A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles (`quantum bundles') over the spacetime manifold yield a quantum ``formalism'' along any 1-dimensional timelike submanifold (a `detector'); in the flat, inertial case this turns out to reproduce the basic results of the usual quantum field theory, while in general it could be seen as a local, ``linearized'' description of the actual physics.

2000 MSC: 58B99, 53C05, 81Q99, 81T20

 Keywords: distributional bundles, quantum particles.

Quantisation, intended as the construction of a quantum theory by applying suitable rules to classical systems, is perhaps the most common approach to the study of the foundations of quantum physics; indeed, this philosophy has produced an immense physical and mathematical literature. There is, however, a widespread opinion that the true relation between classical and quantum theories should rather go in the opposite sense: at least in principle, classical physics should derive from quantum physics, thought to be more fundamental.

As a first step in that direction, one could try and build a stand-alone mathematical model, not derived from a quantisation procedure, which should reproduce (at least) the basic observed facts of elementary particle physics. The present article is a proposal in this sense, based on two main ingredients: \emph{free states}, and \emph{interaction}. A further interesting feature of the model is its freedom from the requirement of spacetime flatness.

The fundamental mathematical tool of my exploration is the geometry of \emph{distributional bundles}, that is bundles over classical (finite-dimensional Hausdorff) manifolds whose fibres are distributional spaces. These arise naturally from a class of finite-dimensional 2-fibred bundles, which turns out to contain the most relevant physical cases. The basics of their geometry have been exposed in two previous papers~\cite{C00a,C04a} along the line of thought stemming from Fr\"olicher's notion of smoothness~\cite{Fr,FK,KM,MK,CK95}.

While I do not quantise classical fields, at this stage I do consider certain finite-dimensional geometric structures which are related to classical field theories.\footnote{ In particular gravitation, here, is a fixed background.} From these one can naturally build 2-fibred bundles and, eventually, \emph{quantum bundles}: distributional bundles whose fibres are spaces of one-particle states, and the related \emph{Fock bundles}. It turns out that the underlying, finite-dimensional geometric structure determines a distinguished connection on a quantum bundle; this connection is related to the description of \emph{free-particle states}.

The basic idea about quantum interactions is that they should be described by a new connection on the Fock bundle, obtained by adding an interaction morphism to the free-particle connection. This approach requires the notion of a \emph{detector}, defined to be a timelike 1-dimensional submanifold of the spacetime manifold. Then a natural interaction morphism indeed exists in the fibres of the restricted Fock bundle. It turns out that a detector carries a quantum ``formalism'' which can be seen as a kind of complicate clock; in the flat, inertial case this turns out to reproduce the basic results of the usual quantum field theory,\footnote{ The usual quantum fields can be recovered~\cite{C04b} as certain natural geometric structures of the quantum bundles, but they only play a marginal role in this approach.} while in general it could be seen as a local, ``linearized'' description of the actual physics.

The paper's plan is as follows. In the two first sections I will summarize the basic ideas about distributional bundles and quantum bundles, the latter being defined as certain bundles of generalized half-densities on classical momentum bundles; then I will introduce generalized frames for quantum bundles and the notion of a detector. In section~\ref{S:Quantum interaction} I will illustrate the construction of the quantum interaction from a general (and necessarily sketchy) point of view. In section~\ref{S:Scalar particles} these ideas will be implemented in the simplest case, a theory of two scalar particles; in sections~\ref{S:Electron and positron free states}, \ref{S:Photon free states}, \ref{S:Electromagnetic interaction} and~\ref{S:QED} I will show how to treat QED in the above said setting; in the flat inertial case one then recovers the basic known results. Here, the role of 2-fibred bundles turns out to be specially meaningful.