4/16/2023 0 Comments Standard model particles![]() The spectral formalism of noncommutative geometry, explained above, makes it possible to consider spaces which are more general than ordinary manifolds. Thus, it is natural to wonder whether the transition from the Lagrangian of electrodynamics to the Standard Model can be understood as a further refinement of the geometry of spacetime, rather than the introduction of a zoo of new particles. From the particle physics viewpoint, the Lagrangian of electromagnetism is however just a very small part of the full Standard Model Lagrangian ( cf. This flat geometry is then extended to the curved Lorentzian manifolds of general relativity. In fact, the Maxwell equations are intrinsically relativistic. The “raison d’ etre” of the finite space F Poincaré and Einstein showed how to infer the correct flat spacetime geometry (known as Minkowski space geometry) underlying special relativity from experimental evidence associ- ated to Maxwell’s theory of electromagnetism. As we shall see below such an explanation is now available ( cf. It is crucial, of course, to understand the “raison d’ etre” of the space F and to explain why crossing the ordinary continuum with such a space is necessary from first principles. As we shall see, after a judicious choice of F, one obtains the full action functional (1) as describing pure gravity on the product space M × F. The product space then appears as a 4- dimensional continuum with “two sides”. Just to get a mental picture one may, in first approximation, think of F as a space consisting of two points. This fine texture appears as the product of the ordinary 4-dimensional continuum by a very specific finite discrete space F. Marcolli, ) and shows that the additional term S SM exhibits the fine texture of the geometry of spacetime. Our goal in this expository text is to explain that a conceptual understanding of the full action functional is now available (joint work with A. Figure 3) is quite complicated (it takes about four hours to typeset the formula) and is begging for a better understanding- in the words of an old friend : it is a Shakespearean king disguised as a beggar 2. While the Einstein-Hilbert action S EH has a clear geometric meaning, the additional term S ( cf. The values of these parameters is a great mystery. ![]() It also has some twenty or more parameters put in by hand according to the experimental data, such as the masses of the various particles. ![]() The action S SM encodes what is known so far about this array of particles and in principle describes all the matter and particles of force in the universe, with the exception of gravity, to our best current understanding. These are typically discovered by smashing together highly accelerated beams of other particles in huge particle accelerators such as at CERN ( Figure 2) and what one finds, even at the kinds of energies observable today (about 100 GeV) is a veritable zoo of particles – electrons, quarks, neutrinos of various flavors to name a few – and their antiparticles as well as the force carriers ( cf. the second term S SM is the standard model action which encapsulates our knowledge of all the different kinds of elementary particles to be found in nature.
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