The Quest for Quantum Gravity

1. The Quest for Quantum Gravity Joseph Polchinski KITP, UC Santa Barbara Daniel Heineman Prize Lecture APS, Jacksonville, April 16, 2007

2. Details of history Discovery of D-branes, and their role: Discovery vs. invention Discovery is often messy: you often don’t get to where you expected. The history of string theory is full of unexpected twists and turns.

3. The Veneziano formula: Semi-phenomenological formula for meson-meson scattering, t s

4. Veneziano amplitude (1968) meson = infinite tower of harmonic oscillators = string (Nambu, Susskind, Nielsen, 1969-70)

5. supersymmetry extra dimensions gravity supergravity duality D-branes M theory braneworlds and warping black hole entropy matrix theory gauge/gravity duality the landscape Veneziano string

6. Veneziano string supersymmetry extra dimensions gravity supergravity duality D-branes M theory braneworlds and warping black hole entropy matrix theory gauge/gravity duality the landscape A messy history, but the underlying object seems to be uniquely determined. We still do not have the complete form of the theory: there are more surprises in store.

7. Some of the tools of theoretical discovery: Thought experiments Einstein Galileo Maxwell Black holes: Bekenstein, Hawking and many others Consistency

8. An example: the weak interaction: The Fermi of a local interaction (plus V-A), m ne `nm e works well to a point, but gives uncontrollable divergences at higher order. (High precision or high energy)

9. The solution is to resolve the local interaction into the exchange of an intermediate vector boson: ne m `nm W e This IVB must have very specific properties: it must originate from a spontaneously broken Yang-Mills theory.

10. A correct theory must be • Renormalizable • Unitary • Lorentz invariant It is very hard to satisfy all three (e.g., a spatial smearing + Lorentz invariance would imply smearing in time as well, and loss of causality). (There are many solutions if we impose only two of the three). Spontaneously broken gauge symmetry works because there are gauges satisfying any two (Coulomb, Feynman, unitary), but all are equivalent!

11. For gravity it is harder. It seems that the infinities of quantum gravity are not cured simply by adding new particles, but by changing the nature of the particles or even the nature of spacetime. One solution: graviton electron . Instant of time: point loop ) (or strand

12. Is there one theory of quantum gravity or many? • If one imposes only two of the three consistency conditions, one can find many theories of quantum gravity. • Many attempts give up Lorentz invariance at the start, and it has even been argued that this is a necessary feature of quantum gravity. • It is hard to see how the successes of Special Relativity can then be maintained. E.g., the Standard Model would have ~20 extra parameters (different speeds of light for every particle), and even Planck-scale breaking will feed down into the low energy Lagrangian.

13. Towards D-branes: T-duality An interesting thought experiment is to put strings in a periodic space, length 2pR, and then take Rto be very small (Kikkawa & Yamasaki, 1984, Sakai & Senda, 1986): 2pR

14. 2pR • For a point particle the effect is to quantize the momentum, pcompact = n/Rwith integer n. This is a con-tribution to the effective mass seen by a lower-dimensional observer: -pm pm= M2 implies Meff2 = -(pm pm)noncomp.= M2 + (pcompact)2= M2 + n2/R2 As R goes to zero, any state with momentum in the extra dimension becomes infinitely massive, so we just lose a dimension. (The zero radius limit of a cylinder is a line).

15. For strings there is an additional effect -- they can wind: 2pR w = 0 w = 1 w = 3 It costs energy to wind around the cylinder: Meff2 =M2 + n2/R2 + w2R2/a’ (a’= string length-scale2) • As R goes to infinity, the winding states get massive and the momentum states form a continuum. • As R goes to zero, the momentum states get massive and the winding states form a continuum. • There is a symmetry, T-duality: n « w, R« a’/R • The zero R limit is the same as the infinite R limit.

16. The zero R limit is the same as the infinite R limit! • R= √a’ is an effective minimum radius. • This is a simple example of `stringy geometry,’ the fact that strings seem spacetime differently than point particles. Further applications lead to topology change and the resolution of some spacetime singularities. • This is a simple example of emergent spacetime: the dimension that reemerges in the R to zero limit is not manifest.

17. This is for closed strings. Open strings do not have a conserved winding number: There is no quantum number w, so the mass formula is just Meff2 =M2 + n2/R2 As R goes to zero, there is no continuum: the cylinder effectively becomes a line.

18. No, it’s a D-brane*: *Dirichlet membrane In a theory with open and closed strings, if we start with Ddimensions, compactify, and take R to zero, the closed strings live in Ddimensions and the open strings live in D-1dimensions. Is this an inconsistency?

19. Phase diagram of highly supersymmetric string backgrounds (today): M theory heterotic E8 xE8 Type IIA Type IIB heterotic SO(32) Type I

20. Phase diagram of highly supersymmetric string backgrounds (in 1989): heterotic E8 xE8 Type IIA Dine, Huet, Seiberg Dai, Leigh, JP Narain Type IIB heterotic SO(32) Dai, Leigh, JP Horava Type I (all perturbative dualities).

21. Phase diagram of highly supersymmetric string backgrounds (in 1989): heterotic E8 xE8 Type IIA Dine, Huet, Seiberg Dai, Leigh, JP Narain Type IIB heterotic SO(32) Dai, Leigh, JP Horava Type I Weak-strong dualities connect the two sides, and lead to new phases.

22. D-branes provide a simple description of the half-BPS objects required by the dualities: M theory heterotic E8 xE8 Type IIA Type IIB heterotic SO(32) Type I (Almost obvious: T-duality preserves SUSY, and SUSY of Type I is half that of Type II).

23. Unexpected consequences: • New connections to mathematics (K theory, non-commutative geometry, topological string theory …). • New possibilities for phenomenology (braneworlds, warped spaces, brane inflation). • The first counting of black hole entropy. • Gauge/gravity duality (see Stan and Juan’s talks). • New ideas about the nature of `string’ theory: strings are present only in certain classical limits, D-branes seem to come closer to the fundamental degrees of freedom.

24. 1 - 2 One example of new degrees of freedom: Let us try something seemingly completely different, starting with the idea that a minumum length scale could arise from noncommutative coordinates, [xi, xj] ≠ 0. To implement this, let us make them matrices in some N-dimensional space, xiab. We want to recover the usual commutative coordinates at low energy, but have noncommutativity at high energy. We can do this with . H = M ∑i,a,b (xiab)2 + M3∑i,j,a,c |xiab xjbc - xjab xibc |2 Nonrelativistic kinetic energy plus potential energy for commutator.

25. Quantum corrections destroy the hierarchy, so add supersymmetry, M2∑i,a,b yabgiybc xica (The most supersymmetry theory, 16 supercharges, has 9 spatial directions i). At low energy, virtual noncommutative matrices give a long-range force between the particles: aa bb

26. Veneziano string supersymmetry extra dimensions gravity supergravity duality D-branes M theory braneworlds and warping black hole entropy matrix theory gauge/gravity duality the landscape This can be interpreted as a theory of quantum gravity, in light-cone gauge. In fact, it is the full Hamiltonian for M theory, with boundary condition of flat spacetime with a null circle (=DLCQ). (de Wit, Hoppe, Nicolai; Banks, Fischler, Shenker, Susskind). The connection comes because it is the low-energy Hamiltonian for N D0-branes.

27. In contrast to string perturbation theory, which does not converge, Matrix Theory is a complete nonperturbative description, which could even be simulated on a computer. It describes, e.g. black hole formation and decay, spacetime topology change, trans-Planckian scattering, and resolution of some spacetime singularities.

28. Where to from here? Matrix theory defines string/M theory with a certain simple boundary condition: it is `almost background independent.’ AdS/CFT defines it for other boundary conditions. Quantum gravity is a holographic theory, so is very sensitive to its boundary conditions. Our current understanding allows us to answer many questions, but there are many others that remain to be answered: the nonperturbative formulation for realistic compactifi-cations, and especially for cosmology.

29. In cosmology, the only natural boundaries are in the past and future, rather than in spatial directions. Apparently we need to extend our understanding of emergent space to emergent time…