Can one hear the shape of spacetime (at the attoscale) ?

May be ... thanks to a quantum educated extension of spectral geometry  
The discovery of the Higgs particle at CERN in Geneva in 2012 formed the crown on the so-called Standard Model of particle physics. Despite its enormous phenomenological success, much of the underlying mathematics remains still to be understood. Walter van Suijlekom, Assistant Professor in mathematical physics at IMAPP, here lifts the curtain of what noncommutative geometry can already say about the Standard Model, offering an intriguing perspective of what space looks like at scales analysed by particle accelerators... 
... we have sketched how the full Standard Model of particle physics can be derived from a noncommutative space, using not more than basic linear algebra. Even though this is quite an achievement, there is still the formidable problem to give a mathematically rigorous description of the quantization of the above system. At present, the derivation of the spectral action functional for the Standard Model, including e.g. the Higgs potential, is a mathematical derivation. However, the translation of it to realistic quantum particles and fields follows a more physics-style approach. It is clear that in order to have a proper understanding of the Standard Model of particle physics this aspect should be improved. It is the goal of my Vidi-research project to take a step in this direction. 
We will analyse the quantization of gauge fields — such as the electromagnetic field — on a discrete space instead of in the continuum. That is, we replace M by a lattice, construct the quantum theory there, and then analyse the limit of small lattice spacing... The main challenge is to do this in a mathematically rigorous way, for which we intend to exploit the powerful functional analytical techniques coming from noncommutative geometry. One of the intriguing links with the above description of noncommutative finite spaces is that the replacement of M by a lattice is very similar to analysing the structure of the discrete spaces F using matrix algebra. In [17] we present a first exploration of this exciting interplay between noncommutative geometry, lattice gauge theory and quantization. 
Walter van Suijlekom, 4 december 2014


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