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

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.

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”!

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.

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?

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.

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

[ ] ( + ) 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:

( + ) 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.”

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] = ( ) [ ]

( + )( ) ( , + , ) 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?

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) = ( , , ) + ( , , )

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.

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 ~ 10-30 K. Cosmic No-Hair: the quantum state approaches a static vacuum state (de Sitter space).

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]

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]

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]

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]

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]

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.