Recent developments in the geometry of distributional bundles yield a natural way of describing quantum fields on a curved spacetime background.

2000 MSC: 53C05, 81T20.

 Keywords: quantum bundles, quantum connections, quantum fields.

This work is addressed mainly to mathematicians and mathematical physicists having a background in differential geometry, who wish to understand fundamental notions of quantum field theory on curved spacetime in a rigorous geometric framework. The attention here is focused on the general notion of a quantum field rather than on particular instances, though in the last section a basic example is given of how the described ideas can be put to work in practice. Note, however, that other pieces need to be added in order to obtain a complete geometrical QFT framework; above all, the still open question of the description of particle interactions is left untouched here. I plan to address at least some of the remaining pieces in forthcoming papers.

Basically, quantum fields are certain geometric structures naturally arising on quantum bundles; these, on turn, are functional bundles derived from the `classical' finite-dimensional bundles where the corresponding classical field theory is formulated. The method used for studying the geometry of functional bundles is based on the notion of smoothness introduced by Fr\"olicher (or \emph{F-smoothness}) and studied by him and several authors. This method was enlarged by myself to include a treatment of \emph{distributional bundles}, namely bundles over classical (i.e.\ finite-dimensional, Hausdorff) manifolds, whose fibres are distributional spaces.

In order to recover the usual notion of a quantum field, namely that of a distribution (on configuration or phase space) valued into a space of operators and obeying a classical field equation, one has to introduce the notion of a \emph{quantum connection}. One already finds a notion of quantum connection in geometric formulations of Quantum Mechanics, in particular in the standard geometric quantization approach, as well as in developments such as the `covariant quantization' approach. There, the term under consideration refers to a connection, on a finite dimensional (`classical') bundle, related to the PDE obeyed by wave functions. This equation, however, can be reinterpreted as the equation of motion for `quantum histories', sections of a `Hilbert functional bundle' over time describing the evolution of a quantum state; on turn, one can view such sections as covariantly constant relatively to a connection on the functional bundle.

Now the method of F-smoothness allows to introduce and study, in the context of functional bundles, several usual notions of differential geometry. In the distributional case, a connection in the underlying finite-dimensional structure determines a distributional connection, while other interesting distributional connections do not arise from classical ones. In this context, a quantum connection on a distributional bundle $\boldsymbol{\mathcal{V}}\to\boldsymbol{M}$ (where $\boldsymbol{M}$ is the classical spacetime manifold) is defined to be an F-smooth linear connection such that horizontal transport along any timelike curve determines continuous isomorphisms among the fibres. Then, a geometric formulation of the basic notions of quantum field theory can be achieved by starting from certain classical structures, which naturally yield quantum bundles and various connections on them. The usual notion of a quantum field, in the form of a section of the quantum state bundle valued into a space of operators, can be recovered from the above said quantum structures through certain bundle splittings; so called \emph{free fields} and \emph{interpolating fields} are also recovered.

The plan of the paper can be summarized as follows: after a short review of the F-smooth geometry of distributional bundles, I introduce quantum bundles, quantum connections and briefly discuss the traditional quantum `pictures' from this point of view. Next, the geometrical structures corresponding to the objects traditionally called `quantum fields' are introduced, and their main properties discussed. Finally, I describe a possible practical implementation of the above ideas on a curved spacetime, by using the notion of `detector'; a new characterization of the connections naturally induced on quantum phase bundles is also provided.