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Yasuhiro Abe, Ph.D.

Cereja Technology Co., Ltd.
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Physics on fuzzy four-sphere

Why fuzzy S4   Have you ever imagined an extremely hot world where everything evaporates away? The world is so hot and so high in energy that even particles may evaporate to vanish. At the present, we consider that there exists a certain energy limit beyond which the energy can not be made any higher. Beyond such an energy level, no physical laws are applicable and even a notion of spacetime disappears. This energy limit is called the Planck energy, expressed uniquely by a set of physical constants, and is about 1.22×1019 GeV (1 GeV = 109 eV = 1.602×10-10 J). A main task of theoretical high energy physicists is to unveil the physics in this Planck scale world. Some researchers consider that in the Planck scale there remain nothing but strings which will create everything. It sounds fascinating. However, the true picture is yet to be confirmed since no experimental or observational data are available for such a high-energy region.

In terms of geometry, the very existence of the Planck energy leads to a limit length beyond which the length can not be made any shorter. The corresponding length is called the Planck length and is about 1.62×10-33cm. Coordinates need to be discretized by the unit of the Planck length. This is similar to what we require for the phase space in quantum mechanics. Thus, analogously, in the Planck scale we need an uncertainty principle for coordinates themselves. This means that the geometry is noncommutative in the Planck scale.

Fuzzy spaces are realizations of noncommutative geometry in terms of N-dimensional square matrices. Matrix realization of a space is sometimes called fuzzification of the space. Use of fuzzy spaces in physics then provides a novel approach to spacetime discretization in addition to the previously known approaches, e.g., trianglization and latticization. Apart from a conceptual issue of discretization, fuzzy spaces are also useful for practical purposes, i.e., they are suitable for numerical simulations. This is simply because calculations on fuzzy spaces reduce to matrix calculus and recent developments of computational technologies facilitate these calculations more than ever. It is therefore natural to seek for four-dimensional fuzzy spaces as a "noncommutative" platform for the study of the Planck scale physics. Fuzzification of four-sphere S4 is particularly interesting since S4 is the simplest four-dimensional compact space that allows spin structure. Notice (a) that fuzzy spaces are constructed for compact spaces as non-compact spaces can not be described by finite size matrices and (b) that the spin structure is necessary for construction of realistic physical models. Fuzzy S4 is therefore expected to be useful in the research of high energy physics.

Construction   Fuzzy S4 can be constructed by use of the fact that Penrose's twistor space (or the rank-3 complex projective space) is an S2-bundle over S4. For details of the construction, see arXiv:hep-th/0406135.

Simulation   An example of R scripts for construction of fuzzy S4 is here. (You can change the matrix dimension N by choosing an arbitrary integer for n in the first.) R is a free software environment for statistical computation, which is one of the GNU projects. There are useful sets of packages, including those for matrix calculations, available in R. For a concrete calculation, this example (on fluctuations from fuzzy two-sphere) may be useful.

Interests   Emergence of fuzzy S4 as a brane solution to a compactifcation model of M(atrix) theory (see arXiv:hep-th/0512174)
Black hole solutions on fuzzy S4 (under investigation)
Computation of multigraviton amplitudes on fuzzy S4 (plan)


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