Does a nontrivial notion of integral under logarithmic deformations of $\mathbb{R}_+$ exist?

Question

Background

The upper and lower Maslov dequantizations are respectively the limits $h \downarrow 0$ and $h \uparrow 0$ of deformations of the semifield $(\mathbb{R}_+,+,\cdot)$ defined for $0 \ne h \in \mathbb{R}$ by the map \begin{equation} \Phi_h(x) := h \log x \end{equation} which sends $\mathbb{R}_+$ to $\mathbb{R} \cup \{-\sigma(h) \cdot \infty\}$. Since the map $\Phi_h$ is invertible, it defines deformed addition and multiplication operations: i.e., \begin{equation} \label{eq:MaslovArithmetic} u +_h v := \Phi_h (\Phi_h^{-1}(u) + \Phi_h^{-1}(v)); \quad u \cdot_h v := \Phi_h (\Phi_h^{-1}(u) \cdot \Phi_h^{-1}(v)). \end{equation} Explicitly, we have \begin{equation} u +_h v := h \log \left ( e^{u/h} + e^{v/h} \right ); \quad u \cdot_h v := h \log \left ( e^{u/h} \cdot e^{v/h} \right ) \equiv u+v. \end{equation} Now writing $\lozenge$ for $\vee$ when $h > 0$ and $\wedge$ when $h < 0$, it is easy to see that $(\sum_{(h)})_{j \in [n]} u_j := u_1 +_h \dots +_h u_n = h \log \sum_{j \in [n]} e^{u_j/h} = \lozenge_j u_j + h \log \sum_j e^{(u_j-\lozenge_k u_k)/h}$. Clearly, as $h \rightarrow 0$, this tends to $\lozenge_j u_j$. This is manifested in the figure below:

There is a well-established theory of analysis in the dequantization limit $h \rightarrow 0$: this is essentially the Ansatz \begin{equation} ^\lozenge : \sum \mapsto \lozenge; \quad ^\lozenge : \prod \mapsto \sum \end{equation} where again $\lozenge$ is either $\vee$ or $\wedge$, as appropriate. The first of the correspondences above naturally leads to \begin{equation} ^\lozenge : \int \mapsto \text{ext}^\lozenge, \end{equation} where here the extremum is given by $\text{ext}^\vee := \sup$ and $\text{ext}^\wedge := \inf$. This turns out to have a rigorous interpretation, so that, e.g., \begin{equation} ^\lozenge : \int K(t,u) f(u) du \mapsto \text{ext}^\lozenge_u \left ( K(t,u) + f(u) \right ) \end{equation} is not only suggestive but correct, and in fact such integral transforms exhaust the class of linear operators. There is also a correspondence between the Fourier and Legendre transformations that is a bit trickier.

Question

Is there a reasonable notion of deformed integral for the case where $h$ is small but nonzero?

The Ansatz suggests something which diverges as $\Delta x \downarrow 0$ unless simultaneously $h \rightarrow 0$ in a controlled way, in which event the extremum results. For my purposes this is degenerate.

Answer 1

So my original thinking was that a Riemann sum over $[a,b]$ with $x_0 := a$, $x_n := b$ and $x_j := (n-j)x_0/n + jx_1/n$ would deform to $(\sum_{(h)})_{j = 1}^n (f(x_j) + \Delta x)$, where $\Delta x = (a-b)/n = x_j-x_{j-1}$. This deformed sum equals $\Delta x + h \log \sum_j \exp(f(x_j)/h)$, which as I suggested in my question diverges for fixed $h$ as $\Delta x \rightarrow 0$. If instead we take $h \rightarrow 0$, then this just yields the dequantization limit, i.e., the extremum of $f$.

A better approach seems to be to perform a "Euclidean Wick rotation" and set $\beta := -1/h$: then the deformed sum is just the "free energy", which yields a heuristic for the deformed integral, viz. $\left ( \int_\beta \right ) f \equiv -\frac{1}{\beta} \log \int e^{-\beta f(x)} dx$. Of course it is generally very difficult to get good behavior out of an integral of this form.