Paolo Budinich

The Role of Mathematics in Physical Sciences and Dirac’s Methodological Revolution

In our paper, avoiding any strong metaphysical commitment on the world, we face the topic of the interplay between mathematics and physics by starting from a semiotic approach. It will be shown that it allows us to insert in a unitary and coherent framework answers to questions such as: Why mathematics is physics? What is the role of mathematics in physics? Why is mathematics effective in physical sciences? In the second part of the paper, and by utilizing what discussed in the first one, we analyse what we call Dirac’s methodological revolution, according to which to do good and new physics we must first work on good and promising mathematics. Finally, we exemplify Dirac’s methodological revolution by recalling the role of the mathematical theory of simple spinors in constructing new perspectives for theoretical physics.

Conformal compacifications from spinor geometry

Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartans definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S3 × S1)/Z2, one may obtain in the conformally flat stereographic projection field theories both infrared and ultraviolet regularized. On the compact manifold themselves instead, Fourier integrals of wave-field oscillations would have to be replaced by Fourier series summed over indices of spherical eigenfunctions: n, l, m, m′. Tentatively identifying those wave structures with spacetime itself (in the frame of Big-Bang) and/or with matter and radiation distribution, some large-scale (hydrogenic) and small-scale (lattice) space structures are conjectured.

Conformal space-times—The arenas of physics and cosmology

The mathematical and physical aspects of the conformal symmetry of space-time and of physical laws are analyzed. In particular, the group classification of conformally flat space-times, the conformal compactifications of space-time, and the problem of imbedding of the flat space-time in global four-dimensional curved spaces with non-trivial topological and geometrical structure are discussed in detail. The wave equations on the compactified space-times are analyzed also, and the set of their elementary solutions constructed. Finally, the implications of global compactified space-times for cosmology are discussed. It is argued that the recent discovery of periodic structure of matter distribution on large distances strongly suggests that the global cosmological space-time should be close. Next we analyze the inflation scalar field in the inflationary model of universe evolution considered on the spatially compact Robertson-Walker space-time. It is shown that the energy distribution in this model is periodic and the periods and density decrease with increasing distance, in striking agreement with experimental data. Our model of the universe also provides a definite predictions for the energy distribution, polar and azimuthal, considered as a function of angles θ and φ. These predictions should be tested with the new astronomical data.

Conformally compactified homogeneous spaces. Possible observable consequences

Some arguments, based on the possible spontaneous violation of the cosmological principle (represented by the observed large-scale structures of galaxies), on the Cartan geometry of simple spinors, and on the Fock formulation of hydrogen atom wave equation in momentum space, are presented in favor of the hypothesis that space-time and momentum space should be both conformally compactified and should both originate from the two four-dimensional homogeneous spaces of the conformai group, both isomorphic (S3×S1)/Z2and correlated by conformal inversion, but should not necessarily be identified with them. Within this framework, the possible common origin for the SO(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits, and of the H atom is discussed. One of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional spacetime, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S1×S1)/Z2,while the possible physical spaces could be two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums, may be well defined on them.

Eigenvibrations of the expanding universe

A theoretical interpretation of the observed periodicity of large-scale (∼128 Mpc) correlations of galaxies is proposed as due to eigenvibrations of the closed expanding universe. Eigensolutions of the equations of motion for a scalar field in an inflationary model allow one to compute the energy density, interpreted as matter density. Isotropic eigensolution give rise to a matter density distribution having a periodic structure centered at the north pole of the closed Robertson-Walker universe represented by S3/Z2. It is able to reproduce well the striking periodicity of the observational data, in the galactic north-south directions. The dipole and quadrupole eigensolutions and the location of the co-moving observer in a point of S3/Z2 different from the center of the vibrational structure would imply, in a theoretically well predictable way, a decrease of the observed periodicity in some other directions.

Clifford Algebras, Pure Spinors and the Physics of Fermions

The equations defining pure spinors are interpreted as equations of motion formulated on the lightcone of momentum space P = ℝ1, 9. Most of the equations for fermion multiplets, usually adopted by particle physics, are then naturally obtained and their properties, such as internal symmetries, charges, and families, appear to be due to the correlation of the associated Clifford algebras with the three complex division algebras: complex numbers, quaternions and octonions. Pure spinors could be relevant not only because the underlying momentum space is compact, but also because they may throw some light on several problematic aspects of particle physics.

On the Geometry of Large-Scale Distribution of Galaxies. (Spontaneous Violation of the Cosmological Principle.)

Recent observations on the distribution of galaxies in the Universe have revealed large-scale structures whose origin cannot be ascribed to gravitation. Therefore they might represent unexpected violations of the Cosmological Principle (stating: the spatial isotropy and uniformity of the Universe). Here we propose a simple model, based on a wave equation which foresees eigenvibrations of a Robertson-Walker universe. Just one of these allows the reproduction of the main features of most of the observed large-scale structures. It also allows predictions of wave structures in the distribution of galaxies at large distances which, if observed (or not), could represent a proof (or a disproof) of the model. In this case the observed structures would represent a natural violation of the Cosmological Principle: one of those is well known in physics with the name spontaneous symmetry breaking. Some possible further consequences of the model are shortly outlined.

ICTP – an Italian crucible for the developing world

The Second World War left Trieste in an awkward position – literally. Close to the Yugoslav border, which coincided with the Iron Curtain, the region was exposed to strong political and nationalist pressures, and associated difficulties.