1. Extracting the Universe
from the Wave Function
Sean Carroll, Caltech
Collaborators: Ning Bao, Kim Boddy, Charles Cao,
Aidan Chatwin-Davies, Liam McAllister, Spiros Michalakis,
Jason Pollack, Charles Sebens, Ashmeet Singh

2. General relativity + cosmology = an initial singularity
The Big Bang – Lemaître’s Primeval Atom.
Precisely where GR is not
reliable. Need a quantum
theory that includes gravity.
[Donald Menzel, Popular Science, 1932]
Approach advocated here:
Take quantum mechanics seriously.
Wave-function-first.
Don’t “quantize gravity.” Rather, look
for gravity within quantum mechanics.

3. What is the “state” of a system?
Classical: position x and velocity v:
an element of phase space. v x
Y(x)
Quantum: the wave function, Y(x),
an element of Hilbert space. x
Position and velocity are what you can observe.
But until you measure them, they don’t exist.
Only the wave function does.
The wave function tells you the probability of
measuring different values of position or velocity.
There is no such thing as “position” or “velocity”!

4. Textbook QM vs. Everettian (Many-Worlds) QM
Hilbert space H.
Schrödinger equation:
Measurements associated with an operator A return eigenvalues:
Born Rule: probability of an is given by
Collapse: after measurement, system is in state
Hilbert space H.
Schrödinger equation:
That’s all.
Measurement etc.
is all derived from
evolution of wave
functions in
Hilbert space.

5. We classically-oriented human beings construct
quantum theories by quantizing classical theories.
Nature doesn’t “quantize” anything; it works directly with
wave functions – elements of Hilbert space.
One quantum theory can
have multiple classical
interpretations
(e.g. AdS/CFT).
Starting with |Y>, how do we extract classical reality?

6. Wave-function-first quantum mechanics:
, not
Elements of classical reality – including space and fields – emerge from the quantum state. (Time is a trickier issue.)
Wave functions don’t really collapse. Apparent collapse happens when branches decohere.
Dimensionality of Hilbert space is crucial.

7. Was the Big Bang the beginning of the universe?
time is emergent
universe may or may not have had a beginning
time is emergent
universe had a beginning
time is fundamental
universe is eternal, and need never experience recurrence
time is fundamental
universe is eternal, with a finite recurrence time

8. [ ] ( + ) Schrödinger’s Cat Erica Schrödinger: “I think my father
just didn’t like cats.”
Sleeping Gas
[cat] =
[ ]
A classical cat is in a definite awake/asleep state:
-or-
(cat) =
( )
A quantum cat can be in a superposition:

9. ( + ) Thinking Quantum-Mechanically Classical intuition:
“Cats are either awake or asleep, but quantum mechanics
allows for superpositions. That’s weird.”
(cat) =
( )
Quantum intuition:
“Cats are in arbitrary quantum superpositions, but when we
look we only see them either awake or asleep. That’s weird.”

10. Textbook (Copenhagen) version
Schrödinger’s Cat:
Textbook (Copenhagen) version
The cat is in a superposition of (awake) and (asleep),
then observed. (Hilbert space = all such superpositions.)
(cat)[observer] =
( ) [ ]
observation/collapse
(cat)[observer] =
( ) [ ]
-or-
(cat)[observer] =
( ) [ ]

11. ( + )( ) ( , + , ) Schrödinger’s Cat: Many-Worlds version
Now the cat and the observer are both quantum.
(cat)(obs) =
( )( )
measurement
(cat,obs) =
( , , )
Why don’t we ever “feel like” we’re in a superposition?

12. No wave function collapse – rather,
Decoherence implies “branching” into distinct worlds
Consider Schrödinger’s cat, an observer, and an environment.
(cat)(obs)(env) =
( )( )( )
measurement
(cat,obs)(env) =
( , , )( )
decoherence
(cat,obs,env) =
( , , ) + ( , , )

13. Cosmology cares about quantum mechanics: the Boltzmann Brain problem
1998 discovery: the
universe is accelerating.
Simplest explanation:
vacuum energy, with a fixed
density through time.

14. Consequences of Vacuum Energy
distant objects disappear: we are surrounded by
an horizon (recession velocity > c).
horizon
the universe expands forever at a fixed rate
our patch inside the horizon has a temperature: T = (Evac)2/Eplanck ~ K.
Cosmic No-Hair: the quantum state approaches a static vacuum state (de Sitter space).

15. Classical thermal fluctuations
In thermal equilibrium there
will be rare fluctuations
downward in entropy, with
rate
If equilibrium lasts forever,
these fluctuations can produce
Boltzmann Brains: freak observers
with unreliable memories/beliefs,
inducing cognitive instability.
Naively: Boltzmann Brain problem
implies vacuum energy isn’t eternal.
[Dyson, Kleban & Susskind; Albrecht & Sorbo;
Page; Bousso & Freivogel; Linde]

16. In Everettian quantum mechanics,
equilibrium doesn’t actually “fluctuate” at all
We talk about “quantum fluctuations”
because that’s what we see when
we observe quantum systems.
But unobserved systems in stationary
states don’t fluctuate. They’re stationary.
No “popping in and out of existence.”
Thermal states are statistical mixtures of stationary states.
In EQM, thermal states of closed systems don’t fluctuate.
No decoherence, no branching.
Therefore, Boltzmann Brains don’t pop into existence.
[Boddy, Carroll & Pollack 2014]

17. In progress: Emergent space from quantum mechanics
Traditional QFT: nearby regions are
more entangled than faraway ones. So let’s try defining distance via
entanglement of factors of Hilbert
space, using mutual information.
We use classical multidimensional
scaling to recover the dimensionality
of space in known, simple examples.
[Cao, Carroll, & Michalakis 2016]

18. Emergent gravity in the bulk?
We have space; don’t have a Hamiltonian, dynamics, time.
But a statistically mixed quantum state is defined by a
density matrix, r ,which defines a “modular energy”:
Perturbing the quantum state changes both the entropy and
the modular energy. They obey the Entanglement First Law:
Change in the entropy of a region is proportional to the
change in modular energy.
[cf. Swingle 2009; Van Raamsdonk 2010; Faulkner et al. 2014; Jacobson 2015]

19. Decreasing entropy decreases entanglement, pushing points
away, creating spatial curvature.
Therefore, spatial curvature at p is related to modular energy:
In long-distance (infrared) limit, our theory should behave
like a CFT. Then modular energy is related to ordinary energy:
Covariant version would be Einstein’s equation:
[Cao, Carroll, & Michalakis 2016]

20. Many (many) steps remain to be filled in, but:
Spatial geometry can plausibly related to quantum entanglement of degrees of freedom in Hilbert space.
In that case, geometry is automatically dynamical, changing in response to changes of quantum state.
In the emergent long-distance limit, the relationship between energy and geometry plausibly looks like Einstein’s equation.
Perhaps gravity naturally emerges from the quantum mechanics of locally finite systems.