DGZ and Lewis on probabilities in deterministic theories Barry Loewer Rutgers

“What we regard as the obvious choice of primitive ontology- the basic kinds of entities that are the building blocks of everything else (except, of course, the wave function)—should by now be clear: Particles, described by their positions in space, changing with time—some of which, owing to the dynamical laws governing their evolutions, perhaps combine to form the familiar objects of daily experience.” (1992 p10 )

Creation Myth God made the world by creating 1. space-time 2. the Bohmian dynamical laws (Schrödinger and the guidance equation) 3. the initial wave function 4. the initial particle positions Every physical thing is composed of particles and every contingent physical fact supervenes on the actual and counterfactual motions of particles.

Probabilities? But if this is all God does to create the world it is not clear that he has done enough or whether he has done the right thing to get quantum mechanics as it is used by physicists. Quantum mechanics is a probabilistic theory that makes claims about the probabilities of events; e.g. that the probability that a certain particle will hit a screen in a particular region is such an such, that a spin measurement will result in such and such an outcome, and so on. Probabilities are an integral part of the kinds of phenomena typically taken to be quantum mechanical, e.g. the uncertainty principles and the violations of Bell’s inequalities.. These don’t follow from the Bohmian dynamical laws alone.

Bohmian probabilities Where do the probabilities in Bohmian mechanics come from? And what do they mean? In orthodox qm probabilities come in via the collapse postulate which- if understood realistically- is an indeterministic law that governs the evolution of the quantum state when measurements- whatever they are- are made. In GRW they also come in via an indeterministic law although without measurement figuring in the law. But in Bohmain mechanics the dynamical laws- Shroedinger’s and the guidance equation- are deterministic. In a deterministic theory it seems that there is only one place for probabilities to enter. – as a distribution over the initial conditions of the universe.

Initial condition probabilities? But what is the meaning of a probability distribution over the possible initial positions compatible with the initial wave function? There seem to be two answers: 1 it is a subjective probability - perhaps one meeting certain rationality constraints- that represents ignorance of the initial positions of particles. 2. it is an objective probability distribution over the initial positions of particles corresponding to some fact about the world..

DGZ argue that it makes no sense to construe the initial distribution as a probability. The reason is that “there is only one universe.” Evidently they have in mind frequency accounts of probability on which an event has a probability only relative to an ensemble of experiments or trials. The probability of e.g. heads on a coin flip is the frequency of heads on coin flips; or perhaps the frequency in the infinite limit were the coin to be tossed infinitely many times.

DGZ approach First off they drop talk of probability for talk of typicality. Even after having discussed this with Shelly I am not sure I understand their notion of typicality. But here is my take on it. Typical behavior is behavior that occurs almost all the time. But not any almost aways behavior is typical. For behavior to be typical laws have to be in some way responsible or partly responsible for the behavior occurring almost always. One can rely on typical behavior so it is rational to believe that behavior will be typical. So, for example, the Chicago cubs early exist from the playoffs was typical or appears so. There is a feeling of inevitability-at least among Cubs fans- that they will never win the world series- or if they do it will be a fluke.

Typical Bohmian Worlds DGZ claim that a typical Bohmian world is one that behaves quantum mechanically.

Problem There are as many (two to the alpha 0) initial conditions that lead to unQM worlds as there are that lead to QM worlds (world in which the QM frequencies are manifested).

DGZ reply DGZ say instead that “majority” should be understood in terms of a measure of typicality. Specifically they suggest that Ψ squared be understood as measuring the typicality of possible initial particle positions compatible with the initial universal wave function. The idea is that the greater the value of this measure the more typical is the initial condition. The typicality of a proposition is then the sum of the typicality of the worlds at which it is true. A typical proposition has typicality 1 or very close to 1. DGZ prove

DGZ In Bohmian mechanics, a property P is typical if it holds true for the overwhelming majority of histories Q(t) of a Bohmian universe. More precisely, suppose that Ψt is the wave function of a universe governed by Bohmian mechanics; a property P, which a solution Q(t) of the guiding equation for the entire universe can have or not have, is called typical if the set S0(P) of all initial configurations Q(0) leading to a history Q(t) with the property P has size very close to one, S0(P) |Ψ0(q)|2dq = 1 − ε, 0 ≤ ε 1, with “size” understood relative to the |Ψ0|2 distribution on the configuration space of the universe. For instance, think of P as the property that a particular sequence of experiments yields results that look random (accepted by a suitable statistical test), governed by the appropriate quantum distribution. One can show, using the law of large numbers, that P is a typical property.

With typicality measured relative to Ψ squared DGZ show that a certain property P of solutions Q(t) is typical. The property P is having a configuration such that, when subsystems of it merit effective wavefunctions, the actual subensembles have probability density of positions given by |  |2. This pattern is precisely what needs to be the case for Born’s rule to be appropriate. The claim that P is typical is another way of saying that a Law of Large Numbers result has been proven such that most (with respect to measure (6)) Bohmian histories Q(t) have property P. They have proved that- given a particular construal of “most”- most Bohmian universes are such that Born’s rule works in them.

Two questions 1. Why should anyone care about this account of “most” and its companion notion of typical? i.e. why should we believe that typical behavior will occur. Afterall a proposition can be typical but false and there are continummly many Bohmian worlds in which it false. 2. What facts about the world make it the case that (given an answer to the first question) Ψ squared is the correct measure of typicality.

What makes a measure the correct typicality measure 1. Nothing objective about the world. Rather it is subjective matter what one finds typical. 2. It is a matter of rationality what measures are typical. 3. The typicality measure(s) supervene on the dynamical laws. 4. The typicality measure(s) supervene on the dynamical laws and the initial condition (the initial wave function and intitial particle position) i.e. supervenes on the total physical history of the world and the laws.. 5. The typicality measure(s) is a fundamental law or principle over and above the laws and total history.

Neither 1, 2,5 are plausible I don’t think either 1 or 2 are tenable for the same reason that it isn’t tenable to construe the world’s initial probability distribution as subjective or dictated by rationality principles like the principle of indifference. Which measures are typical will ultimately determine what behaviors are lawful- e.g. that the uncertainty relations are laws- and that isn’t a matter of subjective belief or even rationality alone. 5 doesn’t have this problem but is quite unbelievable since it severs the connection between typicality and the laws and facts.

Another grounding for typicality measures There is another line of thought that points to the claim that what is typical is a consequence of the dynamical laws. According to it we don’t have to rely on Ψ squared to tell us what is typical. Many other measures – every measure that is continuous with Ψ squared- delivers the same verdict concerning typicality of frequencies in infinite sequences of experiments.

Nevertheless there are infinitely many worlds satisfying the Bohmian dynamics that behave utterly unquanntum mechanical. It looks like more than the laws are required to ground the correct measure.

Lewis’ account of laws. On Lewis’ account the laws that obtain in a world are specified by what he calls the Best Theory of that world. The Best Theory of a world is the true theory that Best combines simplicity, informativeness, (perhaps other scientific virtues e.g. comprehnsiveness). Lewis observes that simplicity and informativeness are typically at odds – increase in informativeness may come at a cost in simplicity. But his idea is that our world may be such that there is a uniqe true theory that Best combines these two virtues so that increases in informativeness come only at a great cost in simplicity and increases in simplicity come only at a great cost in informativenss.

The idea is that probabilities assignment can be introduced into a theory so as to make a great gain in informativeness while keeping the theory fairly simple. For example, given a long sequence hhthhhtthtttththhtththhth….. a very informative description of the sequence will be very complex. A simple description will be quite uninformative. But it may be that the description… sequence of outcomes of independent trials each with probability.5 for heads is both highly informative and also simple.

Lewis applied to Bohmian Mechanics What God creates is the space time- perhaps a high dimensional space to accommodate the wave function- or perhaps he can get away with 3+1 space time. He then distributes the particle positions and the values of the wave function (or whatever corresponds to the wave function that can be fit into 3-d) throughout all of space time. The laws then are given by the Best theory of this world. Of course, what laws there are will depend on the distribution. But it may be that the laws are a package of the two dynamical laws and the initial probability distribution Ψ squared. Note that on this account Ψ squared is not derived from the other laws- or dependent on a prior notion of typicality. And it is on an equal footing as far as lawfulness is concerned with the dynamical laws. This is why it can ground the lawfulness of the qm frequencies, the uncertainty relations, the impossibility of superluminal signaling and so on. It is not surprising that we have guessed that the initial probability distribution is the equivariant one. Its being equivariant makes it enormously simple. Any other distribution would be incredibly complicated wince it would change over time. Of course there is no guarantee that God created a world whose Best Theory is Bohmian Mechanics.

Comparisons 1. DGZ: the initial distribution or family of distributions measures typicality not probability Lewis: the initial distribution is a probability distribution but not probability understood in terms of frequencies. 2. The DGZ account is not tied to any particular account of laws. But if laws are understood along Maudlin’s lines then it is not clear that the typical quantum mechanical frequencies should count as lawful since they are not consequences of the dynamical laws alone. They have the status of special science laws that depend on special intial conditions Lewis: The initial probability distribution has the same nomological status as the dynaical laws. 3. DGZ: prove that the frequencies obtained in measurement situations (and other quantum mechanical frequencies) are typical. This underlies the usual applications of Born’s rule. Lewis can take over DGZ’s argument but now as an argument directly for the probabilities of the outcomes of experiments and other events. There is no need to take a detour through typicality and frequencies.