Why does quantum superposition not cancel out the whole universe? Sorry if this is a stupid question. My layman's understanding is that quantum mechanics describes the universe as a wave function, which can be understood as a weighted superposition (linear combination) of multidimensional sine waves (or equivalently, complex vectors) with particular frequencies and relative phases, and where the terms can be said to correspond to physically-valid "universes", weighted by complex probability amplitude. In general, one of the weird results is that when the phase matches precisely, the waves of different "universes" may cancel each other out, eg yielding an interference pattern in the two-slit experiment.
My vague intuition says that by some sort of symmetry in physical laws, for every such valid "universe", there should be some equivalent universe which is precisely identical except for sign, and which should thus cancel out. Of course this isn't expected to be the case locally in the presence of boundary conditions, but for the universe as a whole I'm not so sure.
Does my question make sense, or is it based on a fundamental misunderstanding? Are there particular asymmetries in our physical laws which break my intuition? Is this related to phenomena like matter/anti-matter asymmetry?
 A: It's a very good question!
The sum-over-histories formulation is based on the path integral formulation, where the condition is that the action (phase change) is stationary over the interval between start and end points. The phase can take any value at the start point, $0-2\pi$, and the phase at the end point after following each path is some offset from it. It is the differences between start and end phases that get summed, and that can cancel out. All the possible start phases result in the same start-end differences, and so the same physical results.
In a sense, the phase of the wavefunction isn't a real physical quantity - it's an artefact of using the integral of something real because that's more mathematically convenient, giving a nice intuitive mental picture, but requiring an arbitrary constant of integration to give it a specific value. It's like defining the zero of potential energy - the absolute altitude of the hill. The physics only depends on potential differences, gradients, and so on. There is no physical difference between the effect of gravitational potentials being measured from mean sea level, the WGS84 ellipsoid, the centre of the Earth, or infinity (the energy needed to blast off into space and reach escape velocity) in terms of which way water flows or a ball rolls down hill. Only the slope is physically meaningful. But we find hill heights easier to think about and work with than an altitude gradient field. Quantum phase is the same. The absolute phase is a meaningless concept. The phase gradient is integrated, and an arbitrary constant applied to give it a definite value.
