Or: what symmetry can teach us about quantum gravity and the unification of physics. Technische Universität Dresden, 5 July 2011

Transcription

1 bla Symmetry and Unification Or: what symmetry can teach us about quantum gravity and the unification of physics. Technische Universität Dresden, 5 July 2011 Hermann Nicolai MPI für Gravitationsphysik, Potsdam (Albert Einstein Institut)

2 Truly exceptional: projection of E 8 root system (an 8- dimensional diamond ) onto a two-dimensional plane.

3 Main theme: Symmetry... arguably the most successful principle of physics! Space-time symmetries Rotations and translations in Newtonian physics Special Relativity: unifying space and time [Einstein (1905)] General Relativity and general covariance [Einstein (1915)] Internal symmetries Isospin SU(2) symmetry: m neutron = m proton [Heisenberg] Flavor symmetry, strong interactions and the quark model Standard Model and (non-abelian) Gauge Theories The two fundamental theories of modern physics, General Relativity and the Standard Model of Particle Physics, are based on and largely determined by symmetry principles!

4 Where we stand Known laws of physics successfully describe observed phenomena over a huge range of distances from m all the way to the visible horizon of our universe! General Relativity : gravity from space-time curvature (general covariance and equivalence principle) Standard Model of Particle Physics: combines quantum mechanics and special relativity to describe Matter = 3 generations of 16 spin- 1 2 fermions Forces = electromagnetic, weak and strong via G = SU(3) c SU(2) w U(1) Y gauge symmetry.

5 Where we stand Known laws of physics successfully describe observed phenomena over a huge range of distances from m all the way to the visible horizon of our universe! General Relativity : gravity from space-time curvature (general covariance and equivalence principle) Standard Model of Particle Physics: combines quantum mechanics and special relativity to describe Matter = 3 generations of 16 spin- 1 2 fermions Forces = electromagnetic, weak and strong via G = SU(3) c SU(2) w U(1) Y gauge symmetry.... but both theories are incomplete and possibly even inconsistent: singularities and infinities!

6 Symmetry and Unification Like a ferromagnet: symmetry is broken more and more with decreasing temperature as universe expands.

7 But where do we go from here? Idea: symmetry enhancement as a guiding principle! Grand Unification: look for a simple group SU(3) c SU(2) w U(1) Y SU(5) SO(10) E 6...? quark lepton unification, proton decay,... Fusion of space-time and internal symmetries? E.g. from pure gravity in higher dimensions?[kaluza,klein (1921)] (Electromagnetic) Dualities: E+iB e iα (E+iB), q +ig e iα (q +ig) Quantum symmetry and quantum space-time?

8 But where do we go from here? Idea: symmetry enhancement as a guiding principle! Grand Unification: look for a simple group SU(3) c SU(2) w U(1) Y SU(5) SO(10) E 6...? quark lepton unification, proton decay,... Fusion of space-time and internal symmetries? E.g. from pure gravity in higher dimensions?[kaluza, Klein (1921)] (Electromagnetic) Dualities: E+iB e iα (E+iB), q +ig e iα (q +ig) Quantum symmetry and quantum space-time? QUESTION: is it possible to pin down the right theory simply by imposing a symmetry principle?

9 Supersymmetry Anew kind ofsymmetry relating Bosons Fermions. Or: Forces (vector bosons) Matter (quarks & leptons)? Probably one of the most important developments in mathematical physics over the last 40 years. Supersymmetry is generally believed to be essential for constructing a perturbatively consistent (finite) theory of quantum gravity. Experimental signatures of (low energy) supersymmetry might show up at LHC accelerator at CERN.

10 Supersymmetry in a Nutshell Simple example: supersymmetric quantum mechanics with supercharge Q,Q and Hamiltonian H: H = 1 2 {Q, Q } [Q,H] = [Q,H] = 0 Hence for any eigenstate E (with E > 0) we have H E = E E H(Q E ) = E(Q E ) degeneracy of bosonic/fermionic energy levels SUSY particles come in pairs of opposite statistics, e.g. Photon γ (s = 1) Photino γ (s = 1 2 ) Equal masses for bosons and fermions in supersymmetric models of elementary particles (unless supersymmetry is broken).

11 Supersymmetric (Quantum) Field Theory For (semi-)realistic field theories need to marry supersymmetry with other symmetries of relativistic quantum field theory (Poincaré and internal symmetries). Supercharges Q i α and Q αj are now space-time spinors. The key relation of the relativistic superalgebra is {Q i α, Q βj } = 2δ i jσ µ α β P µ N-extended supersymmetry (for i,j = 1,...,N) merges spacetime and internal symmetries when N 2. In QFT, supersymmetry entails (partial) cancellations of UV infinities in Feynman diagrams Supersymmetry softens the infinities of QFT! But can it remove them altogether?

12 Supermultiplets... a general analysis of what types of particles (of different spin s) can be put together in superfamilies. There are many models with low degree of supersymmetry but they become more and more restricted with increasing N, and unique for maximal supersymmetry. Distinguish between two types of maximal theories: Supersymmetric matter interactions (no gravity) N = 4 multiplet: 1 [1] 4 [ 1 2] 6 [0] Supersymmetric gravity = supergravity N = 8 multiplet: 1 [2] 8 [ 3 2] 28 [1] 56 [ 1 2] 70 [0]

13 N = 8 Supergravity (I) Unique theory (modulo gauging ) based on multiplet 1 [2] 8 [ 3 2] 28 [1] 56 [ 1 2] 70 [0] This theory incorporates all the known and expected symmetries. In addition it has an unexpected hidden exceptional E 7 duality symmetry. [Cremmer,Julia (1979)] It is thus the most symmetric known field theoretic extension of Einstein s General Relativity! General covariance and local Lorentz symmetry, N = 8 local supersymmetry and SU(8) R-symmetry (local or rigid).

14 N = 8 Supergravity (II)

15 N = 8 Supergravity (III) In the late 1970s this theory was thought to be a promising candidate for a unified theory of quantum gravity and matter interactions. However, Existence of supersymmetric counterterms suggested the appearance of non-renormalizable infinities from three loops onwards; Its properties [absence of chiral fermions, huge negative cosmological constant for gauged theory] seem to be in obvious conflict with observations. And: in both regards superstring theory seemed to do much better! BUT: remarkable recent progress on UV finiteness...

16 Finiteness: to be or not to be? Einstein gravity is perturbatively non-renormalizable Γ (2) div = dvc ε2880(16π 2 ) 2 µνρσ C ρσλτ µν C λτ where C µνρσ = Weyl tensor. [Goroff, Sagnotti(1986); van de Ven(1992)] NB: calculation of the coefficient requires consideration of O( ) Feynman diagrams! Although this 2-loop counterterm does not allow for a supersymmetric extension, supersymmetry is not enough to rule out UV infinities from three loops onwards! However, for N = 8 supergravity computation of 3-loop counterterm coefficient would require consideration of O(10 xxx ) Feynman diagrams hopeless???...

17 Finiteness: to be or not to be? Einstein gravity is perturbatively non-renormalizable Γ (2) div = dvc ε2880(16π 2 ) 2 µνρσ C ρσλτ µν C λτ where C µνρσ = Weyl tensor. [Goroff, Sagnotti(1986); van de Ven(1992)] NB: calculation of the coefficient requires consideration of O( ) Feynman diagrams! Although this 2-loop counterterm does not allow for a supersymmetric extension, supersymmetry is not enough to rule out UV infinities from three loops onwards! However, for N = 8 supergravity computation of 3-loop counterterm coefficient would require consideration of O(10 xxx ) Feynman diagrams hopeless???... NO!

18 3&4-loop finiteness of N = 8 supergravity [Bern,Carrasco,Dixon,Johansson,Roiban, PRL103,081301(2009)] Exploit Gravity = (Yang Mills) 2. Use unitarity based arguments to reduce all amplitudes to integrals over products of tree amplitudes. All particles are on-shell only 3-point vertices. Instead of O(10 xxx ) Feynman diagrams need only calculate O(50) Mondrian-like diagrams!

19 Beyond L = 4 loops There is now mounting evidence from different sources that N = 8 supergravity is UV finite at L < 7 loops: spectacular computational advances (see above...) exploiting nonlinear symmetries (SUSY, E 7(7),...)... and could thus be UV finite to all orders! BUT L = 7 requires 10 xxxx Feynman diagrams! AND: Even if N = 8 supergravity is finite, (non-linear) E 7 and SUSY may not be enough to prove it. Even if it is finite we still do not understand what happens to space and time at the Planck scale.

20 What about real physics? Even if finite it is usually taken for granted that N = 8 supergravity does not match with particle physics This seems hard to argue with but after complete breaking of N = 8 supersymmetry we are left with 48 = 3 16 basic spin- 1 2 fermions equal to the number of quarks and leptons in three families! This could be either a deep truth or a mirage but again: what you see in your Lagrangian may be very different from what you see in the detector! Famous example: QCD would have been considered obviously wrong if it had been proposed in 1950.

21 Exceptional Mathematics and Supergravity Continuing on the road towards more symmetry! N = 8 supergravity has more symmetry than meets the eye (SUSY, E 7 duality), but still not enough! Also true for D 4: hidden E n for maximal supergravity in D space-time dimensions with n = 11 D! Below D = 3 symmetries become infinite-dimensional: E 9 E (1) 8 for maximal supergravity in D = suggests E 10 for D = 1: no space, only time?!? But why should we care about physics in one dimension if we live in D = 4 space-time dimensions???

22 Physics near the Big Bang For T 0 causal decoupling of spatial points Physics becomes effectively one-dimensional in vicinity of cosmological singularity! [Belinski,Khalatnikov,Lifshitz (1972)] IDEA: Full symmetry only visible at the singularity!

23 What is E 10? (No one knows, really...) E 10 is the group associated with the Kac-Moody Lie algebra g e 10 defined via the Dynkin diagram [e.g. Kac] Defined by generators {e i,f i,h i } and relations via Cartan matrix A ij ( Chevalley-Serre presentation ) [h i,h j ] = 0, [e i,f j ] = δ ij h i, [h i,e j ] = A ij e j, [h i,f j ] = A ij f j, (ade i ) 1 A ij e j = 0 (adf i ) 1 A ij f j = 0. e 10 is the free Lie algebra generated by {e i,f i,h i } modulo these relations infinite dimensional as A ij is indefinite Lie algebra of exponential growth! 2 0

24 Infinite Complexity from simple recursion A Mandelbrot set generated from z n+1 = f c (z n ).

25 Vistas into E [from: Teake Nutma (formerly University of Groningen, now at AEI)]

26 What is so special about E 10? E 10 occupies a uniquely distinguished place among all infinite-dimensional Lie groups (much like E 8 among the finite-dimensional Lie groups). But it is monstrously complicated... [see hep-th/ ] Near the initial singularity (for 0 < T < T Planck ) the dynamics of maximal supergravity asymptotes to a cosmological billiards system based on E 10. E 10 may provide an algebraic mechanism for the de-emergence of space and time near the big bang. E 10 knows all about maximal supersymmetry. Only for the specialists: this also works for the fermions!

27 E 10 Versatility sl(10) e D = 11 SUGRA so(9,9) e miia D = 10 SUGRA 2 sl(9) sl(2) e IIB D = 10 SUGRA 2 sl(3) e 7 e N = 8, D = 4 SUGRA

28 Main Conjecture For0 < T < T P space-time de-emerges, and space-time based (quantum) field theory is replaced by purely algebraic description in terms of E 10. [Cf. DN, ]

29 Outlook Symmetry byno means exhausted asaguiding principle of physics but many open questions remain. Both N = 8 supergravity and E 10 are uniquely distinguished by their symmetry properties. E 10 knows all about maximal supersymmetry and unifies many known (S, T, U,...) string dualities. [so supersymmetry may not be as fundamental as we thought...] Exponentially increasing complexity of E 10 algebra an element of non-computability for T 0? what is the role of INFINITY in Physics?

30 Outlook Symmetry byno means exhausted asaguiding principle of physics but many open questions remain. Both N = 8 supergravity and E 10 are uniquely distinguished by their symmetry properties. E 10 knows all about maximal supersymmetry and unifies many known (S, T, U,...) string dualities. [so supersymmetry may not be as fundamental as we thought...] Exponentially increasing complexity of E 10 algebra an element of non-computability for T 0? what is the role of INFINITY in Physics? THANK YOU