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Bertfried Fauser

Publications and source records attributed to Bertfried Fauser.

At least 37 records · Page 2Linked to original sources
arXiv2005

The Dirichlet Hopf algebra of arithmetics

In this work, we develop systematically the ``Dirichlet Hopf algebra of arithmetics'' by dualizing addition and multiplication maps. We study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, obeying only a weakened (multiplicative) homomorphism axiom. The consequences of the weakened structure, called a Hopf gebra, e.g. on cohomology are explored. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an `unrenormalized' coproduct and an `unrenormalized' pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra. This can be modelled alternatively by employing Rota-Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.

math-ph↗
arXiv2005

A Hopf algebraic approach to the theory of group branchings

We describe a Hopf algebraic approach to the Grothendieck ring of representations of subgroups $H_π$ of the general linear group GL(n) which stabilize a tensor of Young symmetry $\{π\}$. It turns out that the representation ring of the subgroup can be described as a Hopf algebra twist, with a 2-cocycle derived from the Cauchy kernel 2-cocycle using plethysms. Due to Schur-Weyl duality we also need to employ the coproduct of the inner multiplication. A detailed analysis including combinatorial proofs for our results can be found in math-ph/0505037. In this paper we focus on the Hopf algebraic treatment, and a more formal approach to representation rings and symmetric functions.

math-ph↗
arXiv2004

Quantum field theory and Hopf algebra cohomology

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.

hep-th↗
arXiv2004

Products, coproducts and singular value decomposition

Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, ie generalized eigenvalues, to these maps. We show, for the case of Grassmann and Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, ie make them noncommutative, we hope to be able to learn more about such geometries. Remarkably the coproduct for positive singular values of eigenvectors in $A$ yields directly corresponding eigenvectors in A\otimes A.

math-ph↗
arXiv2003

A Hopf laboratory for symmetric functions

An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.

math-ph↗
arXiv2002

Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple

Hopf algebraic structures will replace groups and group representations as the leading paradigm in forthcoming times. K-theory, co-homology, entanglement, statistics, representation categories, quantized or twisted structures as well as more geometric topics of invariant theory, e.g., the Grassmann-Cayley bracket algebra are all covered by the Hopf algebraic framework. The new branch of 'experimental mathematics' allows one to easily enter these fields through direct calculations using symbolic manipulation and computer algebra system (CAS). We discuss problems which were solved when building the BIGEBRA package for Maple and CLIFFORD [see math-ph/0212031] to handle tensor products, Grassmann and Clifford algebras, coalgebras and Hopf algebras. Recent results showing the usefulness of CAS for investigating new and involved mathematics provide us with examples. An outlook on further developments is given.

math-ph↗
arXiv2002

Grade free product formulae from Grassmann Hopf gebras

In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector (span of the generators) on homogeneous i.e. sums of decomposable (Grassmann), multivectors and later extended by bilinearity. The Hestenesian 'dot' product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need for applying grade operators at various times. Moreover, the multivector grade is not a generic Clifford algebra concept. The situation becomes even worse in geometric applications if a meet, join or contractions have to be calculated. Starting from a naturally graded Grassmann Hopf gebra, we derive general formulae for the products: meet and join, comeet and cojoin, left/right contraction, left/right cocontraction, Clifford and co-Clifford products. All these product formulae are valid for any grade and any inhomogeneous multivector factors in Clifford algebras of any bilinear form, including non-symmetric and degenerated forms. We derive the three well known Chevalley formulae as a specialization of our approach and will display co-Chevalley formulae. The Rota--Stein cliffordization is shown to be the generalization of Chevalley deformation. Our product formulae are based on invariant theory and are not tied to representations/matrices and are highly computationally effective. The method is applicable to symplectic Clifford algebras too.

math-ph↗
arXiv2002

A Treatise on Quantum Clifford Algebras

Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Grassmann-Cayley algebras including co-meet and co-join for Grassmann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of non-local products and interacting Hopf gebras which are generically non-perturbative. A `spinorial' generalization of the antipode is given. The non-existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially generated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordization is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory.

math.QA↗
arXiv2001

Clifford geometric parameterization of inequivalent vacua

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras --as Clifford algebras-- by different filtrations resp. induced gradings. The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*-algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non-statistical, non-definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U(2)-symmetry producing a gap-equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov-Valatin transformations is given.

hep-th↗
arXiv2000

Clifford Hopf gebra for two-dimensional space

A Clifford algebra Cl(V,η\in V^*\otimes V^*) jointly with a Clifford cogebra Cl(V,ξ\in V\otimes V) is said to be a Clifford biconvolution Cl(η,ξ). We show that a Clifford biconvolution for dim_R Cl=4 does possess an antipode iff det(id-ξ\circη)\neq 0. An antipodal Clifford biconvolution is said to be a Clifford Hopf gebra. We study the Clifford Hopf gebra and examples of antipode less Clifford bigebras including the Grassmann case.

math.QA↗
arXiv2000

Quantum Clifford Hopf Gebra for Quantum Field Theory

We give arguments for the necessity to employ Quantum Clifford Hopf Gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for further research.

hep-th↗
arXiv2000

Projective Relativity: Present Status and Outlook

We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms. Much more in the case of newer higher dimensional Kaluza--Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition R5_{μν}=0 not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint.

gr-qc↗
arXiv2000

On the relation of Manin's quantum plane and quantum Clifford algebras

One particular approach to quantum groups (matrix pseudo groups) provides the Manin quantum plane. Assuming an appropriate set of non-commuting variables spanning linearly a representation space one is able to show that the endomorphisms on that space preserving the non-commutative structure constitute a quantum group. The non-commutativity of these variables provide an example of non-commutative geometry. In some recent work we have shown that quantum Clifford algebras --i.e. Clifford algebras of an arbitrary bilinear form-- are closely related to deformed structures as q-spin groups, Hecke algebras, q-Young operators and deformed tensor products. The natural question of relating Manin's approach to quantum Clifford algebras is addressed here. Some peculiarities are outlined and explicite computations using the Clifford Maple package are exhibited. The meaning of non-commutative geometry is re-examined and interpreted in Clifford algebraic terms.

math.QA↗
arXiv2000

On the Hopf algebraic origin of Wick normal-ordering

A combinatorial formula of G.-C. Rota and J.A. Stein is taken to perform Wick re-ordering in quantum field theory. Wick's theorem becomes a Hopf algebraic identity called Cliffordization. The combinatorial method relying on Hopf algebras is highly efficient in computations and yields closed algebraic expressions. AMS Subject Classifications 2000: 81R50; 16W30

hep-th↗
arXiv1999

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) <-> \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford algebras'. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1} \otimes Cl_{1,1}. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q-deformed structures through nontrivial vacuum states in quantum theories is outlined.

math.QA↗
arXiv1999

On the equivalence of Daviau's space Clifford algebraic, Hestenes' and Parra's formulations of (real) Dirac theory

Recently Daviau showed the equivalence of ordinary matrix based Dirac theory -formulated within a spinor bundle S_x \simeq C^4_x-, to a Clifford algebraic formulation within space Clifford algebra CL(R^3,delta) \simeq M_2(C) \simeq P \simeq Pauli algebra (matrices) \simeq H \oplu H \simeq biquaternions. We will show, that Daviau's map theta : C^4 \mapsto M_2(C) is an isomorphism. Furthermore it is shown that Hestenes' and Parra's formulations are equivalent to Daviau's space Clifford algebra formulation, which however uses outer automorphisms. The connection between such different formulations is quite remarkable, since it connects the left and right action on the Pauli algebra itself viewed as a bi-module with the left (resp. right) action of the enveloping algebra P^e \simeq P\otimes P^T on P. The isomorphism established in this article and given by Daviau's map does clearly show that right and left actions are of similar type. This should be compared with attempts of Hestenes, Daviau and others to interprete the right action as the iso-spin freedom.

hep-th↗
arXiv1999

Hecke Algebra Representations in Ideals Generated by q-Young Clifford Idempotents

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multi-particle systems since they contain spinor and vector representations within the same formalism, and, moreover, allow for a complete study of all classical Lie groups. In this work, together with an accompanying work also presented at this conference, an analysis of q-symmetry -- for generic q's -- based on the ordinary symmetric groups is given for the first time. We construct q-Young operators as Clifford idempotents and the Hecke algebra representations in ideals generated by these operators. Various relations as orthogonality of representations and completeness are given explicitly, and the symmetry types of representations is discussed. Appropriate q-Young diagrams and tableaux are given. The ordinary case of the symmetric group is obtained in the limit q \to 1. All in all, a toolkit for Clifford algebraic treatment of multi-particle systems is provided. The distinguishing feature of this paper is that the Young operators of conjugated Young diagrams are related by Clifford reversion, connecting Clifford algebra and Hecke algebra features. This contrasts the purely Hecke algebraic approach of King and Wybourne, who do not embed Hecke algebras into Clifford algebras.

math.QA↗