mathematical abstractions only. From this perspective they don't describe processes that are actually happening out-there-in-the-world, they're just mathematically convenient and maybe also intuitively useful. (I haven't thought about whether or how this position can be reconciled with what I think is the standard LW position on many-worlds.)
My limited impression of physics is that there is a tendency for mathematically convenient but "not real" descriptions to turn out to be either subtly inaccurate, or to actually correspond to something real. For example, negative frequency photons seem to have some element of reality to them, along with the quantum wave function and virtual particles. I assign some non-trivial probability weight to "either sums over histories are inaccurate descriptions of what happens, or they correspond to something that acts a lot like a real thing", even when knowledgeable physicists say they aren't a real thing.
Me too, but almost all of it would be concentrated at "sums over histories are inaccurate descriptions of what happens." Sums-over-histories are conceptually unsatisfying to me in that they use the classical concept of a history in order to describe quantum phenomena. My vague intuition is that a truer theory of physics would be more "inherently quantum."