Physical interpretation of differential forms with values in $E$ when $E$ is a vector bundle whose sections are fields Disclaimer: I'm much more a mathematician then a physicist and this question is slightly mathematical however it made more sense to ask it in this site then any other.
Let $M$ be a manifold of dimension $n$ which we think of as some space(-time). For any $0\le k \le n$ denote by $\Omega_M^k$ the vector bundle whose sections are differential $k-$forms and let $E \to M$ be a vector bundle whose sections we think of as some field on $M$ (photons, electrons, gluons, quarks etc...).

What is the physical interpretation of sections of $E \otimes \Omega_M^k$?

I know several satisfying geometrical interpretations and this is not what i'm looking for. What i'm interested in ideally is an interpretation in terms of physical objects (fields, forces, potentials, actions etc...).
It may be relevant to point out that if $M$ and $E$ are both endowed with connections then there's a unique induced conection on $E \otimes \Omega_M^k$ so in a sense the dynamics in this bundle is controlled by the dynamics of $E$ and $M$.
 A: Consider the case of a massive spin-1/2 particle; the classical field configurations corresponding to this particle are sections of the spinor bundle $S_M$ on $M$, $x\in M \mapsto \psi(x) \in S_x$. There are several contexts in which the fibers are expanded by some extra stuff. 
One way how this hapens is through the necessity of formulating dynamical equations for the spinor field, because the field theory is formulated for the spinor field and "its first derivatives" $\partial_\nu\psi$. Physically, this is analogous to the fact that the state of a classical particle is not determined only by its position but also by its velocity (first time derivative of position), its kinetic degree of freedom. This then means that the field theory is in fact formulated on the first jet bundle. Informally speaking, a fiber of the first jet bundle at $x \in M$ can be understood as the direct product of the tangent space at x times the spinor fiber, $JS_x \sim T_x M \otimes S_x$, because (in a somewhat abused notation) $\partial_\nu \psi \sim \frac{\partial}{\partial x^\nu} \otimes \psi$. Since $\Omega^1_x \sim T_x M$, we could say that the Lagrangian field theory of the non-interacting spin-1/2 particle is formulated on fibers of the form $\sim S_x \otimes S_x \otimes \Omega^1_x$. 
However, the fun part is gauge theory, where an additional gauge field is introduced and a relevant $\Omega^2_x$ appears on the fibre. Lets say our spin-1/2 particle is an electron and we are talking about the $U(1)$ Abelian gauge group. Then an extra piece of the fibre $A(x) \in \Omega^1_x$ appears which is known as the electromagnetic vector potential (or the gauge potential). It corresponds to the photon, the carrier of the electromagnetic forces, and it does all sorts of beautiful things in the gauge theory. 
Since the field $A(x)$  also needs its proper dynamics, the field theory is also formulated on the field fibres plus their first jet. The funny thing, however, is that only a subspace of the jet $J\Omega^1_x \sim \Omega^1_x \otimes \Omega^1_x$ turns out to be gauge invariant and it is always a subspace of $\Omega^2_x$. In the Abelian case it is the subspace of exact two-forms which we will denote as $\bar{\Omega}^2_x$. 
The full Lagrangian of spinor electrodynamics is then a function on the bundle of fibres $\sim S_x \otimes (S_x \otimes \Omega^1_x) \otimes \Omega^1_x \otimes  \bar{\Omega}^2_x$ where the members of the fibre correspond respectively to the basic spinor field (spin-1/2 particle) and its kinetic degrees of freedom, and the gauge potential (the force carrier particle) and its kinetic degrees of freedom.
