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- 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.

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