 Another area where questions of mathematical rigor arise within perturbative QFT is the use of path integrals. The S-matrix power series expansion contains integrals over momentum space and this is where path integrals / Feynman diagrams have been helpful for making calculations. The key concept is the partition function \(Z\), which is defined as a functional integral involving the action, which is itself an integral of the Lagrangian. The following details come mainly from Hancox-Li (2017). More specifically, the action is a functional of quantum fields. The functional integral over the action ranges over all possible combinations of the quantum fields values over spacetime. Informally, the sum is being taken over all possible field configurations. As Swanson (2017) notes, the path integral requires choosing a measure over an infinite dimensional path space, which is only mathematically well-defined in special cases. For example, if the system is formulated on a hypercubic lattice, then the measure can be defined (see section 1.2 of James Fraser 2016). Another way of having a well-defined measure is to restrict attention to a finite dimensional subspace. But if functions are allowed to vary arbitrarily on short length scales, then the integral ceases to be well-defined (Wallace 2006, p. 42). All of the correlation functions (i.e., vacuum state expectation values of the fields at different spacetime points), can be derived from the partition function \(Z\). So, given \(Z\), all empirical quantities associated with the Lagrangian can be calculated, e.g., scattering cross-sections. Finding \(Z\) amounts to a solution of LQFT. \(Z\) can be expanded in a Taylor series in the coupling constant. When this is done, two types of divergences can occur: (1) individual terms of the perturbation series can diverge and/or (2) the perturbation series itself is divergent, though the series may be an asymptotic series. To deal with (1), physicists do the following procedures (Hancox-Li 2017, pp. 344–345): (i) regularization, which involves reducing the number of degrees of freedom via dimensional regularization, momentum cutoffs, or using a lattice formulation and (ii) add counterterms to compensate for the regularization in (i). But this construction is purely formal and not mathematically defined. The rules used to manipulate the Lagrangian, and hence the partition function, are not well-defined.