The Role of Emergence

One might ask why, if the fundamental theory of everything is fixed by the spectrum of some Hamiltonian, we don’t simply imagine writing the state of the universe in the energy eigenbasis, where its evolution is trivial? The answer is the one that applies to any example of emergence: there might be other descriptions of the same situation that provide useful insight or computational simplification.

Consider the classical theory of N particles moving under the influence of some multi-particle potential in 3 dimensions of space. The corresponding phase space is 6N-dimensional, and we could simply think of the theory as that of one point moving in a 6N-dimensional structure. But by thinking of it as N particles moving in a 3-dimensional space of allowed particle positions, we gain enormous intuition; for example, it could become clear that particles influence each other when they are nearby in space, which in turn suggests a natural way to coarse-grain the theory. Similarly, writing an abstract vector in Hilbert space as a wave function over some classical variables can provide crucial insight into the most efficient and insightful way to think of what is happening to the system.

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