We review some work done with C. Rovelli on the use of the eigenvalues of the Dirac operator on a curved spacetime as dynamical variables, the main motivation coming from their invariance under the action of diffeomorphisms. The eigenvalues constitute an infinite set of “observables” for general relativity and can be taken as variables for an invariant description of the gravitational field dyn...

In [5] Herzlich proved a new positive mass theorem for Riemannian 3-manifolds (N, g) whose mean curvature of the boundary allows some positivity. In this paper we study what happens to the limit case of the theorem when, at a point of the boundary, the smallest positive eigenvalue of the Dirac operator of the boundary is strictly larger than one-half of the mean curvature (in this case the mass...

Based upon the lattice Dirac operator satisfying the Ginsparg-Wilson relation, we investigate canonical formulation of massless fermion on the spatial lattice. For free fermion system exact chiral symmetry can be implemented without species doubling. In the presence of gauge couplings the chiral symmetry is violated. We show that the divergence of the axial vector current is related to the chir...

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with L boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new res...

For the q-deformation Gq , 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of Ug ⊗ Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist...

In a series of papers, we will develop systematically the basic spectral theory of (self–adjoint) boundary value problems for operators of Dirac type. We begin in this paper with the characterization of (self–adjoint) boundary conditions with optimal regularity, for which we will derive the heat asymptotics and index theorems in subsequent publications. Along with a number of new results, we ex...

When chiral symmetry is spontaneously broken, the low-energy part of the Dirac operator spectrum can be computed analytically in the chiral limit. The tool is effective field theory or, equivalently in this case, Random Matrix Theory.

We study the Lp-spectrum of the Dirac operator on complete manifolds. One of the main questions in this context is whether this spectrum depends on p. As a rst example where p-independence fails we compute explicitly the Lp-spectrum for the hyperbolic space and its product with compact spaces.

The zero modes and zero resonances of the Dirac operator H = α ·D+Q(x) are discussed, where α = (α1, α2, α3) is the triple of 4× 4 Dirac matrices, D = (1/i)∇x, and Q(x) = `

On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended over the trace of a surgery of codimension at least three. We then prove that if non-empty the space of metrics with invertible Dirac operators is disconnected in dimensions n ≡ 0, 1, 3, 7 mod 8, n ≥...