Bertfried Fauser

On the Hopf algebraic origin of Wick normal ordering

A combinatorial formula of Rota and Stein is taken to perform Wick reordering in quantum field theory. Wicks 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.

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 formulae 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.

New branching rules induced by plethysm

We derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor T? of Young symmetry type ? where ? is an integer partition. The branchings and fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = ?fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function s? ? {?} by the basic M series of complete symmetric functions and the L = M?1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains ?-generalized Newell?Littlewood formulae and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for and , showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine and in some instances non-reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2-cocycles. The algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters. Modification rules are beyond the scope of the present paper, but are briefly discussed.

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 focused 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.

On an easy transition from operator dynamics to generating functionals by Clifford algebras

Clifford geometric algebras of multivectors are treated in detail. These algebras are built over a graded space and exhibit a grading or multivector structure. The careful study of the endomorphisms of this space makes it clear that opposite Clifford algebras have to be used for obtaining all endomorphisms. Based on these mathematics, we give a fully Clifford algebraic account on generating functionals, which is thereby geometric. The field operators are shown to be Clifford and opposite Clifford maps. This picture, relying on geometry, does not need positivity in principle. Furthermore, we propose a transition from operator dynamics to generating functionals, which is based on the algebraic techniques. As a calculational benefit, this transition is considerably short compared to standard ones. The transition is not injective (unique) and depends additionally on the choice of an ordering. We obtain a direct and constructive connection between orderings and the explicit form of the functional Hamiltonian. These orderings depend on the propagator of the theory and thus on the ground state. This is invisible in path integral formulations. The method is demonstrated within two examples, a nonlinear spinor field theory and spinor QED. Antisymmetrized and normal-ordered functional equations are derived in both cases.

Mathematics of CLIFFORD -A Maple Package for Clifford and Graßmann Algebras

Abstract.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.

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Clifford algebras are naturally associated with quadratic forms. These algebras are ℤ2 -graded by construction. However, only a ℤn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) ↔ Λ 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 ℤn-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 ℤn -grading which we now call Clifford algebras of multi-vectors or quantum Clifford algebras. These quantum Clifford algebras are in fact Clifford algebras of a bilinear form in a functorial way. 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 diagonaliz-ability 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 Clp,q which can be decomposed in the symmetric case into a tensor product Clp-1,q-1 ⊕ Cl1,1. 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.

Hecke algebra representations within Clifford geometric algebras of multivectors

We introduce Clifford geometric algebras of multivectors which exhibit a bilinear form which is not necessarily symmetric. Looking at a subset of bi-vectors in , we prove that these elements provide a representation of the Hecke algebra if the bilinear form B is chosen appropriately. This shows that q-quantization can be generated by Clifford multivector objects which usually describe composite entities. This contrasts current approaches which give deformed versions of Clifford algebras by deforming the one-vector variables. Our example shows that it is not evident, from a mathematical point of view, that q-deformation is in any sense more elementary than the undeformed structure.

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V, Q) with a non-degenerate quadratic form Q of any signature (p, q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra Cℓ(V*, Q) of linear functionals (multiforms) acting on the universal Clifford algebra Cℓ(V, Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of Cℓ(V, Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of Cℓ(V, Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Abłamowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].

Plethysms, replicated Schur functions and series, with applications to vertex operators

Specializations of Schur functions are exploited to define and evaluate the Schur functions sλ[αX] and plethysms sλ[αsν(X))] for any α—integer, real or complex. Plethysms are then used to define pairs of mutually inverse infinite series of Schur functions, Mπ and Lπ , specified by arbitrary partitions π. These are used in turn to define and provide generating functions for formal characters, s(π) λ , of certain groups Hπ , thereby extending known results for orthogonal and symplectic group characters. Each of these formal characters is then given a vertex operator realization, first in terms of the seriesM = M(0) and various L⊥σ dual to Lσ , and then more explicitly in the exponential form. Finally the replicated form of such vertex operators are written down. The characters of the orthogonal and symplectic groups have been found by Schur [34] and Weyl [35] respectively. The method used is transcendental, and depends on integration over the group manifold. These characters, however, may be obtained by purely algebraic methods, . . . . This algebraic method would seem to offer a better prospect of successful application to other restricted groups than the method of group integration. Littlewood D E 1944 Phil. Trans. R. Soc. London, Ser. A 239 (809) 392 PACS numbers: 02.10.−v, 02.10.De, 02.20.−a, 02.20.Hj Mathematics Subject Classification: 05E05, 17B69, 11E57, 16W30, 20E22, 33D52, 43A40 1751-8113/