A semi-technical talk on deriving the Born Rule for quantum probabilities in the Everett (many-worlds) formulation of quantum mechanics.

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.