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Copyright © 2003 Institute for Scientific Information From the geometry of pure spinors with their division algebras to fermion physics SO: Foundation of physics Abstract: The Cartan equations defining simple spinors (renamed "pure" by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to invariant-mass-spheres imbedded in each other, since the signatures result steadily Lorentzian; starting from dimension four (Minkowski) 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: from Weyl equations for neutrinos (and Maxwell's) to Majorana ones, to those for the electroweak model and for the nucleons interacting with the pseudoscalar pion, up to those for the 3 baryon-lepton families, steadily progressing from the description of lower energy phenomena to that of higher ones. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with Clifford algebras and then with this spinor-geometrical approach, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation: at the third step complex numbers generate U(1), possible origin of the electric charge and of the existence of charged-neutral fermion pairs, explaining also easily the opposite charges of proton-electron. Another U(1) appears to generate the strong charge at the fourth step. Quaternions generate the signature of space-time at the first step, the SU(2) internal symmetry of isospin and, in the gauge term, the SU(2)L one, of the electroweak model at the third step; they are also at the origin of 3 families; in number equal to that of quaternion imaginary units. At the fifth and last step octonions generate the SU(3) internal symmetry of flavour, with SU(2) isospin subgroup and, in the gauge term, the one of color, correlated with SU(2)L of the electroweak model. These 3 division algebras seem then to be at the origin of charges, 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 by the four Poincare translations, and dimensional reduction from U(1, 9) to U(1, 3) is equivalent to decoupling of the equations of motion. This spinor-geometrical approach is compatible with that teased on strings, since these may he expressed bilinearly (as integrals) in terms of Majorana-Weyl simple or pure spinors which are admitted by CL(1,9)= R(32). BP 1347 |
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Copyright © 2003 Institute for Scientific Information On conformally compacted phase space SO: Foundations of physics letters Abstract: Conformally compactified phase space is conceived as an automorphism space for the global action of the extended conformal group. Space time and momentum space appear then as conformally dual, that is conjugate with respect to conformal reflections. If now the former, as generally agreed, is appropriate for the description of classical mechanics in euclidean geometrical form, then the latter results appropriate for the description of quantum mechanics in spinor geometrical form. In such description, fermion multiplets will naturally appear as consequence of higher symmetries and furthermore, the euclidean geometry, bilinearly resulting from that of spinors, will a priori guarantee the absence of ultraviolet divergences when dealing with quantum field theories. Some further possible consequences of conformal reflections of interest for physics, are briefly outlined. BP 455 |
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On dual lattices in compactified phase space PT Journal Abstract: It is conjectured that space-time and momentum space may be both conformally compactified and correlated by conformal inversion, rendering a priori impossible the empirical realization of the concept of both infinity and infinitesimal. It appears that in such a world momentum space is appropriate for the description of quantum mechanics in spinorial form. An exactly soluble, two-dimensional model is presented and discussed. BP 905 |
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PT Journal Abstract: The hypothesis of simultaneous conformal compactification of both space-time and momentum space, possibly identified as two homogeneous spaces of the conformal group, leads to the need to define on them two dual finite lattices correlated by conformal inversions. It is shown that, with the help of orthonormal sets of (Hahn) polynomials on S-n identifying with spherical harmonics in the limit of dense lattices, they may be effectively constructed. In fact, they build up examples of orthonormal basis allowing the formulation of both discrete Fourier transforms on the finite, dual lattices and of scale-invariant propagators, which, in the Euclidean case, are shown to be free from both infrared and ultraviolet divergences. In the one-dimensional case, an exactly soluble toy model is presented where the truncation of infinite sums is correlated with the origin of dual, finite lattices. Such lattices carry an action of quantum deformation of the conformal algebra SU(1, 1) with q being a root of unity. BP 1035 |
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The spontaneous violation of the cosmological principle and the possible wave structures of the universe
PT Journal AB Recent redshift surveys reveal large structures of galaxies with sizes in excess of 100 h(-1) Mpc. These structures appear to be two-dimensional sheets (''walls'') that are perhaps periodically spaced. Here we propose that these ''walls'' are the manifestation of a spontaneous breaking of the symmetry implied by the cosmological principle. We present a model of a Robertson-Walker universe where the geometry of the large-scale matter-distribution is determined by the most symmetric eigenmode Y-n,Y-0,Y-0 of S-3. This model reproduces the geometry of the observed large-scale structures for an appropriate choice of the center of vibration and wavelength of the eigenmode Y-n,Y-0,Y-0. We also formulate predictions on the distribution of galaxies that should be observed as soon as new deep and wide-angle redshift surveys will become available. The observation of the predicted wave structures would confirm our model, which in turn could have far-reaching consequences for cosmology and also for physics. In fact, as shown by Fock in 1935, the most symmetric eigenfunction of the hydrogen atom represented in momentum space is the same Y-n,Y-0,Y-0. BP 10 |
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On the geometry of large-scale distribution of galaxies (spontaneous violation of the cosmological principle) PT Journal SO EUROPHYSICS LETTERS Abstract: Recent observations on the distribution of galaxides 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. BP 373 |
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Conformal space-times - the arenas of physics and cosmology
PT Journal SO: FOUNDATIONS OF PHYSICS Abstract: The mathematical and physical aspects of tire 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 nontrivial 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 Ii 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. II 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 theta and phi. These predictions should be tested with the new astronomical data. BP 1461 |
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Possible evidences from wave cosmology PT Journal Abstract: An explanation of the recently observed periodic structure of the large-scale distribution of galaxies is proposed as a manifestation of the validity of classical wave field theory applied to cosmology. The periodic wave-structure of energy density distribution provided by the eigensolutions of the field equations, in an inflationary model of the universe, may provide, in fact, the same periodic structure as the observed one. BP 295 |
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Conformal compactifications from spinor geometry
PT Journal Abstract: Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartan's definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S3 x 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 BP 949 |
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Global properties of conformally flat momentum-space and their implications PT Journal Abstract: We consider the global structure of momentum space PI3,1 in a field theory which is covariant with respect to the action of global conformal group G. We show that PI3,1 is a homogeneous space for G which coincides with (S3 x S1)/Z2 compact space. The radius of momentum space determines the natural invariant ultraviolet cutoff which may take the form of a Pauli-Vilars form factor in perturbation theory. We demonstrate in the case of the massless lambdaphi4 theory how the conventional ultraviolet divergences which appear in the flat momentum space are regularized in PI3,1 global momentum space. BP 599 |
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Eigenvibrations of the expanding universe PT Journal SO: FOUNDATIONS OF PHYSICS Abstract: A theoretical interpretation of the observed periodicity of large-scale (approximately 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. BP 225 |
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ICTP - An Italian cricible for the developing-world
PT Journal BP 49 |
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Relativistic spinor quasi-particles behind hubbard antiferromagnetism for space dimension-3
PT Journal SO: PHYSICA A Abstract: The magnetization Lagrangian density of a lightly doped, large U Hubbard - or more properly "t-J" - antiferromagnet is extended to include charge fluctuations. As a result of this extension, it is argued that the usual restriction m2 = 1 for the local magnetization magnitude m should be replaced by the more general restriction m2 = rho-2 where rho is the local charge magnitude. The theory may be naturally expressed in spinorial form where m and rho build up a 4-vector represented by the familiar current density of weak interactions: m-mu = psi-approximately-gamma-mu (1 + gamma-5)psi. The corresponding space-time is x-mu = {x, v0t} with v0 different from c, the velocity of light. In this frame the restriction m2 = rho-2 appears as an identity characterizing m-mu as a null-vector defined by the simple Weyl spinor 1/2 (1 + gamma-5)psi. The theory then assumes the form of a "relativistic", massless, parity breaking vector field theory. Speculations are advanced on the quasiparticle nature of these "relativistic" fermionic excitations, and on consequences of their spinorial structure.
BP 348 |
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Conformally compactified homogeneous spaces (Possible observable consequences) by P. Budinich Abstract: Some arguments based on the possible spontaneous violation of the Cosmological Principle (represented by the observed large-scale structures of galaxies), the Cartan-geometry of simple spinors and on the Fock-formulation of hydrogen-atom wave-equation in momentum-space, are presented in favour of the hypothesis that space-time and momentum-space should be both conformally compactified and represented by the two four-dimensional homogeneous spaces of the conformal group, both isomorphic to (S^3\times S^1)/Z_2 and correlated by conformal inversion. Within this framework, the possible common origin for the S0(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, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional space-time is presented where the conjectured conformally compactified space-time and momentum-space are both isomorphic to (S^1\times S^1)/Z_2, while the physical spaces are two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums may be well defined on them. Furthermore, a q-deformed SU_q(1,1) may be represented on them if q is a root of unity. Full article is available from ICTP |
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PT Journal Conformally compactified homogeneous spaces - possible observable consequences SO: FOUNDATIONS OF PHYSICS AB 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 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 conformal group, both isomorphic (S-3 X S-1)Z(2) and 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 test described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional space-time, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S-1 X S-1)/Z(2), 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. BP 969 |
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On Possible Wave Structures of the
Universe By P. Budinich, P. Nurowski, R. Raczka and M. Ramella Abstract: Recent redshift surveys reveal large structures of galaxies with sizes in excess of 100 $h^{-1}$ Mpc. These structures appear to be twodimensional sheets ("walls", perhaps periodically spaced. Here we propose that these "walls"are the manifestation of a spontaneous breaking of the symmetry implied by the Cosmological Principle. We present a model of a Robertson-Walker universe where the geometry of the large scale matter-distribution is determined by the most symmetric eigenmode $Y_{n,0,0}$ of ${\bf S}^3$. This model reproduces the geometry of the observed large scale structures for an appropriate choice of the center of vibration and wavelength of the eigenmode $Y_{n,0,0}$. We also formulate predictions on the distribution of galaxies that should be observed as soon as new deep and wide-angle redshift surveys will become available. The observation of the predicted wave structures would confirm our model, which in turn could have far reaching consequences for cosmology and also for physics. In fact, as shown by Fock in 1935, the most symmetric eigenfunction of the Hydrogen atom represented in momentum space is the same $Y_{n,0,0}$. Full article is available from ICTP |
High Energy Physics - Theory, abstract
From: Oleksandr Pavlyk <pavlyk@sissa.i> |
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Paolo BUDINICH
Internal symmetry from division algebra in pure spinor geometry Abstract: The E'. Cartan's equations defining "simple" spinors (renamed "pure" by C. Chevalley) are interpreted as (quantum) equations of motion for fermion multiplets in momentum spaces. The Cartan's conjecture on the non elementary nature of euclidean geometry is adopted; it conceives euclidean vectors as sums (or integrals) of null vectors bilinearly constructed in terms of pure or simple spinors. Consequentely those momentum spaces, constructed with pure spinors, result lorentzian and compact, isomorphic to invariant mass spheres imbedded in each other. The equations found are most of those traditionally adopted ad hoc by theoretical physics in order to represent the observed phaenomenology of elementary particles. In particular it is shown how, the known internal symmetry groups, might derive from the 3 complex division algebras correlated with the associated Clifford algebras. Precisely complex numbers generate U(1), at the origin of charges of fermions, which steadly appear in charged-neutral pairs of fermions, or of fermion multiplets; quaternions generate SU(2) isospin and SU(2)L of the electroweak model; octonions generate SU(3) both of flavour and of color. They also explain some of the elementary particle properties such as the 3 lepton-hadron families and the 3 colors, both in number equal to the 3 immaginary units of quaternions (or Pauli matrices). The possible role of pure spinors, such as those correlated the constraint relations, are presented and discussed. The adopted Cartan's conjecture allows a striking parallelism between geometry and physics, in so far, while notoriously classical mechanics of macroscopic bodies is well represented with euclidean geometry in space-time (example: celestial mechanics), neither macroscopic bodies nor, according to Cartan, euclidean geometry are elementary and then the mechanics of the "elementary constituents" of matter: the fermions, has to be represented with pure spinors: the "elementary constituents" of euclidean geometry, and what is obtained is wave - or quantum - mechanics (in first quantization): the "elementary constituent" of classical mechanics, as it should, and, in this frame, the euclidean concept of point-event has to be abandoned; it could be sobstituted by a continuous sum, or integral, of null vectors (bilinear in pure spinors) which happens to be, as it is shown, precisely a string. In this way this approach is not at all orthogonal to the now prevailing one of strings and superstrings, as it could have appeared at first sight. Some further consequences are drawn from this parallelism. Full article (PDF) |
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Paolo Budinich (of Trieste, Italy) and Andrzej Trautman (of the University of Warszaw), The Spinorial Chessboard, Springer-Verlag, Berlin, 1988. This text is from the Trieste Notes in Physics series. While it is still of print, it can be ordered from the publisher at: http://www.springer-ny.com/staticpages/0387190783.htm Publisher's note: Spinor theory is an important tool in mathematical physics in particular in the context of conformal field theory and string theory. These lecture notes present a new way to introduce spinors by exploiting their intimate relationship to Clifford algebras. The presentation is detailed and mathematically rigorous. Not only students but also researchers will welcome this book for the clarity of its style and for the straightforward way it applies mathematical concepts to physical theory. It includes a great treatment of Clifford algebras and their spinor representations. Excerpt from the discussion of the "Clifford algebra clock": R
R+R C
R H
C H+H
H
This clock
easily lets you remember the real Clifford algebras in every dimension
and signature of spacetime. Bott periodicity explains why it loops
around after
8 hours. The spinorial chessboard presents the same information
in the form of an 8 x 8 grid. I won't draw it here, but it's a
picture of the Clifford algebras with p roots of -1 and q roots
of 1 for p,q = 0,1,2,3,4,5,6,7. The black squares correspond to cases
that admit chiral spinors; the red ones correspond to cases
that don't. Black is when p+q is even; red is when it's odd.
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AMCA Document # cahs-52 3rd International ISAAC Congress ISAAC Board Flavour and colors SU(3) from octonions in CI(1,9) Presented by: Paolo Budinich The É. Cartan's equations defining ``simple'' spinors, renamed ``pure'' by C. Chevalley, are interpreted as equations of motion in momentum spaces. The pure spinors associated with Clifford algebra CI(1, 9) may be equivalentely represented as a quadruplet of Majorana-Weyl fermions or as a doublet of octonions o since CI(1, 9) = R(3, 2) and Spin (1, 9) = SL(2, o). The corresponding equation of motion presents, once the octonion unit e7 is fixed, the possibility of two SU(3) internal symmetries; one interpretable as flavour (having SU(2)-isospin as subgroup) and the other, in the dynamical sector, as color, correlated with the electroweak model. In this approach dimensional reduction from CI(1, 9) down to CI(1, 3) identifies with decoupling of the corresponding equations of motion and there seems to be non need of higher dimensional ( > 4) space-times. On the way of the dimensional reduction one finds most of the equations (including electro weak's and Maxwell's ones) and internal symmetries, traditionally postulated ad hoc, for barions and 3 families of leptons. The origin of both charges (electric, and strong from U(1)) and internal symmetries (SU(2), SU(3)) may be envisadged in the division algebras naturally contained in the corresponding Clifford algebras (U(1) from complex numbers, SU(2) from quaternians, SU(3) from octonions). Both the geometrical construction and the physical interpretation require the spinors to be simple or pure and then subject to constraint equations whose possible role is breefly discussed. This approach is compatible with the one based on strings which may be bilinearly expressed (as integrals) in terms of Majorana-Weyl spinors admitted by CI(1, 9). Date received: May 31, 2001 Copyright © 2001 by Paolo Budinich.The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts. |
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Geometrical Aspects of Quantum Mechanics in Compactified Momentum Space by Paolo Budinich Abstract: It is shown how, from the Cartan's equations defining simple (or pure) spinors, several of the basic equations of quantum mechanics in first quantization may be naturally derived in momentum space $P=I\!\!R^{3,1}$. In particular the equation for the nucleon doublet interacting with the pseudo-scalar pion triplet is obtained, where the $SU(2)$ isospin symmetry is generated by a reflection group in a six dimensional space. This possible reflection origin of isospin may be extended to other instances of internal symmetry. Simple (or pure) spinor geometry could then be at the basis not only of Euclidean geometry, in accordance with Cartan's conjecture, but also of quantum mechanics in momentum space, while the dual configuration space remains the one appropriate for the description of classical mechanics. Since simple (or pure) spinors are equivalent to totally null planes laying in compact manifolds (grassmanians), it is supposed that the above momentum space $P$ should be densely imbedded in conformally compactified momentum space $P_c=(S_3\times S_1)/Z_2$ and then the $SO(4)$ symmetry of the non relativistic, stationary, H-atom in momentum space, discovered by V. Fock, could be identified with the maximal compact group of the conformal group, thus allowing its purely geometrical interpretation. The corresponding classical system could be represented by planetary motions but also, possibly, by the expanding universe, as some recent observations of large structures of distant galaxies seem to suggest, in accordance with an old hypothesis due to Eddington and Schroedinger. Should this, after further astronomical observations, turn out to be true, it would support the hypothesis of simultaneous conformal compactification of both space-time and momentum spaces with the consequent elimination of both ultraviolet and infrared divergences from any field theory in such compact manifolds (provided an appropriate algorithm may be found to deal with it). A soluble toy model of conformal compactification of both space-time and momentum space is presented for the particular case of compactified two dimensional space-time and momentum space. |
High Energy Physics - Theory, abstract
From: Paolo Budinich <milazzi@ictp.trieste.it> |
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Fock space description of simple spinors Paolo Budinich and Andrzej Trautman Journal of Mathematical Physics Vol 30(9) pp. 2125-2131. September 1989 Abstract: Cartan's simple—often called pure—spinors corresponding to even-dimensional complex vector spaces are defined in terms of the associated maximal totally null planes. Their geometrical properties are derived and described using notions familiar to physicists: Dirac and Weyl spinors, gamma matrices, tensors formed bilinearly from pairs of spinors, and creation and annihilation operators of Fermi states. A new theorem characterizes a simple spinor Full article is available for purchase at: http://content.aip.org/JMAPAQ/v30/i9/2125_1.html |
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Spinors as fundamental objects Paolo Budinich and Krystyna Bugajska Abstract: We define, on the algebraic Dirac spinor space
Full article is available for purchase at: http://content.aip.org/JMAPAQ/v26/i4/588_1.html |
by the properties of the vector
t
B
µ
is an arbitrary spinor and
B is the matrix connecting the gamma matrices with their transposes. The Cartan constraint equations on the components of simple spinors are given a new, geometrically transparent derivation based on the action on simple spinors of a maximal Abelian subgroup of the group Spin.
± between
Ds,t to belong to some minimal left ideal of Ds,t. Next we use the decomposition of the Dirac algebra
Ds,t onto the Dirac spinor spaces to demonstrate two different ways of an action of any element
s