Dirac Operators on Taub-NUT Space: Relationship and Discrete Transformations

It is shown that the N = 4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant constant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N = 4 supersymmetry.     

Symplectic Symmetries of Hamiltonian Systems

Abstract

We prove a generalization to the case of s × s matrix linear differential operators of the classical theorem of E. Cotton giving necessary and sufficient conditions for equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger operator are derived and formulated geometrically in terms of vanishing conditions on the curvature of a gl(s, R)-valued connection. These conditions are illustrated on a class of matrix differential operators of physical interest, arising by symmetry reduction from Dirac’s equation for a spinor field minimally coupled with a cylindrically symmetric magnetic field.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

Can one ADM quantize relativistic bosonicstrings and membranes?

The standard methods for quantizing relativistic strings diverge significantly from the Dirac-Wheeler-DeWitt program for quantization of generally covariant systems and one wonders whether the latter could be successfully implemented as an alternative to the former. As a first step in this direction, we consider the possibility of quantizing strings (and also relativistic membranes) via a partially gauge-fixed ADM (Arnowitt, Deser and Misner) formulation of the reduced field equations for these systems. By exploiting some (Euclidean signature) Hamilton-Jacobi techniques that Mike Ryan and I had developed previously for the quantization of Bianchi IX cosmological models, I show how to construct Diff(S ¹)-invariant (or Diff(Σ)-invariant in the case of membranes) ground state wave functionals for the cases of co-dimension one strings and membranes embedded in Minkowski spacetime. I also show that the reduced Hamiltonian density operators for these systems weakly commute when applied to physical (i.e. Diff(S ¹) or Diff(Σ)-invariant) states. While many open questions remain, these preliminary results seem to encourage further research along the same lines.     

Deformation Quantization for Hilbert Space Actions

Rieffel's theory of deformations of C^*-algebras for -actions can be extended to actions of infinite-dimensional Hilbert spaces. The CCR algebra over a Hilbert space H can be exhibited in this manner as a deformation of a commutative C^*-algebra of almost periodic functions on H. Received: 26 August 1996 / Accepted: 28 January 1997     

The Spin Interaction of a Dirac Particle in an Aharonov-Bohm Potential in First Order Scattering

For a Dirac particle in an Aharonov-Bohm (AB) potential, it is shown that the spin interaction (SI) operator which governs the transitions in the spin sector of the first order S-matrix is related to one of the generators of rotation in the spin space of the particle. This operator, which is given by the projection of the spin operator Σ along the direction of the total momentum of the system, and the two operators constructed from the projections of the Σ operator along the momentum transfer and the z-directions close the SU(2) algebra. It is suggested, then, that these two directions of the total momentum and the momentum transfer form some sort of natural intrinsic directions in terms of which the spin dynamics of the scattering process at first order can be formulated conveniently. A formulation and an interpretation of the conservation of helicity at first order using the spin projection operators along these directions is presented.     

Principal Fibrations from Noncommutative Spheres

We construct noncommutative principal fibrations S_θ⁷→S_θ⁴ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. “The algebra inclusion is an example of a not-trivial quantum principal bundle.”     

Generalized Weierstrass Relations and Frobenius Reciprocity

This article investigates local properties of the further generalized Weierstrass relations for a spin manifold S immersed in a higher dimensional spin manifold M from the viewpoint of the study of submanifold quantum mechanics. We show that the kernel of a certain Dirac operator defined over S, which we call a submanifold Dirac operator, gives the data of the immersion. In the derivation, the simple Frobenius reciprocity of Clifford algebras S and M plays an important role.      

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.     

Holomorphy of the scattering matrix with respect toc −2 for Dirac operators and an explicit treatment of relativistic corrections

We prove holomorphy of the scattering matrix at fixed energy with respect toc ^–2 for abstract Dirac operators. Relativistic corrections of orderc ^–2 to the nonrelativistic limit scattering matrix (associated with an abstract Pauli Hamiltonian) are explicitly determined. As applications of our abstract approach we discuss concrete realizations of the Dirac operator in one and three dimensions and explicitly compute relativistic corrections of orderc ^–2 of the reflection and transmission coefficients in one dimension and of the scattering matrix in three dimensions. Moreover, we give a comparison between our approach and the firstorder relativistic corrections according to Foldy-Wouthuysen scattering theory and show complete agreement of the two methods.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich by an E. Schrödinger Fellowship and by Project No. P7425     

Riemannian Geometry of Quantum Groups¶and Finite Groups with Nonuniversal Differentials

We construct noncommutative “Riemannian manifold” structures on dual quasitriangular Hopf algebras such as ℂ _q [SU ₂] with its standard bicovariant differential calculus, using the quantum frame bundle approach introduced previously. The metric is provided by the braided-Killing form on the braided-Lie algebra on the tangent space and the n-bein by the Maurer–Cartan form. We also apply the theory to finite sets and in particular to finite group function algebras ℂ[G] with differential calculi and Killing forms determined by a conjugacy class. The case of the permutation group ℂ[S ₃] is worked out in full detail and a unique torsion free and cotorsion free or “Levi–Civita” connection is obtained with noncommutative Ricci curvature essentially proportional to the metric (an Einstein space). We also construct Dirac operators in the metric background, including on finite groups such as S ₃. In the process we clarify the construction of connections from gauge fields with nonuniversal calculi on quantum principal bundles of tensor product form. Received: 22 June 2000 / Accepted: 26 August 2001     

Parallel transport in the determinant line bundle: The zero index case

For a product family of invertible Weyl operators on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator onR×X.Supported in party by NSF Grants PHY8605978 and PHY-82-15249 and the Robert A. Welch FoundationSupposed in part by NSF Grant PHY-82-15249     

The Invertible Double of Elliptic Operators

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderón projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderón projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderón projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.     

Lower bounds for eigenvalues of the Dirac operator on n-spheres with SO(n)-symmetry

In this paper we derive estimates for the eigenvalues of the Dirac operator and their multiplicity on manifolds diffeomorphic to Sⁿ with an isometric SO(n)-action. Especially we prove a new lower bound for the first eigenvalue and show an example, where this new bound coincides in the limit with the known upper bounds.     

Logarithmic Intertwining Operators and Vertex Operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg and lattice vertex algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1) _a , of central charge 1 − 12a ². We classify these operators in terms of depth and provide explicit constructions in all cases. Our intertwining operators resemble puncture operators appearing in quantum Liouville field theory. Furthermore, for a = 0 we focus on the vertex operator subalgebra L(1, 0) of M(1)₀ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of hidden logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1, 0)-module.     

Monopoles and maps fromS 2 toS 2; the topology of the configuration space

The configuration space for the SU(2)-Yang-Mills-Higgs equations on ³ is shown to be homotopic to the space of smooth maps fromS ² toS ². This configuration space indexes a family of twisted Dirac operators. The Dirac family is used to prove that the configuration space does not retract onto any subspace on which the SU(2)-Yang-Mills-Higgs functional is bounded.National Science Foundation Postdoctoral Fellow in Mathematics     

On the reduction of the Dirac equation to non-relativistic form: the dipole-dipole interaction in the hydrogen atom

Finite nucleus models are used in the calculation of the dipole-dipole part of the second-order hyperfine energy in the ground state of the hydrogen atom. The results are used as a guide to bring the non-relativistic calculation for a point nucleus into agreement with the relativistic calculation. This necessitates the introduction of delta function operators in the dipole-dipole operator and the zeroth-order hamiltonian. It is concluded that the reduction of the Dirac equation to non-relativistic form is valid for the hyperfine interaction up to order mc ²α⁶.     

On Bilinear Invariant Differential Operators Acting on Tensor Fields on the Symplectic Manifold

Abstract

Let M be an n-dimensional manifold, V the space of a representation ρ : GL(n) → GL(V). Locally, let T (V ) be the space of sections of the tensor bundle with fiber V over a sufficiently small open set U ? M, in other words, T (V ) is the space of tensor fields of type V on M on which the group Diff(M) of diffeomorphisms of M naturally acts. Elsewhere, the author classified the Diff(M)-invariant differential operators D : T (V ₁) ? T (V ₂) → T (V ₃) for irreducible fibers with lowest weight. Here the result is generalized to bilinear operators invariant with respect to the group Diff_ω(M) of symplectomorphisms of the symplectic manifold (M, ω). We classify all first order invariant operators; the list of other operators is conjectural. Among the new operators we mention a 2nd order one which determins an “algebra” structure on the space of metrics (symmetric forms) on M.     

Twisted Diff S 1-action on loop groups and representations of the virasoro algebra

A modified Hamiltonian action of Diff S ¹on the phase space LG ^C /G^C, where LG is a loop group, is defined by twisting the usual action by a left translation in LG. This twisted action is shown to be generated by a nonequivariant moment map, thereby defining a classical Poisson bracket realization of a central extension of the Lie algebra diff_C S ¹. The resulting formula expresses the Diff S ¹generators in terms of the left LG translation generators, giving a shifted modification of both the classical and quantum versions of the Sugawara formula.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the National Science Foundation.     

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).