\section{Path Integrals}
\label{section:path-integrals}

\begin{definition}[Rectifiable Path]
\label{definition:rectifiable-path}
    Let $[a, b] \subset \real$, $F$ be a locally convex space over $K \in \RC$, and $\gamma \in C([a, b]; F)$ be a path, then $\gamma$ is \textbf{rectifiable} if $\gamma \in BV([a, b]; F)$.
\end{definition}

\begin{definition}[Path Integral]
\label{definition:path-integral}
    Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $\gamma \in C([a, b]; F)$ be a path. For any $f: \gamma([a, b]) \to E$, $f$ is \textbf{path-integrable with respect to $\gamma$} if $f \circ \gamma \in RS([a, b], \gamma; E)$. In which case, 
    \[
    \int_\gamma f = \int_a^b f(\gamma(t)) \gamma(dt)
    \]

    is the \textbf{path integral} of $f$ with respect to $\gamma$. The set $PI([a, b], \gamma; E)$ is the space of all functions path-integrable with respect to $\gamma$. 
\end{definition}

\begin{proposition}[Change of Variables]
\label{proposition:path-integral-change-of-variables}
    Let $[a, b], [c, d] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $\gamma \in C([a, b]; F)$ be a path, and $\varphi: C([c, d]; [a, b])$ be non-decreasing with $\varphi(c) = a$ and $\varphi(d) = b$, then for any $f \in PI([a, b], \gamma; E)$, $f \in PI([c, d], \gamma \circ \varphi; E)$, and
    \[
    \int_\gamma f = \int_{\gamma \circ \varphi} f
    \]
\end{proposition}
\begin{proof}
    Since $\varphi(c) = a$, $\varphi(d) = b$, and $\varphi$ is continuous, it is surjective. As $\varphi$ is also non-decreasing, for any tagged partition $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, there exists a tagged partition $(Q = \seqfz{y_j}, d = \seqf{d_j}) \in \scp_t([c, d])$ such that $\varphi(y_j) = x_j$ for each $0 \le j \le n$ and $\varphi(d_j) = c_j$ for each $1 \le j \le n$. In addition,  
    \begin{align*}
        S(P, c, f \circ \gamma, \gamma) &= \sum_{j = 1}^n f \circ \gamma(c_j)[\gamma(x_j) - \gamma(x_{j - 1})] \\
        &= \sum_{j = 1}^n f \circ \gamma \circ \varphi (d_j)[\gamma \circ \varphi(y_j) - \gamma \circ \varphi(y_{j-1})] \\
        &= S(Q, d, f \circ \gamma \circ \varphi, \gamma \circ \varphi)
    \end{align*}

    Therefore if $f \in PI([a, b], \gamma; E)$, then $f \in PI([c, d], \gamma \circ \varphi; E)$, with $\int_\gamma f = \int_{\gamma \circ \varphi} f$.
\end{proof}

\begin{definition}[Curve]
\label{definition:rs-curve}
    Let $[a, b], [c, d] \subset \real$, $F$ be a locally convex space over $K \in \RC$, and $\gamma \in C([a, b]; F)$ and $\mu \in C([c, d]; F)$ be paths, then $\gamma$ and $\mu$ are \textbf{equivalent} if there exists a continuous, strictly increasing bijection $\varphi \in C([c, d]; [a, b])$ such that $\mu = \gamma \circ \varphi$. In which case, $\varphi$ is a \textbf{change of parameter} between $\gamma$ and $\mu$. 

    A \textbf{curve} in $F$ is then an equivalence class of paths. 
\end{definition}

\begin{lemma}
\label{lemma:rectifiable-piecewise-linear}
    Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $\gamma \in C([a, b]; F)$ be a rectifiable path, and $U \in \cn_F(\gamma([a, b]))$, then for any continuous seminorm $[\cdot]_G: G \to [0, \infty)$, $\eps > 0$, and $f \in C(U; E)$, there exists a piecewise linear path $\Gamma \in C([a, b]; F)$ such that:
    \begin{enumerate}
        \item $\Gamma(a) = \gamma(a)$ and $\Gamma(b) = \gamma(b)$. 
        \item $\braks{\int_\gamma f - \int_\Gamma f}_F < \epsilon$. 
    \end{enumerate}
\end{lemma}
\begin{proof}
    Let $[\cdot]_E: E \to [0, \infty)$ and $[\cdot]_F: F \to [0, \infty)$ such that for any $x \in E$ and $y \in F$, $[xy]_G \le [x]_E[y]_F$. Since $\gamma([a, b])$ is compact, by modifying $[\cdot]_F$, assume without loss of generality that there exists $V \in \cn_F(\gamma([a, b]))$ such that for any $x, y \in V$ with $[x - y]_F \le 1$, $[f(x) - f(y)]_E \le \eps$.

    Since $f \in C(U; E)$, $f \in PI([a, b], \gamma; E)$ by \autoref{proposition:rs-bv-continuous}. Given that $\gamma$ is continuous, there exists $(P  = \seqfz{x_j}, c) \in \scp_t([a, b])$ such that:
    \begin{enumerate}[label=(\alph*)]
        \item For each $1 \le j \le n$, 
        \[
        \gamma([x_{j-1}, x_j]) \subset \bracs{y \in F|[y - x_{j-1}]_F \le 1}
        \]
        \item $\braks{\int_\gamma f - S(P, c, f \circ \gamma, \gamma)}_G < \epsilon$.
    \end{enumerate}

    Let $\Gamma$ be the piecewise linear path formed by linearing $f$ at points in $P$. For any $(Q, d) \in \scp_t([a, b])$ with $(Q, d) \ge (P, c)$, 
    \[
    \braks{S(P, c, f \circ \gamma, \gamma) - S(Q, d, f \circ \Gamma, \Gamma)}_G \le \eps [\gamma]_{\text{var}, [\cdot]_F}
    \]

    As $\Gamma$ is also of bounded variation, $f \in PI([a, b], \Gamma; E)$. Since the above holds for all refinements of $(Q, d)$, 
    \[
    \braks{\int_\gamma f - \int_\Gamma f}_G < \eps(1 + [\gamma]_{\text{var}, [\cdot]_F})
    \]

    
\end{proof}

\begin{remark}
\label{remark:piecewise-linear-remark}
    Past me made the mistake of believing that in \autoref{lemma:rectifiable-piecewise-linear}, it is possible to approximate rectifiable curves with piecewise linear curves in \textit{total variation distance}. However, this is not possible: as every piecewise linear curve is absolutely continuous, and the limit of these curves in total variation distance must also be absolutely continuous. As such, this strong approximation exists if and only if the curve is absolutely continuous. 
\end{remark}



\begin{theorem}[Fundamental Theorem of Calculus for Path Integrals]
\label{theorem:ftc-path-integrals}
    Let $[a, b] \subset \real$, $E, F$ be separated locally convex spaces, $\gamma \in C([a, b]; F)$ be a rectifiable path, $U \in \cn_F(\gamma([a, b]))$.
    
    Let $\sigma \subset \mathfrak{B}(F)$ be an ideal containing all compact sets, then for any $f \in C^1_\sigma(U; E)$, 
    \[
    \int_\gamma D_\sigma f = f(\gamma(b)) - f(\gamma(a))
    \]

    In particular, if $\gamma(a) = \gamma(b)$, then $\int_\gamma D_\sigma f = 0$. 
\end{theorem}
\begin{proof}
    Using \autoref{lemma:rectifiable-piecewise-linear}, assume without loss of generality that $\gamma$ is piecewise smooth. By the \hyperref[Chain Rule]{proposition:chain-rule-sets}, $f \circ \gamma \in C^1([a, b]; F)$ with $D(f \circ \gamma)(t) = Df(\gamma(t)) \cdot D\gamma(t)$. In which case, by \autoref{proposition:lebesgue-stieltjes-differentiable} and the \hyperref[Fundamental Theorem of Calculus]{theorem:ftc-riemann},
    \begin{align*}
    \int_\gamma D_\sigma f &= \int_a^b D_\sigma f (\gamma(t)) \cdot D\gamma(t)dt \\
    &= \int_a^b D(f \circ \gamma)(t) dt =  f(\gamma(b)) - f(\gamma(a))
    \end{align*}
\end{proof}