Unraveling the fine structure of spacetime
Walter D. van Suijlekom
waltervs@math.ru.nl
http://www.math.ru.nl/~waltervs
ncgnl
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 century
Einstein's gravitational theory
• Gravity is a consequence of the curvature of
spacetime
Einstein (1879-1955)
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 to represent
– decay:
Matrix product is noncommutative:
idem
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
square
sphere
Hearing the shape of (some) drums


The spectrum of some (commutative) drums:
Higher frequencies:
disk
square
sphere
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)