A general approach to the geometry of distributional bundles is presented. In particular, the notion of connection on these bundles is studied. A few examples, relevant to quantum field theory, are discussed.

2000 MSC: 46F99, 53C05, 58B10.

 Keywords: distributional bundles, Frölicher smoothness, connections.

The notion of smoothness introduced by Frölicher provides a general setting for calculus in functional spaces and differential geometry in functional bundles. An important aspect of that approach is that the essential results can be formulated in terms of finite-dimensional spaces and maps, without heavy involvement in infinite-dimensional topology and other intricated questions. In particular, the notion of a smooth connection on a functional bundle has been applied in the context of the ``covariant quantization'' approach to Quantum Mechanics.

In a previous paper I applied these ideas to the differential geometry of certain bundles whose fibres are distributional spaces, more specifically scalar-valued generalized half-densities. The main purpose of the present paper is to extend those results to the general case of the bundle of generalized `tube' sections of a $2$-fibred `classical' (i.e.\ finite dimensional) bundle; basic notions of standard differential geometry---such as tangent space, jet space, connection and curvature---are introduced for this case; adjoint connections and tensor product connections are shown to exist. Furthermore, a suitable connection on the underlying classical bundle is shown to yield a connection on the corresponding distributional bundle; some particularly important cases are the vertical bundle and its tensor algebra, which turn out to be closely related to the notion of adjoint connection. Finally, I consider a few examples which are relevant in view of applications to quantum field theory: the `Dirac connection' on the bundle of $1$-electron states for a given observer, and the connections induced on the phase-distributional bundles describing electron and photon fields.