On Marshak's and Connes' views of chirality

I render the substance of the discussions I had with Robert E. Marshak shortly before his death, wherein the kinship between the ``neutrino paradigm'' ---espoused by Marshak--- and the central notion of K-cycle in noncommutative geometry (NCG) was found. In that context, we give a brief account of the Connes--Lott reconstruction of the Standard Model (SM).Comment: 10 pages, Plain Te

Improved Epstein-Glaser Renormalization II. Lorentz invariant framework

The Epstein--Glaser type T-subtraction introduced by one of the authors in a previous paper is extended to the Lorentz invariant framework. The advantage of using our subtraction instead of Epstein and Glaser's standard W-subtraction method is especially important when working in Minkowski space, as then the counterterms necessary to keep Lorentz invariance are simplified. We show how T-renormalization of primitive diagrams in the Lorentz invariant framework directly relates to causal Riesz distributions. A covariant subtraction rule in momentum space is found, sharply improving upon the BPHZL method for massless theories.Comment: LaTeX, 15 pages, no figure. Version to be published in J. Math. Phys. (Section 7 on the Massive Case and some references have been withdrawn). To the Memory of Laurent Schwart

Internal Space for the Noncommutative Geometry Standard Model and Strings

In this paper I discuss connections between the noncommutative geometry approach to the standard model on one side, and the internal space coming from strings on the other. The standard model in noncommutative geometry is described via the spectral action. I argue that an internal noncommutative manifold compactified at the renormalization scale, could give rise to the almost commutative geometry required by the spectral action. I then speculate how this could arise from the noncommutative geometry given by the vertex operators of a string theory.Comment: 1+22 pages. More typos and misprints correcte

The Kirillov picture for the Wigner particle

We discuss the Kirillov method for massless Wigner particles, usually (mis)named "continuous spin" or "infinite spin" particles. These appear in Wigner's classification of the unitary representations of the Poincar\'e group, labelled by elements of the enveloping algebra of the Poincar\'e Lie algebra. Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to quantization. Here we exhibit for those particles the classical Casimir functions on phase space, in parallel to quantum representation theory. A good set of position coordinates are identified on the coadjoint orbits of the Wigner particles; the stabilizer subgroups and the symplectic structures of these orbits are also described.Comment: 19 pages; v2: updated to coincide with published versio

A new antisymmetric bilinear map for type-I gauge theories

In the case of gauge theories, which are ruled by an infinite-dimensional invariance group, various choices of antisymmetric bilinear maps on field functionals are indeed available. This paper proves first that, within this broad framework, the Peierls map (not yet the bracket) is a member of a larger family. At that stage, restriction to gauge-invariant functionals of the fields, with the associated Ward identities and geometric structure of the space of histories, make it possible to prove that the new map is indeed a Poisson bracket in the simple but relevant case of Maxwell theory. The building blocks are available for gauge theories only: vector fields that leave the action functional invariant; the invertible gauge-field operator, and the Green function of the ghost operator.Comment: 10 page

Monopole-based quantization: a programme

We describe a programme to quantize a particle in the field of a (three dimensional) magnetic monopole using a Weyl system. We propose using the mapping of position and momenta as operators on a quaternionic Hilbert module following the work of Emch and Jadczyk.Comment: Contribution to the volume: Mathematical Physics and Field Theory, Julio Abad, In Memoriam}, M. Asorey, J.V. Garcia Esteve, M.F. Ranada and J. Sesma Editors, Prensas Universitaria de Zaragoza, (2009

Anomalies and Schwinger terms in NCG field theory models

We study the quantization of chiral fermions coupled to generalized Dirac operators arising in NCG Yang-Mills theory. The cocycles describing chiral symmetry breaking are calculated. In particular, we introduce a generalized locality principle for the cocycles. Local cocycles are by definition expressions which can be written as generalized traces of operator commutators. In the case of pseudodifferential operators, these traces lead in fact to integrals of ordinary local de Rham forms. As an application of the general ideas we discuss the case of noncommutative tori. We also develop a gerbe theoretic approach to the chiral anomaly in hamiltonian quantization of NCG field theory.Comment: 30 page

From Peierls brackets to a generalized Moyal bracket for type-I gauge theories

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang--Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to i hbar times the Peierls bracket to lowest order in hbar.Comment: 14 pages, Late