Many worlds,the born rule, and

self-locating uncertainty

Sean Carroll, Caltech

w/ Charles (Chip) Sebens(Philosophy, U. Michigan)

Textbook quantum mechanics

1. Hilbert space H.

2. Schrödinger’s equation:

3. Measurements associated with an operator A give eigenvalues:

4. Born Rule: probability of obtaining an isgiven by

5. Collapse: after measurement, system is in state

Everett quantum mechanics (EQM)a/k/a Many-Worlds

1.The world is represented by a state in a Hilbert space H.

2.Schrödinger’s equation:

That’s it!

Distinct worlds from decoherence

Decoherence explains why EQM is a theory ofseparate “worlds.” Let the system interact witha macroscopic environment via Schrödinger:

If hea|esi ≈ 0, decoherence has occurred and branches no longer interfere. They are distinct, non-interacting worlds.

Problems for EQM

Pseudo-problem: “That’s a lot of universes”!(Well, Hilbert space is big.)

Interesting problems:

1. Why do we “collapse” onto some states and not others?

2. How do space and objects emerge from a wave function?

3. Why are probabilities given by the Born rule, p(a) = |ψa|2? Why are there probabilities at all?

Why some states and not others?The preferred-basis problem

Rough guide: “pointer” states have projectorsthat commute with the interaction Hamiltonian:

Upshot: interactions are local in space, so pointer states are localized in space.

cf. quantum Darwinism.

[Zurek et al.]

How does the wave functionget carved up into worlds with objects?

No complete picture yet. Aspiration: state + Hamiltoniandetermine system/environment split, then decoherencetakes over and fixes semiclassical worlds.

Without decoherence, no “quantum fluctuations”!

x

Energy eigenstates are stationary;they don’t “fluctuate” at all.

Therefore: Boltzmann Brains don’t pop into existence in the de Sitter vacuum.[Boddy, Carroll, & Pollack]

Deriving the Born Rule: Approaches

• Frequency operators. In limit of many observations,eigenstates of frequency obey Born-rule statistics.

• Symmetry (Zurek). Environment-assisted symmetries(“envariance”) imply that the Born Rule is the rightway to calculate probabilities.

• Decision theory (Deutsch, Wallace). Rational actorsseeking to maximize their utility will act as if theBorn Rule is true.

Gleason’s theorem: given the inner product, the Born Rule is the unique probability measure that depends only on amplitudes.

Our approach:Self-Locating Uncertainty

How do you apportion credence when there aremultiple copies of your situation in the universe?

Elga argues for a unique(if unsurprising) answer:“Indifference,” givingequal credence to eachindistinguishable circumstance.

Self-Locating Uncertainty in EQM

In the process of measurement, decoherence precedesknowledge. Wave function branches before you know it.

The quantum state is probed by a macroscopicmeasuring apparatus, which passes the outcome onto the observer.

observerapparatus

observerapparatus

tconsciousness ~ 10-2 s

tdecoherence < 10-23 s

In equations:

(apparatus measures)

(decoherence)

(self-locating uncertainty)

(measurement complete)

Applying “indifference between indistinguishablecopies” seems to imply “counting branches” in EQM,which gets the probability rule completely wrong.

Resolution: dig into why indifference was justified in the first place.

A worry: Indifference vs. EQM

Epistemic Separability Principle: Credence you assign to being at different indistinguishable locations shouldn’t depend on what’s happening elsewhere in the universe.

What’s happening far away doesn’t matter.

ESP in the context of quantum mechanics

Take a factorizable Hilbert space, which can bedivided up into Observer, System, and Environment:

Consider unitary transformations acting on the environment:

ESP implies that probabilities are invariant:

The Born Rule for equal amplitudes

If states Ψ1 and Ψ2 only differ in the environment, their branch probabilities should be equal.

But P(awake, 2) = P(friendly, 2), since that’s one branch.

Therefore: P(awake, 1) = P(asleep, 1)= ½.

Ignoring aliens: P(awake, 1) = P(awake, 2).

Ignoring cats: P(friendly, 1) = P(friendly, 2).

Unequal amplitudes: trick doesn’t work!

In this example, treating cats as part of the environmentyields two different states. No reason for them to betreated equally under ESP.

New trick: Decompose the state into equal-length components with identical observer circumstances.

Each one gets assignedequal probability.

Branches with larger amplitude correspond to twice as many equal-length components, and therefore twice the probability. That’s the origin of the Born Rule.

It’s just “counting,” but not branches -- equal-lengthcomponents with indistinguishable observers.

[cf. Zurek]

Naïve idea: “The Born Rule gives the probability thatI will end up as any particular observer.”

Problem: you don’t “end up as some observer.”You evolve, with certainty, into several observers.

But all of them should assign Born-Rule credences.

Why probability at all?

all your descendants use the Born Rule to assign credence in their present

therefore you shoulduse the Born Rule to assign

probabilities to the future

Conclusions

• Probability in EQM can be thought of as arising fromsmooth evolution into self-locating uncertainty.

• The Epistemic Separability Principle says that features of observer+system are independent of the state of the environment.

• Using ESP, we show that self-locating credences shouldbe apportioned via the Born Rule.

• Interesting questions remain regarding the nature ofbranching and semiclassical worlds.