HOW TO OPEN TWO LOCKS WITH ONE KEY

Two of the most difficult problems in the foundations of physics are (1) what gives rise to the arrow of time (Penrose [1999]; Callender [2021]) and (2) what the ontology of quantum mechanics is. We have the first puzzle because the fundamental dynamical laws do not specify an arrow of time. We have the second puzzle because the quantum mechanical wave function describes a mysterious reality that is dramatically different from the world of our ordinary experiences (Ney and Albert [2013]).

In my article, I propose a new solution that solves both problems. The key is what I call density matrix realism. This is the idea that the quantum state of the universe is objective but impure. The new solution to the problem of time asymmetry, which I call the initial projection hypothesis, makes the quantum state a good candidate for a simple law. If I am right, quantum mechanics and statistical mechanics are made for each other and are supposed to be understood in a unified way. Nature is so unified that we can open both locks with one key.

In the past, many people have accepted that to solve the problem of the arrow of time, we need to posit a low-entropy boundary condition of the universe, now called the past hypothesis. Many people have also accepted that to solve the mysteries of quantum mechanics, we should accept one of the three kinds of realist quantum theory, namely, Bohmian mechanics, Ghirardi–Rimini–Weber spontaneous collapse (GRW) theory, or Everettian quantum mechanics (the many-worlds interpretation).

These theories all share a common assumption: that the quantum state of the universe is objective (that is, it is not something that merely represents our ignorance) and pure (that is, it is represented by a wave function). In Bohmian mechanics, the quantum state determines particle motion; in GRW theory, it undergoes spontaneous collapses; in Everettian quantum mechanics, it realizes an emergent multiverse. Let’s call such a view wave function realism (Chen [2019]).

The past hypothesis and wave function realism are popular proposals, but there remain several puzzles. It’s these that motivated the search for a new theory and my article.

The first puzzle is that the past hypothesis is not enough. We also need to posit an equal-probability distribution over all wave functions compatible with the past hypothesis. But what justifies this distribution is a difficult philosophical question. Moreover, this probability distribution will be distinct from the usual Born rule probability that comes from quantum theory. So, we have two sources of objective probabilities: statistical mechanics and quantum mechanics. It would be nice to understand one in terms of the other.

The second puzzle is that to really understand how the past hypothesis explains law-like regularities, we need to understand the past hypothesis as a fundamental law that is not explained any further. That’s a point stressed by Feynman ([1964]) and Albert ([2000]), among others. But as a fundamental law, it’s a bit odd because it is vague. It is vague because it does not precisely determine the boundary of nomological possibilities (Chen [2022]). Some people will be uncomfortable with a fundamental yet vague law.

Another puzzle is that if we are serious about the quantum state represented by the wave function, we run into interpretational difficulties. Does it represent a material object in spacetime? But how can it, when it’s defined over a vastly high-dimensional configuration space? It’s not impossible to squeeze it into physical spacetime, but it would look awkward. A better proposal is that the quantum state represents a law that determines how matter moves. Call it the nomological interpretation of the quantum state. This faces a difficulty too, because we expect the quantum state of the universe to be highly complicated, which does not make the proposal a good candidate for a fundamental law.

It turns out that these problems are artefacts of wave function realism. We often assume that the real quantum state of the universe must be pure. But quantum mechanics also allows us to associate mixed states or impure states with the universe, and this is represented by density matrices. If, instead, we adopt ‘density matrix realism’, the situation is transformed. Let us modify the realist quantum theories mentioned earlier. In Bohmian mechanics, it’s not a pure state (the wave function) that determines particle motion, but a density matrix, in accordance with the density-matrix version of the guidance equation. In GRW theory, it’s not a pure state (the wave function) that undergoes collapses, but a density matrix, in accordance with a generalized version of the collapse equation. And, finally, in Everettian quantum mechanics, it’s not a wave function evolving according to the Schrödinger equation, but a density matrix evolving according to the von Neumann equation. The fundamental mixed state realizes an emergent multiverse, which has more branches than the one realized by a pure state.

Now, density matrix realism was introduced much earlier but it’s not explicitly endorsed by anyone (Bell [1980]; Dürr et al. [2005]; Wallace [2012], [unpublished]; Allori et al. [2013]). After all, it was not obvious that there is any clear advantage in adopting density matrix realism over wave function realism. What I show in my article is that density matrix realism has many unexpected advantages, but only after we posit a new version of the past hypothesis. My new version states that the quantum state of the universe is not just a density matrix merely compatible with the constraint, but the most natural one given the constraint—the normalized projection onto the past-hypothesis subspace. I call this the initial projection hypothesis. Let’s call this new framework (that is, density matrix realism plus the initial projection hypothesis) ‘the Wentaculus’. We have three versions of the Wentaculus, corresponding to the three realist quantum theories: the Bohmian Wentaculus, the GRW Wentaculus, and the Everettian Wentaculus. (There is a story about the name ‘Wentaculus’, but I won’t get into that right now.)

What differences does the Wentaculus make? A lot, as it turns out.

First, the Wentaculus removes the central difficulty facing the nomological interpretation of the quantum state—namely, the complexity problem. If the past hypothesis is simple enough to be a law, then so is the initial projection hypothesis. If the past-hypothesis subspace is sufficiently simple to characterize, then so is its normalized projection. So, the initial quantum state is sufficiently simple to be nomological. The initial quantum state, which is now nomological, together with the dynamical laws entail the nomologically possible histories (or probability distributions in the GRW theory case). This has the additional advantage of ensuring what Albert ([2015]) calls the ‘narratability’ of the fundamental material ontology in a fully Lorentz-invariant theory, such as Everettian quantum mechanics (Chen [forthcoming]).

Second, the Wentaculus achieves probability monism (Chen [2020]). While the past hypothesis selects an infinity of quantum states, the initial projection hypothesis chooses a unique one. We no longer need a probability distribution over initial quantum states, because there is exactly one nomologically possible quantum state. Thus, the only objective probabilities are from quantum mechanics.

Third, the Wentaculus offers more theoretical unity. Both the universe and the subsystems are described by mixed states, and they evolve according to similar equations.

Fourth, the Wentaculus gets rid of the vagueness of the boundary condition (Chen [2022]). The exact specification of the initial density matrix has dynamical significance, manifesting in the motion of objects in spacetime. This is not so for the exact boundary of traditional versions of the past hypothesis.

Finally, the Wentaculus, in its Everettian version, is the first realistic and simple example of strong determinism; its fundamental laws permit exactly one possible world. The actual multiverse could not have been different without violating the laws.

In summary, the Wentaculus is a new class of theories that solves the puzzles of time’s arrow and quantum ontology, together and in a more satisfactory way. This has implications for several central debates in philosophy of physics. There are no longer two sources of objective probabilities; the vagueness of the boundary condition law is removed; and we don’t need to reify the high-dimensional configuration space as the fundamental space. The Wentaculus shows that the foundations of quantum mechanics and foundations of statistical mechanics are intimately connected. It’s satisfying to see that nature is so unified, and two puzzles can be unlocked with a single key.