Infinite Dimensional Clifford Algebra Research Articles

A new perspective on quantum field theory revealing possible existence of another kind of fermions forming dark matter

Quantum fields are considered as generators of infinite-dimensional Clifford algebra [Formula: see text], which can be either orthogonal (in the case of fermions) or symplectic (in the case of bosons). A generic quantum state can be expressed as a superposition of the basis elements of [Formula: see text], with the superposition coefficients being multiparticle complex-valued wavefunctions. The basis elements, that are products of the generators of [Formula: see text] in the Witt basis, act as creation and annihilation operators. They create positive and negative energy states that include the bare and the Dirac vacuum as special cases. It is shown that the nonvanishing electric charge arises from an extra dimension or from doubling the number of creation and annihilation operators, which brings an extra imaginary unit [Formula: see text] into the description. A further extension is to consider [Formula: see text] as one of the quaternionic imaginary units and consider a generic state as having values in the quaternionic algebra or, equivalently, in the complexified two-dimensional Clifford algebra, [Formula: see text]. It contains two distinct fundamental representations of [Formula: see text], one associated with the weak isospin doublet [Formula: see text] and the other one with the doublet of new leptons, denoted by [Formula: see text], that together with the new quarks [Formula: see text] can be identified with dark matter.

Induced map on $K$-theory for certain $\Gamma$-equivariant maps between Hilbert spaces

Higson-Kapsparov-Trout introduced an infinite-dimensional Clifford algebra of a Hilbert space, and verified Bott periodicity on K-theory. To develop algebraic topology of maps between Hilbert spaces, in this paper we introduce an induced Hilbert Clifford algebra, and construct an induced map between K-theory of the Higson-Kasparov-Trout Clifford algebra and the induced Clifford algebra. We also compute its K-group for some concrete case.

The metric nature of matter

We construct a metric structure on a configuration space of gauge connections and show that it naturally produces a candidate for a non-perturbative, 3+1 dimensional Yang-Mills-Dirac quantum field theory on a curved background. The metric structure is an infinite-dimensional Bott-Dirac operator and the fermionic sector of the emerging quantum field theory is generated by the infinite-dimensional Clifford algebra required to construct this operator. The Bott-Dirac operator interacts with the HD(M) algebra, which is a non-commutative algebra generated by holonomy-diffeomorphisms on the underlying manifold, i.e. parallel-transforms along flows of vector fields. This algebra combined with the Bott-Dirac operator encode the canonical commutation and anti-commutation relations of the quantised bosonic and fermionic fields. The square of the Bott-Dirac operator produces both the Yang-Mills Hamilton operator and the Dirac Hamilton operator as well as a topological Yang-Mills term alongside higher-derivative terms and a metric invariant.

The Chiral Anomaly of the Free Fermion in Functorial Field Theory

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory. In this paper, we give a detailed construction of this anomaly theory, as a functor that sends manifolds to infinite-dimensional Clifford algebras and bordisms to bimodules.

3D Current Algebra and Twisted K Theory

Equivariant twisted K theory classes on compact Lie groups [Formula: see text] can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra [Formula: see text] using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

Charged Representations of the Infinite Fermi and Clifford Algebras

The real and quaternionic charge conjugation operators invariant under the infinite-dimensional Clifford algebra, or compatible with the Fermi algebra, are determined. There results a maze of inequivalent irreducible charged representations, all of which are non-Fock. The representation vectors and their charges admit two interpretations besides those of spinors or states of quantum fields: as wavelets on the circle, with charge conjugations acting via ordinary complex conjugation; and as infinite-dimensional numbers, with charge conjugations acting by automorphisms.

On Types of Quasifree Representations of Clifford Algebras

The types of von Neumann algebras generated by quasifree representations of infinite dimensional Clifford algebras are studied in terms of spectral properties of positive operators parametrizing quasifree states. §

Riemann surfaces, Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a “gauge” group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.

Riemann surfaces, Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a “gauge” group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.

The Clifford algebra of an infinite-dimensional space

The Clifford algebra of a real infinite-dimensional vector space is constructed. This algebra contains all finite- and infinite-dimensional Clifford algebras over the real and complex numbers. Some of the mathematical properties of these algebras are discussed; the automorphisms and involutions are specifically addressed. The special case with metric gνμ=δνμ, ν, μ=1,2,..., follows as a corollary to the more general results.

Symplectic reduction, BRS cohomology, and infinite-dimensional clifford algebras: Bertram Kostant and Shlomo Sternberg. The Raymond and Mortimer Sackler Institute for Advanced Study, University of Tel Aviv, Ramat Aviv, Israel

Symplectic reduction, BRS cohomology, and infinite-dimensional clifford algebras: Bertram Kostant and Shlomo Sternberg. The Raymond and Mortimer Sackler Institute for Advanced Study, University of Tel Aviv, Ramat Aviv, Israel

Automorphisms of the infinite dimensional clifford algebra and the atiyah-singer mod 2 index

Automorphisms of the infinite dimensional clifford algebra and the atiyah-singer mod 2 index

Torus invariance for the Clifford algebra, II

The structure of the representations of the infinite-dimensional Clifford algebra generated by states symmetric about a basis is studied. In particular, it is shown where they fit into the Gårding-Wightman classification. These representations have an unusual structure: the fibres are all infinite tensor product spaces, but the fibres corresponding to points on different orbits of the underlying group are different separable subspaces of the same inseparable infinite tensor product space. A procedure is given for constructing a large class of other representations of similar structure in which the torus automorphisms are unitarily implementable. In all cases the torus invariance depends on the geometric structure of the fibres, not on the underlying measure.

Torus Invariance for the Clifford Algebra. I

A problem in Quantum Field Theory leads to the study of a representation of the torus, ${T^3}$, as automorphisms of the infinite dimensional Clifford algebra. It is shown that the irreducible product representations of the Clifford algebra fall into two categories: the discrete representations where the automorphisms are unitarily implementable, and all the others in which the automorphisms are not implementable and which cannot even appear as subrepresentations of larger representations in which the automorphisms are implementable.

Torus invariance for the Clifford algebra. I

A problem in Quantum Field Theory leads to the study of a representation of the torus, ${T^3}$, as automorphisms of the infinite dimensional Clifford algebra. It is shown that the irreducible product representations of the Clifford algebra fall into two categories: the discrete representations where the automorphisms are unitarily implementable, and all the others in which the automorphisms are not implementable and which cannot even appear as subrepresentations of larger representations in which the automorphisms are implementable.