Wave Function Realism vs. Density Matrix Realism in Everettian Quantum Mechanics

Table of Links

Abstract & Introduction

Decoherence and Branching

The Born Rule

Discussion

Conclusion and References

4 Discussion

In the previous sections, we’ve generalized the standard defenses for the Born rule to DMRE. In addition to answering a technical question that has been neglected in the literature, we take our results to have several conceptual implications.

First, in order to set up the stage for the generalized arguments, we were required to contemplate, without presupposing a universal pure state, the ontological structure of the Everettian multiverse. For EQM to allow both WFR and DMR, the story about decoherence and branching should apply to both without prejudice. As we’ve seen, that is indeed the case. This leads us to see that the essence of the Everettian story about the emergence of a branching multiverse is not a universal wave function that gives rise to many branches represented by wave functions, but a universal density matrix (which can be pure or mixed) that gives rise to many branches represented by density matrices. According to the perspective of DMRE, a pure-state multiverse is a very special case.

Second, with DMRE, Everettians can explore new theoretical possibilities of DMR. For example, we can consider a unified treatment of ‘classical’ and ‘quantum’ probabilities in EQM. In WFRE, there are two sources of probabilities: the quantum probability of finding ourselves in a particular branch (or betting preferences in the decisiontheoretic framework), associated with the weight of the branch in the multiverse, and the classical probability of the particular multiverse, associated with a density matrix representing our ignorance of the underlying universal pure state. Their justifications are very different. The latter is not understood in terms of self-locating uncertainties or betting preferences. Instead, it may have a statistical mechanical origin, corresponding to a probability distribution over initial universal quantum states, the so-called Statistical Postulate (Albert 2000). In DMRE, however, the two can be reduced to a single notion of probability, that of finding ourselves in a particular branch (or betting preferences given the actual quantum state), albeit in a more expansive multiverse. Whichever ρ is used by defenders of WFRE to represent their ignorance of the fundamental pure state of the multiverse, defenders of DMRE can regard that ρ as the fundamental mixed state. Insofar as classical and quantum probabilities in EQM can be reduced to a single source, they also can be justified in the same way.

A theory on which we can apply this strategy is the Everettian Wentaculus (Chen 2021, Chen 2022c). This version of DMRE proposes a simple and unique choice of the initial density matrix of the multiverse (as a version of the Past Hypothesis) and regards it as the only nomological possibility. As a matter of physical laws, the history of the Everettian multiverse could not have been different. There is no longer a choice of the fundamental density matrix, beyond the choice of the physical law, because the actual one is nomologically necessary. It is an instance of “strong determinism.” Both classical (statistical mechanical) and quantum probabilities can be understood as branch weights of the Everettian Wentaculus multiverse, represented by a mixed-state density matrix. With the possibility of a unified treatment of probabilities (among other things), the generalization from WFRE to DMRE is theoretically attractive. [5]

Finally, we’ve derived the Born rule in DMRE in essentially the same ways as in WFRE, by appealing to the same epistemic principles (separability, symmetry, decision theoretic axioms) and metaphysical foundations (decoherence and branching). The two theories are empirically equivalent, not just in a mathematical sense, but also conceptually. They give us the same empirical predictions, not just in terms of equal probabilities of measurement outcomes, but also the same kind of probabilities (selflocating uncertainties or betting preferences). We suggest that Everettians, by their own lights, should regard DMRE as a genuine rival to WFRE. Everettians, then, face the question which version of EQM they should accept. What can be the grounds for deciding? It cannot be settled by experiments because of the empirical equivalence. It cannot be based on the insistence that the universal quantum state must be pure, because that would beg the question. It cannot be based on the incompleteness of justifications for DMRE, for the solutions to the ontology problem and the probability problem in WFRE readily extend to DMRE. [6]

[5] Two remarks here: (1) Saunders (2021) has proposed that we can understand quantum probabilities in terms of “branch-counting.” His considerations are analogous to the counting arguments of Boltzmann in the foundations of statistical mechanics. When applied to the Everettian Wentaculus, Saunders’s proposal, if correct, would allow us to justify both classical and quantum probabilities by counting branches in a natural way. (2) For more discussions about the elimination of classical probability in the Wentaculus theories and other theoretical advantages, see Chen (2021, 2020). For two other proposals of eliminating the Statistical Postulate, see Albert (2000, §7) and Wallace (2012, §9).

[6] This adds an interesting wrinkle to the debate about scientific realism and the issues raised by Wallace (2022). Even if Wallace is right that EQM is the only way to make sense of why sky is blue, there is a further question about which version of EQM is correct. Quantum mechanics still leads to in-principle empirical underdetermination.