Unraveling the fine structure of spacetime

Walter D. van Suijlekom

waltervs@math.ru.nl ncgnlhttp://www.math.ru.nl/~waltervs

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 spacetime as consisting of three layers

With the Higgs particle we detect the 2nd layer

A new, sigma particle sees the 3rd layer

Begin 20th centuryEinstein's gravitational theory

Einstein (1879-1955)

•Gravity is a consequence of the curvature of spacetime

Geometrical description of spacetime around us at the large scale. But what does spacetime look like at the smallest scale?

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)

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:

A 3rd layer at even smaller scale ( ) separated by a new, sigma particle

Unification of atomic forces with gravity

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

Noncommuting physical processes

Noncommuting physical processes as matrices

Matrix product is noncommutative:

idem

– matrix to represent – decay:

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:

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

Hearing the shape of (some) drums

The spectrum of some (commutative) drums:

disk spheresquare

Hearing the shape of (some) drums

disk spheresquare

The spectrum of some (commutative) drums:

Higher frequencies:

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

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

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 at http://www.math.ru.nl/~waltervs or http://arxiv.org

Website: http://www.noncommutativegeometry.nl

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)