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
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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.