probability in the everett interpretation: how to live without uncertainty or, how to avoid doing...

Probability in the Everett interpretation: How to live without uncertaintyor, How to avoid doing semantics

Hilary GreavesNew Directions in the Foundations of PhysicsApril 29, 2006

Aims of the talk

Raise and solve the epistemic problem for many-worlds quantum mechanics (MWQM).

Solve it without relying on contentious language of uncertainty.

Conclude that worries about probability do not provide a reason to reject the many-worlds interpretation.

1. The many-worlds interpretation; probability2. First problem of probability (practical)3. Solution to the practical problem

(Deutsch/Wallace)4. Interlude: On the semantics of branching5. Second problem of probability (epistemic)6. Solution to the epistemic problem7. Concluding remarks

Outline of the talk

1.1 Many-worlds interpretations (MWI) introduced

MMeasurement occurs

Pointer goes into mixed

state

Cat goes into mixed state

M

M

Branch 1 Branch 2

Splitting occurs

1.1 Many-worlds interpretations (MWI) introduced

A first pass: “When a quantum measurement is performed, the world splits into multiple branches, and each ‘possible’ outcome is realized in some branch”

1.2 MWI via consistent histories

What there is: ρ (≡|ΨΨ|), undergoing unitary evolution

How the macroworld supervenes on ρ: via a decomposition into histories

Preferred basis problem: which history set? Use dynamical

decoherence (Zurek, Zeh, Gell-Mann and Hartle, Saunders, Wallace)

Emergent branching structure

t

P1(t1)

P2(t2) P2

’(t2)

t

ρ

1.3 The problem of probability

“If one postulates that all of the histories… are realised … then no role has been assigned to the probabilities, and there seems no obvious way of introducing further assumptions which would allow probabilistic statements to be deduced.”(Dowker & Kent (1994))

P1(t1)

P2(t2) P2

’(t2)

|a1|2 |an|2

Quantum weight of ith branch,

|ai|2 := || Ci |Ψ||2

The quantum weights:

satisfy the axioms of probability…

…but mean...…??

1.4 Everett on probability in MWI

Everett (1957), DeWitt (1973): in the limit, the quantum measure of ‘deviant’ branches goes to zero This is true, but it’s

not enough (as in classical case)

Normal statisticsN

1

2

3

4

1.5 My claim (1)

Worries about probability do not provide reasons to reject the many-worlds interpretation.

Outline of the talk


(Deutsch/Wallace)4. Interlude: On the semantics of branching 5. Second problem of probability (epistemic)6. Solution to the epistemic problem7. Concluding remarks

2.1 The practical problem: How to use QM as a guide to life Nuclear power plant

design A:p(disaster) = 0.0000…..07.

Nuclear power plant design B:p(disaster) = 0.9999…..94.

What to do?

|a1|2 |a2|2

Outline of the talk



3.1 Deutsch’s/Wallace’s program Quantum games: (|Ψ, X, P) Utility function, U Probability function, p:

‘decision-theoretic branch weights’

Structural claim: Maximization of expected utility (MEU)

Quantitative claim: decision-theoretic branch weight = quantum branch weight

So: “The rational agent acts as if the Born rule were true.” (Deutsch (1999))

|Ψ = a| + b|

X = (½) || + (-½) ||

P(½)=$100

P(-½) =$1000

$100 $1000

|a|2 |b|2

Outline of the talk



4.1 Semantics A: ‘Subjective uncertainty’

Thing 1 might happen; Thing 2 might happen.

I am uncertain about which will happen.

Thing 1 happens on this branch


‘Thing 1 might happen’ is true iff Thing 1 happens on some

branch to the future.

4.2 Semantics B: The ‘fission program’ (or, How to live without uncertainty)



Thing 1 will (certainly) happen; Thing 2 will (certainly) happen.

There is nothing for me to be uncertain about.

‘Thing 1 will happen’ is true iff Thing 1 happens on some

branch to the future.

4.3 Two ways to understand maximization of expected utility (MEU)

‘Subjective uncertainty’ (SU): despite knowing that the world will undergo branching, the agent is uncertain about which outcome will occur. The probability measure quantifies the rational

agent’s degree of belief in each branch. ‘The fission program’ (FP): There’s nothing for

the agent to be uncertain about: she knows that all branches will be real. The probability measure (‘caring measure’) quantifies

the rational agent’s degree of concern for each branch.

4.4 ‘Mere semantics’?

Two questions:

what is the right

semantics for talking about branching?

is MWQM an

acceptable physical theory?

What hangs on the SU/FP debate?

What hangs on the SU/FP debate? -The applicability of decision theory? (No.) -The epistemological acceptability of a many-worlds

interpretation?? (I will argue: no.)

4.4 ‘Mere semantics’?

Outline of the talk


(Deutsch/Wallace)4. Interlude: ‘Subjective uncertainty’ and the

‘fission program’5. Second problem of probability (epistemic)6. Solution to the epistemic problem7. Concluding remarks

5.1 The epistemic problem: Why believe QM in the first place? e.g. 2-slit experiment:

The problem: Knowing how rationally to bet on the assumption that MWQM is true does not amount to knowing whether or not MWQM is true. We need two things from quantum probability; MEU is only one

of them

This confirms quantum

mechanics

5.2 The confirmation challenge for MWQM Compare and contrast:

“Quantum mechanics predicted that the relative frequency would approximately equal R with very high probability. We observed relative frequency R. This gives us a reason to regard QM as empirically confirmed.”

Seems fine “MWQM predicted that the relative frequency would

approximately equal R on the majority of branches [according to the ‘caring measure’]. We observed relative frequency R. This gives us a reason to regard MWQM as empirically confirmed.”

??? ‘Empirical incoherence’: coming to believe the theory

undermines one’s reason for believing anything like it Is MWQM empirically incoherent?

5.3 My claims (2 & 3)

(2) The epistemic problem (not only the practical problem) can be solved in a way favorable to MWQM

and(3) This is the case regardless of which

is the right semantics for branching.

Outline of the talk




6.1 Strategy for solving the epistemic problem Ask: how exactly do we deal with the epistemic

issue in the non-MW case? Dynamics of rational belief: A Bayesian model of

common-or-garden empirical confirmation Illustrate how 2-slit experiments (etc) confirm QM

Argue that: the same solution (mutatis mutandis) works for MWQM Work out the dynamics of rational belief for an agent

who has non-zero credence in MWQM Deduce that 2-slit experiments (etc) confirm MWQM

6.2 Bayesian confirmation theory (non-branching case) Suppose I have two

theories, QM and T Suppose I perform an

experiment with two possible outcomes, R and R

Four ‘possible worlds’:W={TR, TR, QMR, QMR}

Credence function Cr0 at time t0, prior to experiment Cr0 obeys the ‘Principal

Principle’, i.e.:Cr0(|T) = ChT()Cr0(|QM) = ChQM()

TR TR

TR TR

t0

t2

T

TR TR TR TRQMR QMR

QMR QMR

QM

M M M M

TR TR QMR QMR

6.2 Bayesian confirmation theory (non-branching case)

TR TR

TR TR

t0

t2

T

TR TR TR TRQMR QMR

QMR QMR

QM

M M M M

Centered world in which the agent adopts credence function Cr2

R over W


R over W

TR TR QMR QMR

6.3 How to update beliefs: choosing Cr2

R and Cr2R

Conditionalization on observed outcome: use posterior credence functions Cr2

R=Cr0(|R), Cr2R

= Cr0(|R) IF

Cr0 obeys the Principal Principle, and the agent updates by conditionalization

THEN observing R increases credence in QM at the expense of credence in T

This is why observing R counts as confirmatory of QM

6.4 Generalized Bayesian confirmation theory (‘branching case’) Candidate theories:

MWQM, T Possible worlds:

W = {TR, TR, MWQM}

Centered possible worlds at time t2:

WC = {TR, TR, MWQMR, MWQMR}

TR TR MWQM

TR TR MWR MWR

t0

t2

T

TR TR MWR MWR

M M M

6.4 Generalized Bayesian confirmation theory (‘branching case’)

TR TR MWQM

TR TR

t0

t2

T


R over W


R over W

TR TR MWR MWR

M M M

6.5 Choosing Cr2R and Cr2

R in the branching case Two prima facie plausible updating

policies:Naïve conditionalizationExtended conditionalization

Both of these are generalizations of ordinary conditionalization

6.6 Naïve conditionalization

Some very natural, but pernicious intuitions: ‘Caring measure’ has

nothing to do with credence

The agent’s credence that R occurs is given by: Cr0(R) = Cr0(TR) + Cr0(MWQM)

How to conditionalize:

Cr2R() = Cr0(|R)

≡ Cr0(R)/Cr0(R)

TR TR MWQM

TR TR MWQMR MWQMRTR TR MWR MWR

R definitely happens in this possible world (and so does R)

R does not happen in this possible world

R happens in this possible world

6.7 Naïve conditionalization is bizarre Observation: Naïve conditionalization has the

consequence that: credence in MWQM increases at the expense of credence in T, regardless of whether R or R occurs

i.e. Cr2R(MWQM) > Cr0(MWQM)

and Cr2R(MWQM) > Cr0(MWQM)

This is not surprising Auxiliary premise: No rational updating policy can allow

any theory to enjoy this sort of ‘free ticket to confirmation’ Conclusion: Naïve conditionalization is not the rational

updating policy for an agent who has nonzero credence in a branching-universe theory

6.8 Defining Extended Conditionalization

We have the resources to define an updating policy according to which evidence supports belief in MWQM in the same way that it supports belief in QM: Construct an ‘effective credence function’, Cr'0 (defined on WC), from Cr0

and Car0: Cr‘0(TR) = Cr0(TR) Cr’0(TR) = Cr0(TR) Cr’0(MWQMR) = Cr0(MWQM)Car0(R) Cr’0(MWQMR) = Cr0(MWQM)Car0(R)

Updating policy: obtained by conditionalizing the effective credence function on R and on R

This policy would have the effect that credence in MWQM responds to evidence in just the same way that credence in QM responds to evidence

6.9 Defending Extended Conditionalization

Is Extended Conditionalization the rational updating policy for an agent who thinks the universe might be branching?

Yes: All the arguments we have in favour of

conditionalization in the ordinary case apply just as well in the branching case, and favour Extended Conditionalization over Naïve Conditionalization

6.10 Defending (ordinary) conditionalization: The (diachronic) Dutch Book argument

Assume that degrees of belief give betting quotientsThis holds because the agent is an expected

utility maximizerA fair bet is a bet with zero net expected utility

If the agent updates other than by conditionalization, a Dutch Book can be made against her

6.11 Defending Extended Conditionalization: diachronic) Dutch Book argument

If the agent is an expected-utility maximizer in Deutsch’s/Wallace’s sense (+…), her betting quotients are given by her effective credence function, Cr'0

If the agent updates other than by Extended Conditionalization, a Dutch Book can be made against her

(Other arguments for conditionalization can be generalized in the same sort of way)

6.12 On black magic

How these arguments manage to connect a ‘caring measure’ to credences: Cast the confirmation question in terms of rational

belief-updating Choosing an updating policy is an epistemic action Epistemic action is a species of action The caring measure is relevant to all choices of

actions, including epistemic ones

Outline of the talk




7 Concluding remarks

There exists a natural measure over Everett branches, given by the Born rule (we knew this already)

The measure governs: rational action (Deutsch/Wallace have argued); so we know how to use

the theory as a guide to life rational belief (I have argued); so we are justified in believing the

theory on the basis of our empirical data, just as in the non-MW case The ‘subjective uncertainty’ semantics is not required for any

of the above. This time, we can do philosophy of physics without doing semantics.

Worries about probability are not a reason to reject the many-worlds interpretation.

probability in the everett interpretation: how to live without uncertainty or, how to avoid doing...

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