## A Single 3N-Dimensional Universe: Splitting vs. Decoherence

A common way of viewing Everettian quantum mechanics is to say that in an act of measurement, the universe splits into two. There is a world in which the electron has x-spin up, the pointer points to “x-spin up,” and we believe the electron has x-spin up. There is another world in which the electron has x-spin down, the pointer points to “x-spin down,” and we believe the electron has x-spin down. This is why Everettian quantum mechanics is often called “the many worlds interpretation.” Because the contrary pointer readings exist in different universes, no one notices that both are read. This way of interpreting Everettian quantum mechanics raises many metaphysical difficulties. Does the pointer itself split in two? Or are there two numerically distinct pointers? If the whole universe splits into two, doesn’t this wildly violate conservation laws? There is now twice as much energy and momentum in the universe than there was just before the measurement. How plausible is it to say that the entire universe splits?

Although this “splitting universes” reading of Everett is popular (Deutsch 1985 speaks this way in describing Everett’s view, a reading originally due to Bryce Dewitt), fortunately, a less puzzling interpretation has been developed. This idea is to read Everett’s theory as he originally intended. Fundamentally, there is no splitting, only the evolution of the wave function according to the Shrödinger dynamics. To make this consistent with experience, it must be the case that there are in the quantum state branches corresponding to what we observe. However, as, for example, David Wallace has argued (2003, 2010), we need not view these branches -indeed, the branching process itself- as fundamental. Rather, these many branches or many worlds are patterns in the one universal quantum state that emerge as the result of its evolution. Wallace, building on work by Simon Saunders (1993), argues that there is a kind of dynamical process; the technical name for this process is “decoherence,” that can ground the emergence of quasi-classical branches within the quantum state. Decoherence is a process that involves an interaction between two systems (one of which may be regarded as a system and the other its environment) in which distinct components of the quantum state come to evolve independently of one another. That this occurs is the result of the wave function’s Hamiltonian, the kind of system it is. A wave function that (due to the kind of state it started out in and the Shrödinger dynamics) exhibits decoherence will enter into states capable of representation as a sum of noninteracting terms in particular basis (e.g., a position basis). When this happens, the system’s dynamics will appear classical from the perspective of the individual branches.

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Note the facts about the quantum state decohering are not built into the fundamental laws. Rather, this is an accidental fact depending on the kind of state our universe started out in. The existence of these quasi-classical states is not a fundamental fact either, but something that emerges from the complex behavior of the fundamental state. The sense in which there are many worlds in this way of understanding Everettian quantum mechanics is therefore not the same as it is on the more naive approach already described. Fundamentally there is just one universe evolving according to the Schrödinger equation (or whatever is its relativistically appropriate analog). However, because of the special way this one world evolves, and in particular because parts of this world do not interfere with each other and can each on their own ground the existence of quasi-classical macro-objects that look like individual universes, it is correct in this sense to say (nonfundamentally) there are many worlds.

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As metaphysicians, we are interested in the question of what the world is fundamentally like according to quantum mechanics. Some have argued that the answer these accounts give us (setting aside Bohmian mechanics for the moment) is that fundamentally all one needs to believe in is the wave function. What is the wave function? It is something that as we have already stated may be described as a field on configuration space, a space where each point can be taken to correspond to a configuration of particles, a space that has 3N dimensions where N is the number of particles. So, fundamentally, according to these versions of quantum mechanics (orthodox quantum mechanics, Everettian quantum mechanics, spontaneous collapse theories), all there is fundamentally is a wave function, a field in a high-dimensional configuration space. The view that the wave function is a fundamental object and a real, physical field on configuration space is today referred to as “wave function realism.” The view that such a wave function is everything there is fundamentally is wave function monism.

To understand wave function monism, it will be helpful to see how it represents the space on which the wave function is spread. We call this space “configuration space,” as is the norm. However, note that on the view just described, this is not an apt name because what is supposed to be fundamental on this view is the wave function, not particles. So, although the points in this space might correspond in a sense to particle configurations, what this space is fundamentally is not a space of particle configurations. Likewise, although we’ve represented the number of dimensions configuration space has as depending on the number N of particles in a system, this space’s dimensionality should not really be construed as dependent on the number of particles in a system. Nevertheless, the wave function monist need not be an eliminativist about particles. As we have seen, for example, in the Everettian approach, wave function monists can allow that there are particles, derivative entities that emerge out of the decoherent behavior of the wave function over time. Wave function monists favoring other solutions to the measurement problem can also allow that there are particles in this derivative sense. But the reason the configuration space on which the wave function is spread has the number of dimensions it does is not, in the final analysis, that there are particles. This is rather a brute fact about the wave function, and this in turn is what grounds the number of particles there are.

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The Wave Function: Essays on the Metaphysics of Quantum Mechanics.Edited by Alyssa Ney and David Z Albert (pgs. 33-34, 36-37).