13.5 Path Integrals
Definition 13.5.1 (Rectifiable Path).label 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 rectifiable if $\gamma \in BV([a, b]; F)$.
Definition 13.5.2 (Path Integral).label 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 path-integrable with respect to $\gamma$ if $f \circ \gamma \in RS([a, b], \gamma; E)$. In which case,
is the 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 $\gamma$.
Proposition 13.5.3 (Change of Variables).label 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
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,
Therefore if $f \in PI([a, b], \gamma; E)$, then $f \in PI([c, d], \gamma \circ \varphi; E)$, with $\int_{\gamma} f = \int_{\gamma \circ \varphi}f$.$\square$
Definition 13.5.4 (Curve).label 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 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 change of parameter between $\gamma$ and $\mu$.
A curve in $F$ is then an equivalence class of paths.
Lemma 13.5.5.label 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:
- (1)
$\Gamma(a) = \gamma(a)$ and $\Gamma(b) = \gamma(b)$.
- (2)
$\braks{\int_\gamma f - \int_\Gamma f}_{F} < \epsilon$.
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 Proposition 13.4.3. Given that $\gamma$ is of bounded variation, there exists $(P = \seqfz{x_j}, c) \in \scp_{t}([a, b])$ such that:
- (a)
For each $1 \le j \le n$,
\[\gamma([x_{j-1}, x_{j}]) \subset \bracs{y \in F|[y - x_{j-1}]_F \le 1}\] - (b)
$\braks{\int_\gamma f - S(P, c, f \circ \gamma, \gamma)}_{G} < \epsilon$.
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 $(Q, d) \ge (P, c)$,
As $\Gamma$ is also of bounded variation, $f \in PI([a, b], \Gamma; E)$. Since the above holds for all refinements of $(Q, d)$,
$\square$
Theorem 13.5.6 (Fundamental Theorem of Calculus for Path Integrals).label 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 $f \in C^{1}_{\sigma}(U; E)$,
In particular, if $\gamma(a) = \gamma(b)$, then $\int_{\gamma} D_{\sigma} f = 0$.
Proof. Using Lemma 13.5.5, assume without loss of generality that $\gamma$ is piecewise smooth. By the Chain Rule, $f \circ \gamma \in C^{1}([a, b]; F)$ with $D(f \circ \gamma)(t) = Df(\gamma(t)) \cdot D\gamma(t)$. In which case, by Proposition 13.7.2 and the Fundamental Theorem of Calculus,
$\square$