against space

space is not fundamental.time might be.

Sean Carroll, Caltech

http://preposterousuniverse.com/

“What is and is not fundamental”is not fundamental.

What features will be important ingredients in an ultimate (as yet hypothetical) comprehensive theory of everything.

Theories often have very different-looking butequivalent descriptions (e.g. soliton/particleduality). Who is to say what is “fundamental”?

But some things are certainly not fundamental;e.g. temperature. Theories using them are notcomprehensive. Space is like that.

Classical Mechanics

Start with a set of coordinates .

These obey second-order equations of motion:

Specifying the coordinates alone doesn’t determinea solution; need to give and .

Coordinates qi and momenta pj.

Hamiltonian function H(qi, pj).

Hamilton’s equations:

Together we have a = {qi, pj}, defining phase space .

A single point a(t0) in defines a unique trajectory.

Hamiltonian Mechanics

Phase space is a symplectic manifold.

A symplectic form is a closed, invertible 2-form.

Trajectories are integral curves of the Hamiltonianvector field,

a(t)

Xa

The coordinate/momentum distinction is blurred.

Conventionally:

cotangent bundle T*M = {qi, pi}

= phase space

configuration space M, coordinates qi

symplectic form = dpi dqi

(automatic)

Every cotangent bundle is a symplectic manifold, butnot every symplectic manifold is a cotangent bundle.

Symplecticity is more “fundamental” than coordinate/momentum distinction.

Mechanics is invariant under canonical transformations:

{q, p} {Q(q,p), P(q,p)}

that leave the form of Hamilton’s equations unchanged.

Example:

Nothing “fundamental” about which are the coordinates,which are the momenta.

Qi = pi ,

Pj = -qj .

Why don’t we live in momentum space?

Think of interacting harmonic oscillators.

Interactions are local in position, not in momentum.

Better: position is the thing in which interactions are local.

Quantum mechanics

States are rays in Hilbert space: |.

Evolution is governed by the Schrödinger equation:

Energy eigenbasis:

Dynamics are defined by the eigenvalues {En},

the spectrum of the Hamiltonian.

Where is “space” in the quantum state?

We can define a position operator with eigenstates

in terms of which the state is

But we don’t have to; momentum also works.

These are related by Fourier transform,

Or other bases, e.g. creation/annihilation operatorsfor a simple harmonic oscillator.

Here,

These operators raise and lower energy eigenstates:

Entanglement

For a generic multiparticle state |,

The wave function is not a function of space,but of many copies of space.

Things don’t happen in “space”; they happenin Hilbert space.

Again, it’s locality of interactions that tempts usto speak otherwise.

Quantum Field Theory

QFT would seem to deeply privilege “space”; theHamiltonian is an integral over space. But why?

Interactions are local in space:

not in momentum:

Gravity

Consider a compact dimension on a circle.R

A scalar field can be decomposedinto Kaluza-Klein modes

with energies

From the higher-dimensional perspective, these modes comprise a tower of massive states.

Conversely: if every field has such a tower, that implies an extra dimension.

M-theory’s 11th dimension

Witten 1995: there are supersymmetric particlemultiplets in Type IIA string theory with massesthat depend on the coupling as

Small : states are heavy and decouple.Large : Kaluza-Klein tower, as if an extra dimension.

Q: How many dimensions are there in string theory?

A: It depends.

x11

10 dimensionalIIA string theory

11 dimensionalsupergravity

T-duality: string theory on a small circle isequivalent to string theory on a big circle.Momentum/winding duality.

Mirror symmetries: IIA string theory on one Calabi-Yaumanifold equals IIB string theory on another one.

These are gauge symmetries; exact equivalence.

No such thing as the “true” compactification.

R

Holography

Maximum entropy inside a regionof space doesn’t go as R3, thevolume, but as R2, the area.

Discovered in the context ofblack holes, but believed to be more general.

Significance:

The world is not made of separate degrees offreedom at each point in space.

Emergent space isn’t just a matter of discreteness.

Maldacena, 1997:quantum gravity(string theory) onfive-dimensionalanti-de Sitter spacetimes a five-sphereis equivalent to aconformal field theory without gravity on the four-dimensional boundary.

“The spacetime one is in” is not unambiguously defined.

10 dimensionsAdS5 x S5

4-dimensionalMinkowski space

AdS/CFT

• QM, states, time, & the Schrödinger equation:

Space somehow recovered from |.

• QM, states, & the Wheeler-de Witt equation:

Space and time recovered from |.

• A generalization of, or replacement for, QM.

What might be fundamental?

Closing ruminations

• Space/coordinates are picked out by thespecific Hamiltonian of the world, notby the structure of our theories.

• Investigations of quantum gravity providestrong evidence that space is emergent,and in a deeper way than local discreteness.Degrees of freedom are not local.

• Unwarranted speculation: trying to understandthe early universe will help us understandthe role of space & time.