24 research outputs found

### On conformal reflections in compactified phase space

Some results from arguments of research dealt with R. Raczka are exposed and
extended. In particular new arguments are brought in favor of the conjecture,
formulated with him, that both space-time and momentum may be conformally
compactified, building up a compact phase space of automorphism for the
conformal group, where conformal reflections determine a convolution between
space-time and momentum space which may have consequences of interest for both
classical and quantum physics.Comment: 13 pages, no figures, requires JHEP.cl

### A Spinorial Formulation of the Maximum Clique Problem of a Graph

We present a new formulation of the maximum clique problem of a graph in
complex space. We start observing that the adjacency matrix A of a graph can
always be written in the form A = B B where B is a complex, symmetric matrix
formed by vectors of zero length (null vectors) and the maximum clique problem
can be transformed in a geometrical problem for these vectors. This problem, in
turn, is translated in spinorial language and we show that each graph uniquely
identifies a set of pure spinors, that is vectors of the endomorphism space of
Clifford algebras, and the maximum clique problem is formalized in this setting
so that, this much studied problem, may take advantage from recent progresses
of pure spinor geometry

### From the Geometry of Pure Spinors with their Division Algebras to Fermion's Physics

The Cartan's equations definig simple spinors (renamed pure by C. Chevalley)
are interpreted as equations of motion in momentum spaces, in a constructive
approach in which at each step the dimesions of spinor space are doubled while
those momentum space increased by two. The construction is possible only in the
frame of geometry of simple or pure spinors, which imposes contraint equations
on spinors with more than four components, and the momentum spaces result
compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the
signatures appear to be unambiguously defined and result steadily lorentzian;
up to dimension ten with Clifford algebra Cl(1,9), where the construction
naturally ends. The equations of motion met in the construction are most of
those traditionally postulated ad hoc for multicomponent fermions. The 3
division algebras: complex numbers, quaternions and octonions appear to be
strictly correlated with this spinor geometry, from which they appear to
gradually emerge in the construction, where they play a basic role for the
physical interpretation. In fact they seem then to be at the origin of
electroweak and strong charges, of the 3 families and of the groups of the
standard model. In this approach there seems to be no need of higher
dimensional (>4) space-time, here generated merely by Poincare translations,
and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling
of the equations of motion.Comment: 42 pages Late

### On Fermions in Compact momentum Spaces Bilinearly Constructed with Pure Spinors

It is shown how the old Cartan's conjecture on the fundamental role of the
geometry of simple (or pure) spinors, as bilinearly underlying euclidean
geometry, may be extended also to quantum mechanics of fermions (in first
quantization), however in compact momentum spaces, bilinearly constructed with
spinors, with signatures unambiguously resulting from the construction, up to
sixteen component Majorana-Weyl spinors associated with the real Clifford
algebra \Cl(1,9), where, because of the known periodicity theorem, the
construction naturally ends. \Cl(1,9) may be formulated in terms of the
octonion division algebra, at the origin of SU(3) internal symmetry.
In this approach the extra dimensions beyond 4 appear as interaction terms in
the equations of motion of the fermion multiplet; more precisely the directions
from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions
$(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones
SU(3). There seems to be no need of extra dimension in configuration-space.
Only four dimensional space-time is needed - for the equations of motion and
for the local fields - and also naturally generated by four-momenta as
Poincar\'e translations.
This spinor approach could be compatible with string theories and even
explain their origin, since also strings may be bilinearly obtained from simple
(or pure) spinors through sums; that is integrals of null vectors.Comment: 55 pages Late

### From Pure Spinors to Quantum Physics and to Some Classical Field Equations Like Maxwell's and Gravitational

In a previous paper [1] we proposed a purely mathematical way to quantum
mechanics based on Cartan's simple spinors in their most elementary form of 2
component spinors. Here we proceed along that path proposing, this time, a
symmetric tensor, quadrilinear in simple spinors, as a candidate for the
symmetric tensor of general relativity. This is allowed now, after the
discovery of the electro-weak model and its introduction in the Standard Model
with SU(2)_L.
The procedure resembles closely that in which one builds bilinearly from
simple spinors an antisymmetric "electromagnetic tensor", from which easily
descend Maxwell's equations and the photon can be seen as a bilinear
combination of neutrinos. Here Lorentzian spaces result compact, building up
spheres, where hopefully some of the problems of the Standard Model could be
solved as pointed out in the conclusions.Comment: Slight changes mainly in abstract and conclusions; 11 pages, 10
reference

### The role of mathematics in physical sciences: interdisciplinary and philosophical aspects

Even though mathematics and physics have been related for centuries and this relation appears to be unproblematic, there are many questions still open: Is mathematics really necessary for physics, or could physics exist without mathematics? Should we think physically and then add the mathematics apt to formalise our physical intuition, or should we think mathematically and then interpret physically the obtained results? Do we get mathematical objects by abstraction from real objects, or vice versa? Why is mathematics effective into physics? These are all relevant questions, whose answers are necessary to fully understand the status of physics, particularly of contemporary physics. The aim of this book is to offer plausible answers to such questions through both historical analyses of relevant cases, and philosophical analyses of the relations between mathematics and physics