Abstract: In many approaches to quantum mechanics, it is standard to adopt the eigenstate-eigenvalue link (EEL) to describe how the quantum state of a system represents its properties (Fletcher and Taylor, 2021). Any determinable property $A$ of some system can be represented by a self-adjoint operator $\hat{A}$ with eigenstates $\ket{\psi_i}$, where the corresponding determinates are given by eigenvalues $a_i$. EEL states that the system instantiates the property $a_i$ just in case it is in the eigenstate $\ket{\psi_i}$. In Everettian quantum mechanics (EQM), no system ever enters an eigenstate of any operator that represents a nontrivial determinable of physical relevance; the universal wave-function evolves unitarily without collapse, and decoherence ensures that each possible outcome obtains along a distinct emergent branch. Thus, no Everettian system instantiates nontrivial properties \textit{at the most fundamental level of description}. On the basis of this feature, Wallace argues that if a system does not determinately possess some property at \textit{some} time in the future, then it does not determinately possess that property at \textit{any} future time, even if it possesses that property \textit{in the present}. Consequently, a system can only possess a property for at most one instant or for nearly all instants (Wallace, 2019). Wallace thus dismisses EEL as ``useless." EQM currently lacks a link between its mathematical formalism and the familiar emergent physical properties instantiated along decoherent branches.
We propose a branch- and subsystem-relative EEL: \textit{relative to some emergent branch and subsystem}, the system instantiates the determinate represented by $a_i$ just in case it is in a reduced state stat represents the subsystem as occupying $\ket{\psi_i}$. The branch relativization allows us to link the universal wave-function to the experiences of agents within the branching structure. We show that Wallace's argument presupposes that the system has a time-independent Hamiltonian. To avoid Wallace's conclusion that EEL is ``useless," we restrict the link to subsystems, which generally have time-dependent reduced Hamiltonians. Thus modified, EEL connects the universal wave-function to the branch-relative emergent properties of the subsystems whose behavior we aim to represent.
This is joint work with Sam Fletcher.
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Philosophy of Physics Graduate Lunch Seminar Convenors: Paolo Faglia, Gregor Gajic and Rachel Pederson
