Probability in the Everett interpretation: How to live without uncertainty

Probability in the Everett interpretation: How to live without uncertainty or, How to avoid doing semantics Hilary Greaves New Directions in the Foundations of Physics April 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.

Outline of the talk • The many-worlds interpretation; probability • First problem of probability (practical) • Solution to the practical problem (Deutsch/Wallace) • Interlude: On the semantics of branching • Second problem of probability (epistemic) • Solution to the epistemic problem • Concluding remarks

1.1 Many-worlds interpretations (MWI) introduced Cat goes into mixed state Pointer goes into mixed state Measurement occurs M M

1.1 Many-worlds interpretations (MWI) introduced Branch 1 Branch 2 • A first pass: “When a quantum measurement is performed, the world splits into multiple branches, and each ‘possible’ outcome is realized in some branch” Splitting occurs M

1.2 MWI via consistent histories ρ • What there is: ρ (≡|ΨΨ|), 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 P2(t2) P2’(t2) P1(t1) t

1.3 The problem of probability |a1|2 |an|2 • “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)) P2(t2) P2’(t2) P1(t1) • Quantum weight of ith branch, • |ai|2 := || Ci|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 statistics N 4 3 2 1

1.5 My claim (1) Worries about probability do not provide reasons to reject the many-worlds interpretation.

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

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)) |a|2 |b|2 P(½)=$100 $100 $1000 P(-½) =$1000 X = (½) || + (-½)|| |Ψ = a| + b|

4.1 Semantics A: ‘Subjective uncertainty’ ‘Thing 1 might happen’ is true iff Thing 1 happens on some branch to the future. Thing 1 happens on this branch Thing 2 happens on this branch Thing 1 might happen; Thing 2 might happen. I am uncertain about which will happen.

4.2 Semantics B: The ‘fission program’ (or, How to live without uncertainty) ‘Thing 1 will happen’ is true iff Thing 1 happens on some branch to the future. Thing 1 happens on this branch Thing 2 happens on this branch Thing 1 will (certainly) happen; Thing 2 will (certainly) happen. There is nothing for me to be uncertain about.

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?

4.4 ‘Mere semantics’? • 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.)

Outline of the talk • The many-worlds interpretation; probability • First problem of probability (practical) • Solution to the practical problem (Deutsch/Wallace) • Interlude: ‘Subjective uncertainty’ and the ‘fission program’ • Second problem of probability (epistemic) • Solution to the epistemic problem • 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 Rwith 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 Ron 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.

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={TR, TR, QMR, QMR} • Credence function Cr0 at time t0, prior to experiment • Cr0 obeys the ‘Principal Principle’, i.e.: Cr0(|T) = ChT() Cr0(|QM) = ChQM() TR TR TR TR TR TR QMR QMR TR QMR TR QMR t2 M M M M t0 TR TR QMR QMR T QM

6.2 Bayesian confirmation theory (non-branching case) Centered world in which the agent adopts credence function Cr2R over W TR TR TR TR TR TR QMR QMR TR TR QMR QMR t2 Centered world in which the agent adopts credence function Cr2R over W M M M M t0 TR TR QMR QMR T QM

6.3 How to update beliefs: choosing Cr2R and Cr2R • Conditionalization on observed outcome: use posterior credence functions Cr2R=Cr0(|R), Cr2R = 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 = {TR, TR, MWQM} • Centered possible worlds at time t2: WC = {TR, TR, MWQMR, MWQMR} TR TR TR TR MWR MWR MWR MWR t2 M M M t0 TR TR MWQM T

6.4 Generalized Bayesian confirmation theory (‘branching case’) Centered world in which the agent adopts credence function Cr2R over W TR TR TR TR MWR MWR t2 Centered world in which the agent adopts credence function Cr2R over W M M M t0 TR TR MWQM T

6.5 Choosing Cr2R and Cr2R in the branching case • Two prima facie plausible updating policies: • Naïve conditionalization • Extended 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 credencethat R occurs is given by: Cr0(R) = Cr0(TR) + Cr0(MWQM) • How to conditionalize: Cr2R() = Cr0(|R) ≡ Cr0(R)/Cr0(R) TR TR TR TR MWQMR MWR MWQMR MWR TR TR MWQM R definitely happens in this possible world (and so does R) R happens in this possible world R does not happen 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 Cr2R(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(TR) = Cr0(TR) • Cr’0(TR) = Cr0(TR) • Cr’0(MWQMR) = Cr0(MWQM)Car0(R) • Cr’0(MWQMR) = 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 quotients • This holds because the agent is an expected utility maximizer • A 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

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