MiTopos

Jackson A., Artin M., Mumford D., Tate J., Ladegaillerie Y., Lichtenbaum S., Lochak P., Mazur B., Messing W., Mumford D., Murre J., Poénaru V., Schneps L.

Schmeikal B.

Since Immanuel Kant’s Inaugural Dissertation of 1770 we assume that the concepts of space and time are not abstracted from sensations of external things. But outer experience is considered possible at all only through an inner representation of space and time within the cognitive system. In this work we describe a representation which is both inner and outer. We add to the Kantian imagination that “forms of nature, matter, space and time are intelligible, perceivable and comprehensible”, the idea that these four are indeed intelligent, perceiving, grasping and clear. They are active systems with their own intelligence. In this paper on the mind-matter interface we create the mathematical prerequisites for an appropriate system representation. We show that there is an oriented logic core within the space–time algebra. This logic core is a commutative subspace from which not only binary logic, but syntax with arbitrary real and complex truth classifiers can be derived. Space–time algebra too is obtained from this inner grammar by two rearrangements of four basic forms of connectives. When we conceive the existence of a few features like polarity between two appearances, identification and rearrangement of the latter as basic and primordial to human cognition and construction, the intelligence of space–time is prior to cognition, as it contains within its representation the basic self-reference necessary for the intelligible de-convolution of space–time. Thus the process of nature extends into the inner space.

Schmeikal B.

Here it is shown that the known forces of nature unfold in parallel with an exact decomposition of the geometric algebra Cl 3,1 of spacetime. Up to an important common scalar this decomposition is a partition into a positive definite commutative graded space of strong forces and two negative non-commutative spaces with a quaternion structure for weak and other fields. The 6 fundamental spaces of strong forces are acted on by a rank 2 Lie algebra ℒ = sl Cl (2, ℝ) × so Cl (3, ℝ) of dimension 8 which brings in an isotropy group of the neutrinos, the flavour and colour symmetries and an isotropy of the weaker forces. The standard model, in an improved form, is a feature of the Clifford algebra of spacetime, and relativity as Lorentz invariance reduced to dimension (2, 1) is compatible with quantum theory. The whole Lorentz group cannot be, however, a property of physical motion. Due to the total exploitation of the whole geometric space and its convincing logical structure, the author believes there is at present no better algebraic model for the totality of known symmetries of physical dynamics.

Ablamowicz R., Fauser B.

CLIFFORD performs various computations in Graßmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Graßmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

Kosmann-Schwarzbach Y.

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of ‘Poisson structures with background’.

Schmeikal B.

Recoding base elements in a spacetime algebra is an act of cognition. But at the same time this act refers to the process of nature. That is, the internal interactions with their standard symmetries reconstruct the orientation of spacetime. This can best be represented in the Clifford algebra Cl 3, 1 of the Minkowski spacetime. Recoding is carried out by the involutive automorphism of transposition. The set of transpositions of erzeugende Einheiten (primitive idempotents), as Hermann Weyl called them, generates a finite group: the reorientation group of the Clifford algebra. Invariance of physics laws with respect to recoding is not a mere matter of computing, but one of physics. One is able to derive multiplets of strong interacting matter from the recoding invariance of Cl 3, 1 alone. So the SU(3) flavor symmetry essentially turns out to be a spacetime group. The original quark multiplet independently found by Gell-Mann and Zweig is reconstructed from Clifford algebraic eigenvalue equations of isospin, hypercharge, charge, baryon number and flavors treated as geometric operators. Proofs are given by constructing six possible commutative color spinor spaces ℂh χ, or color tetrads, in the noncommutative geometry of the Clifford algebra Cl 3, 1 Calculations are carried out with CLIFFORD, Maple V package for Clifford algebra computations. Color spinor spaces are isomorphic with the quaternary ring 4ℝ = ℝ ⊕ ℝ ⊕ ℝ ⊕ ℝ. Thus, the differential (Dirac) operator takes a very handsome form and equations of motion can be handled easily. Surprisingly, elements of Cl 3, 1 representing generators of SU(3) bring forth (1) the well-known grade-preserving transformations of the Lorentz group together with (2) the heterodimensional Lorentz transformations, as Jose Vargas denoted them: Lorentz transformations of inhomogeneous differential forms. That is, trigonal tetrahedral rotations do not preserve the grade of a multivector but instead, they permute the base elements of the color tetrad having grades 0, 1, 2 and 3. In the present model the elements of each color space are exploited to reconstruct the flavor SU(3) such that each single commutative space contains three flavors and one color. Clearly, the six color spaces do not commute, and color rotations act in the noncommutative geometry. To give you a picture: Euclidean space with its reorientation group, i.e., the again embed the root spaces of the 6 flavor su(3). Color su(3) is exact because it does not involve relativistic effects. Flavor does and is therefore inexact. It seems that Cl 3, 1 comprises enough structure for both the color- and the flavor-SU(3). In this very first approach both symmetries are reconstructed exact, whereas in reality the flavor SU(3) is only approximate. Here, the only way to make a difference between a color- and a flavor-rotation may be to distinguish between commutative and noncommutative geometry. We conclude that an extended heterodimensional Lorentz invariance of the Minkowski spacetime and the SU(3) of strongly interacting quarks result from each other. This involves a nongrade preserving-degree of freedom of motion taking spatial lines to spacetime areas and areas to spacetime-volumes and back to lines and areas. Although the form of the Dirac equation is preserved, a thorough study of the involved nonlinearities of equations of motion is still outstanding.

Schmeikal B.

Purpose of a minimal theory is to achief most with least. Least may be for example the spacetime algebra. But the symmetric unitary groupSU(3) is not a part of any real Clifford algebra of 4-dimensional space, especially not of the algebraCl 1,3 of the Minkowski spacetime, nor of the algebraCl 3,1 in the opposite metric. Therefore we can ask how quantumchromodynamics enters into the theory. A first answer is that the groupSU(3) is an object of both the complexified algebras C ⊕Cl 1,3 and C⊕Cl 3,1. To show this we first define six color spaces which are spanned by conjugate triples of commuting base elements. These contain the six idempotent lattices that can be located inCl 3,1. Their images exist in both C⊕Cl 1,3 and C⊕Cl 3,1. Further in each color space there is defined an octahedral orientation stabilizer group which fixates one lepton and color rotates the states in its quark family. Thus quantum numbers of strong interacting fields such as isospin, charge, hypercharge and color turn out as geometric properties. Next we ask if the artificialty of complexification can be avoided. The answer is yes. Defining the class of Clifford algebras with proper imaginary unit it turns out thatCl 1,3 andCl 3,1 do not belong to this class. ButCl 4,1 andCl 1,6 do. It is shown that in the latter algebra the whole color space Ansatz can be established and the generators ofSU(3) represented most naturally and without complexification. That the proposed theory becomes a physically true statement requires that there exists a non rank preserving freedom of motion within the constituents of primitive idempotents, that is, transpositions among conjugate triples in color space.

Lounesto P.

In this book, Professor Lounesto offers a unique introduction to Clifford algebras and spinors. The initial chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This book also gives the first comprehensive survey of recent research on Clifford algebras. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing between the Weyl, Majorana and Dirac spinors. Scalar products of spinors are classified by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the analytic side, Brauer-Wall groups and Witt rings are discussed, and Caucy's integral formula is generalized to higher dimensions.

Schmeikal B.

We shall show how elementary logic calculus can be represented within the Clifford algebras Cℓ1,1, Cℓ3,1 ≃ Cℓ2,2 and Cℓm,m , in general. Thus, we shall also understand why logic is connected with the standard model of physics and the quantum structure of spacetime, respectively. “It is not the substance which is in space,” as Alfred North Whitehead had pointed out, “but the attributes”. Accordingly, as we consider the Dirac algebra Mat(4, C), annihilation and time reversal turn out to be logic operations acting on Dirac spinors. This is remarkable as three generations of physicists have calculated time reverted wave functions. In a complete Clifford algebraic image of the Dirac equation, those wave functions do not exist because time reversal, when carried out in the whole algebra, annihilates any Dirac spinor. It is then, essentially in the quantum chromodynamics context of the Majorana algebra, that classical logic becomes incomplete with regard to definite truth. That is, color rotations cannot be represented as products of logic operations. It also follows that once we connect spinor spaces with probability distributions, classical logic naturally turns into quantum logic. Future developments may show how logic derivative procedures can be represented synchronically as elements in neutral signature and, further, how logic quantum computations can be generated practically by reflections of waves in subnuclear arrays.

Salingaros N.

Clifford algebras are traditionally realized in terms of a specific set of representation matrices. This paper provides a more effective alternative by giving the finite group associated with each Clifford algebra. All the representation-independent algebraic results, which are really direct consequences of the underlying group structure, can thus be derived in an easier and more general manner. There are five related but distinct classes of finite groups associated with the Clifford algebras. These groups are constructed from the complex, cyclic, quaternion, and dihedral groups in a way which is discussed here in detail. Of particular utility is a table which lists the order structure of each group: this permits the immediate identification of any Clifford algebra in any dimension.