15.5 Path Integrals
Definition 15.5.1 (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 15.5.2 (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 15.5.3 (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 15.5.4 (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 15.5.5.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 $U \in \cn_{F}(\gamma([a, b]))$.
For each $P \in \scp([a, b])$, let $\Gamma_{P} \in C([a, b]; F)$ be the piecewise linear path obtained by interpolating values of $\gamma$ at points of $P$, then for any continuous seminorm $[\cdot]_{G}: G \to [0, \infty)$, $\eps > 0$, and $f \in C(U; E) \cap PI([a, b], \gamma; E)$, there exists $P \in \scp([a, b])$ such that for any $Q \in \scp([a, b])$ with $Q \ge P$,
- (1)
$\Gamma_{P}(a) = \gamma(a)$ and $\Gamma_{P}(b) = \gamma(b)$.
- (2)
$\braks{\int_\gamma f - \int_{\Gamma_P} f}_{F} < \epsilon$.
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 15.4.3. Given that $\gamma$ is continuous, there exists $(P_{0}, c_{0}) \in \scp_{t}([a, b])$ such that for any $(P = \seqfz{x_j}, c) \in \scp_{t}([a, b])$ with
- (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 $\Gamma = \Gamma_{P}$, then 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 $(Q, d)$,
$\square$
Remark 15.5.1.label Past me made the mistake of believing that in Lemma 15.5.5, it is possible to approximate rectifiable curves with piecewise linear curves in 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.
Lemma 15.5.6.label Let $[a, b] \subset \real$, $E$ be a separated locally convex space over $K \in \RC$, $F$ be a Banach space over $K$, $H$ be a complete locally convex space over $K$, all 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 piecewise $C^{1}$ curve that is constant on $[a, a + \eps)$ and $(b - \eps, b]$, and $U \in \cn_{F}(\gamma([a, b]))$.
Extend $\gamma$ to $\real$ by
For each $\varphi \in C_{c}^{\infty}(\real; \real)$ with $\int_{\real} \varphi = 1$ and $t > 0$, let
then
- (1)
For each $t > 0$, $\gamma_{t} \in C^{\infty}([a, b]; F)$.
- (2)
There exists $t > 0$ such that for any $s \in (0, t)$, $\gamma_{s}(a) = \gamma(a)$ and $\gamma_{s}(b) = \gamma(b)$.
- (3)
For any $f \in C(U; E)$,
\[\int_{\gamma} f = \lim_{t \downto 0}\int_{\gamma_t}f\]
Proof. (1): By the Mean Value Theorem, for each $x, y \in [a, b]$,
By the Dominated Convergence Theorem, $\gamma_{t} \in C^{\infty}([a, b]; F)$.
(2): For sufficiently small $t$, $\supp{\varphi}\subset (-\eps, \eps)$. In which case, by assumption, $\gamma_{t}(a) = \gamma(a)$ and $\gamma_{t}(b) = \gamma(b)$.
(3): Since $\gamma$ is piecewise $C^{1}$ and $\gamma_{t} \in C^{\infty}([a, b]; F)$,
by the Dominated Convergence Theorem.$\square$
Theorem 15.5.7 (Fundamental Theorem of Calculus for Path Integrals).label Let $[a, b] \subset \real$, $E, F$ be separated locally convex spaces, $\sigma \subset \mathfrak{B}(F)$ be an ideal containing all compact sets, $\gamma \in C([a, b]; F)$ be a rectifiable path, and $U \in \cn_{F}(\gamma([a, b]))$, 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 15.5.5, assume without loss of generality that $\gamma$ is piecewise smooth. By the Chain Rule, $f \circ \gamma$ is piecewise $C^{1}$ with $D(f \circ \gamma)(t) = Df(\gamma(t)) \cdot D\gamma(t)$ on all but finitely many points. In which case, by change of variables formula and the Fundamental Theorem of Calculus,
$\square$
Post a Comment