Against Space
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Transcript of 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.