Quantum mechanics bases on states in the Hilbert space; these can be seen like vectors (representable in an orthonormal basis system), where a scalar product can be defined. If one multiplies two different vectors $a_i,b_j$, where the indices $i,j$ run over the complete Hilbert space basis by the scalar product

$<a_i,b_j> = c_{ij}$

one obtains a matrix $c_{ij}$. To obtain higher rank tensors you have to define first a new space of states, e.g. the states of the $L^p$ space with $p>2$. Moreover you have to define e.g. a generalized scalar product

$<a_i,b_j,c_k,...> = d_{ijk ...}$

with orthogonality conditions. Suppose you can generalize the spectral theorem as follows:

$H = \sum_{\lambda \in \Gamma} H_{\gamma} e_\gamma$

where $H$ is an operator, $H_\lambda$ the eigenvalues of it, $\Gamma$ the generalized spectrum and $e_\gamma$ a generalized projector on the eigenstate basis. You will compute now products like $e_\gamma(t) e_\lambda(t+dt)$ with time coordinate $t$. Such products you have to define and in general, these are some higher rank tensors.

At the end you will get a path integral very similar to the ordinary one, but with integrals over the generalized state basis (due to insertion of the $\sum_{\lambda \in \Gamma}$). However, if such a spectral theorem can not be applied, the path integral cannot be defined rigorously.