1.
space is not fundamental.
time might be.
Sean Carroll, Caltech
http://preposterousuniverse.com/

2.
“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 but
equivalent descriptions (e.g. soliton/particle
duality). Who is to say what is “fundamental”?
But some things are certainly not fundamental;
e.g. temperature. Theories using them are not
comprehensive. Space is like that.

3.
Classical Mechanics
Start with a set of coordinates .
These obey second-order equations of motion:
Specifying the coordinates alone doesn’t determine
a solution; need to give and .

4.
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

5.
Phase space is a symplectic manifold.
A symplectic form  is a closed, invertible 2-form.
Trajectories are integral curves of the Hamiltonian
vector field,

a(t)
Xa

6.
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, but
not every symplectic manifold is a cotangent bundle.
Symplecticity is more “fundamental” than
coordinate/momentum distinction.

7.
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 .

8.
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.

9.
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.

10.
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,

11.
Or other bases, e.g. creation/annihilation operators
for a simple harmonic oscillator.
Here,
These operators raise and lower energy eigenstates:

12.
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 happen
in Hilbert space.
Again, it’s locality of interactions that tempts us
to speak otherwise.

13.
Quantum Field Theory
QFT would seem to deeply privilege “space”; the
Hamiltonian is an integral over space. But why?
Interactions are local in space:
not in momentum:

14.
Gravity
Consider a compact dimension on a circle.
R
A scalar field  can be decomposed
into 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.

15.
M-theory’s 11th dimension
Witten 1995: there are supersymmetric particle
multiplets in Type IIA string theory with masses
that 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 dimensional
IIA string theory
11 dimensional
supergravity

16.
T-duality: string theory on a small circle is
equivalent to string theory on a big circle.
Momentum/winding duality.
Mirror symmetries: IIA string theory on one Calabi-Yau
manifold equals IIB string theory on another one.
These are gauge symmetries; exact equivalence.
No such thing as the “true” compactification.

17.
R
Holography
Maximum entropy inside a region
of space doesn’t go as R3, the
volume, but as R2, the area.
Discovered in the context of
black holes, but believed to be more general.
Significance:
The world is not made of separate degrees of
freedom at each point in space.
Emergent space isn’t just a matter of discreteness.

18.
Maldacena, 1997:
quantum gravity
(string theory) on
five-dimensional
anti-de Sitter space
times a five-sphere
is equivalent to a
conformal field theory
without gravity on the
four-dimensional boundary.
“The spacetime one is in” is not unambiguously defined.
10 dimensions
AdS5 x S5
4-dimensional
Minkowski space
AdS/CFT

19.
• 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?

20.
Closing ruminations
• Space/coordinates are picked out by the
specific Hamiltonian of the world, not
by the structure of our theories.
• Investigations of quantum gravity provide
strong evidence that space is emergent,
and in a deeper way than local discreteness.
Degrees of freedom are not local.
• Unwarranted speculation: trying to understand
the early universe will help us understand
the role of space & time.