Many Worlds, the Born Rule, and Self-Locating Uncertainty

Many Worlds, the Born Rule, and Self-Locating Uncertainty

• 1.
  Many worlds, the bornrule, and self-locating uncertainty Sean Carroll, Caltech w/ Charles (Chip) Sebens (Philosophy, U. Michigan)
• 2.
  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 is given by 5. Collapse: after measurement, system is in state
• 3.
  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!
• 4.
  Distinct worlds fromdecoherence Decoherence explains why EQM is a theory of separate “worlds.” Let the system interact with a macroscopic environment via Schrödinger: If hea|esi≈0, decoherence has occurred and branches no longer interfere. They are distinct, non-interacting worlds.
• 5.
  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?
• 6.
  Why some statesand not others? The preferred-basis problem Rough guide: “pointer” states have projectors that commute with the interaction Hamiltonian: Upshot: interactions are local in space, so pointer states are localized in space. cf. quantum Darwinism. [Zurek et al.]
• 7.
  How does thewave function get carved up into worlds with objects? No complete picture yet. Aspiration: state + Hamiltonian determine system/environment split, then decoherence takes 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]
• 8.
  Deriving the BornRule: 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 right way to calculate probabilities. • Decision theory (Deutsch, Wallace). Rational actors seeking to maximize their utility will act as if the Born Rule is true. Gleason’s theorem: given the inner product, the Born Rule is the unique probability measure that depends only on amplitudes.
• 9.
  Our approach: Self-Locating Uncertainty Howdo you apportion credence when there are multiple copies of your situation in the universe? Elga argues for a unique (if unsurprising) answer: “Indifference,” giving equal credence to each indistinguishable circumstance.
• 10.
  Self-Locating Uncertainty inEQM In the process of measurement, decoherence precedes knowledge. Wave function branches before you know it. The quantum state is probed by a macroscopic measuring apparatus, which passes the outcome on to the observer. observer apparatus observer apparatus tconsciousness ~ 10-2 s tdecoherence < 10-23 s
• 11.
  In equations: (apparatus measures) (decoherence) (self-locatinguncertainty) (measurement complete)
• 12.
  Applying “indifference betweenindistinguishable copies” 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
• 13.
  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.
• 14.
  ESP in thecontext of quantum mechanics Take a factorizable Hilbert space, which can be divided up into Observer, System, and Environment: Consider unitary transformations acting on the environment: ESP implies that probabilities are invariant:
• 15.
  The Born Rulefor 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).
• 16.
  Unequal amplitudes: trickdoesn’t work! In this example, treating cats as part of the environment yields two different states. No reason for them to be treated equally under ESP.
• 17.
  New trick: Decompose thestate into equal-length components with identical observer circumstances. Each one gets assigned equal 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-length components with indistinguishable observers. [cf. Zurek]
• 18.
  Naïve idea: “TheBorn Rule gives the probability that I 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?
• 19.
  all your descendantsuse the Born Rule to assign credence in their present therefore you should use the Born Rule to assign probabilities to the future
• 20.
  Conclusions • Probability inEQM can be thought of as arising from smooth 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 should be apportioned via the Born Rule. • Interesting questions remain regarding the nature of branching and semiclassical worlds.