Many Worlds, the Born Rule, and Self-Locating Uncertainty
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The document discusses the foundations of quantum mechanics focusing on the many-worlds interpretation (Everett quantum mechanics), highlighting decoherence as a mechanism that explains distinct worlds. It examines the Born rule and its derivations through frameworks like decision theory and frequency operators, emphasizing self-locating uncertainty in measuring multiple copies of a quantum state. The text raises questions about probabilities, the emergence of classical worlds from quantum states, and the epistemic separability principle in relation to observers in different environmental conditions.
![Why some states and 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.]](https://image.slidesharecdn.com/aps-bornrule-150317182227-conversion-gate01/85/Many-Worlds-the-Born-Rule-and-Self-Locating-Uncertainty-6-320.jpg)
![How does the wave 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]](https://image.slidesharecdn.com/aps-bornrule-150317182227-conversion-gate01/85/Many-Worlds-the-Born-Rule-and-Self-Locating-Uncertainty-7-320.jpg)
![New trick:
Decompose the state 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]](https://image.slidesharecdn.com/aps-bornrule-150317182227-conversion-gate01/85/Many-Worlds-the-Born-Rule-and-Self-Locating-Uncertainty-17-320.jpg)
Many Worlds, the Born Rule, and Self-Locating Uncertainty
- 1. Many worlds, the born rule, 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 from decoherence 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 states and 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 the wave 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 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 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 How do 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 in EQM 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-locating uncertainty) (measurement complete)
- 12. Applying “indifference between indistinguishable 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 the context 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 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).
- 16. Unequal amplitudes: trick doesn’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 the state 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: “The Born 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 descendants use 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 in EQM 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.