Unraveling the fine structure of spacetime

1. Unraveling the fine structure of spacetime Walter D. van Suijlekom waltervs@math.ru.nl ncgnl http://www.math.ru.nl/~waltervs

2. Abstract mathematics vs. experimental physics – Institute for Mathematics, Astrophysics and Particle Physics (IMAPP):direct contact with experiment – Collaboration with Connes (Paris) and Chamseddine (Beirut) – Fine structure of spacetimeas consisting of three layers • With the Higgs particle we detect the 2nd layer • A new, sigma particle sees the 3rd layer

3. Begin 20th centuryEinstein's gravitational theory • Gravity is a consequence of the curvatureof spacetime Einstein (1879-1955) • Geometrical description of spacetime around us at the large scale. • But what does spacetime look like at the smallest scale?

4. Zooming in at the smallest scale • How to measure an atom ( ) and smaller, if the ruler itself consists of atoms... • In practice, measuring at this scale is spectral, leading to a much more exotic geometry noncommutative geometry • Relatively young field of mathematics, founded by French mathematician Alain Connes (Fields Medal, 1982)

5. What is the fine structure of the universe? • Our model:spacetime consists of two layers ( ) and the Higgs particle(CERN!) can separate them • Both the experiment (CERN) and our model demands there to be more: A3rd layerat even smaller scale ( ) separated by a new, sigma particle

6. Unification of atomic forces with gravity

7. What is noncommutative about it? Einstein works in his description of spacetime with coordinates Such coordinates are given by numbers and commute, such as In noncommutative geometry coordinates do not commute anymore, allowing for a geometrical description of noncommuting physical processes

8. Noncommuting physical processes

9. Noncommuting physical processes as matrices – matrix to represent – decay: Matrix product is noncommutative: idem

10. The noncommutative geometry of elementary particles This noncommutativity of physical processes can be built into the geometry Coordinates are extended to become matrix-valued matrices (electromagn.) matrices ( ) matrices (quark colors) corresponding to three layers of spacetime Higgs and sigma fields jump between layers:

11. Hearing the shape of the (noncommutative) drum Noncommutative geometry takes a spectral standpoint, just as experiment Forces in nature are described by the spectrum of noncommutative spacetime

12. Hearing the shape of (some) drums The spectrum of some (commutative) drums: sphere square disk

13. Hearing the shape of (some) drums • The spectrum of some (commutative) drums: • Higher frequencies: sphere square disk

14. Spectrum of noncommutative spacetime Einstein Equations can be described purely from the spectrum of spacetime (eigenfrequencies of wave equation) The spectrum of noncommutative spacetime is shifted from the spectrum of ordinary spacetime and couples matter to gravity: fine structure of spacetime

15. Detecting the three layers of spacetime: from high to 'low'-energy

16. Collaboration Nijmegen-Paris-Beirut Ali Chamseddine, Alain Connes and WvS – Inner Fluctuations in Noncommutative Geometry without the first order condition. J. Geom. Phys. 73 (2013) 222-234. – Beyond the Spectral Standard Model: Emergence of Pati-Salam Unification. JHEP 11 (2013) 132. Preprint download athttp://www.math.ru.nl/~waltervsor http://arxiv.org Website: http://www.noncommutativegeometry.nl

17. Meet the others: noncommutative geometry in Nijmegen (IMAPP) • Mathematical Physics: quantization (Boeijink, Landsman), gauge theories and noncommutative geometry (Brain, Iseppi, Kaad, Neumann, VIDI), quantum groups (Koelink, Aldenhoven) … • High-energy physics: supersymmetry (Beenakker, van den Broek, Kleiss) • Quantum geometry (Ambjorn, Landsman, Loll, Saueressig)