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<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMBX10">GEOMETRY AND THE QUANTUM<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMCSC10">Alain Connes<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10">Chair of Analysis and Geometry, College de France<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10">Distinguished Professor, The Ohio State University<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10"><o:p> </o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMBX10">Abstract.
<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10">In collaboration with A. Chamseddine and S. Mukhanov, motivated by the construction of spectral manifolds in noncommutative geometry, we intro-<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10">duce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar elds. This commutation relation appears in two versions, one sided
and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras
</span><span style="font-family:CMMI10">M</span><span style="font-size:8.0pt;font-family:CMR8">2</span><span style="font-family:CMR10">(</span><span style="font-family:MSBM10">H</span><span style="font-family:CMR10">) and
</span><span style="font-family:CMMI10">M</span><span style="font-size:8.0pt;font-family:CMR8">4</span><span style="font-family:CMR10">(</span><span style="font-family:MSBM10">C</span><span style="font-family:CMR10">) which are the algebraic constituents of
the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show<o:p></o:p></span></p>
<p class="MsoNormal" style="text-autospace:none"><span style="font-family:CMR10">that any connected Riemannian Spin 4-manifold with quantized volume
</span><span style="font-family:CMMI10">> </span><span style="font-family:CMR10">4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these representations give a seductive model of the
“particle picture" for a theory of quantum gravity in which both the Einstein geometric standpoint and the Standard Model emerge from Quantum Mechanics.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-family:CMR10"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-family:CMR10">This lecture will be presented Tuesday April 28, and Wednesday April 29 in Cockins Hall 240 at 10:00 am each morning</span><o:p></o:p></p>
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