Wiener Measure versus Feynman Measure

Wiener measure is a countably additive probability measure on continuous paths. The free real-time Feynman kernel is an oscillatory distribution and does not provide an analogous finite complex measure with the usual coordinate marginals. This difference concerns the meaning of integration, not the usefulness of either representation. The elementary time-slicing construction belongs to Time Slicing.

Required background. Measure Theory Minimum supplies probability measures and dominated convergence. Free-Particle Path Integral fixes the free real-time kernel.

Gaussian increments define a continuous path law

Let $W_t$ be standard real Brownian motion with $W_0=0$ and covariance $\mathbb E(W_sW_t)=\min(s,t)$. For $D>0$, set $X_t=x+\sqrt{2D}\,W_t$. Its transition density is

$$p_t(y-x)=(4\pi Dt)^{-1/2} \exp\left[-\frac{(y-x)^2}{4Dt}\right],\qquad t>0.$$

For ordered times $0<t_1<\cdots<t_n$, the joint density is the product

$$p_{t_1}(x_1-x)\prod_{j=2}^n p_{t_j-t_{j-1}}(x_j-x_{j-1}).$$

These are nonnegative normalized densities. Integrating out an intermediate coordinate gives the coarser density by the Gaussian convolution law. Their consistency supplies a process law, and Gaussian moment estimates supply a continuous version: for example, $\mathbb E|X_t-X_s|^4=12D^2|t-s|^2$. This yields Wiener measure on $C([0,T])$ with fixed initial position.

The measure is a law of paths, not a product of Lebesgue measures over an uncountable set of times. The limiting continuity statement is part of its construction; consistent finite-dimensional data alone would not establish concentration on continuous functions.

Typical paths do not have finite classical kinetic action

On a uniform partition with $n$ intervals, define

$$Q_n=\sum_{j=1}^n \left(W_{jT/n}-W_{(j-1)T/n}\right)^2.$$

Independent Gaussian increments give

$$\mathbb E Q_n=T,\qquad \operatorname{Var}Q_n=\frac{2T^2}{n}.$$

Thus $Q_n\to T$ in $L^2$. Along dyadic partitions, the variance bounds are summable; Chebyshev’s inequality and Borel–Cantelli give almost-sure convergence. For $X$, the corresponding limit is $2DT$.

By contrast, if a path $h$ is absolutely continuous with $\dot h\in L^2(0,T)$, Cauchy–Schwarz gives

$$\sum_j|h(t_j)-h(t_{j-1})|^2 \leq\max_j(t_j-t_{j-1}) \int_0^T|\dot h(s)|^2\,ds\longrightarrow0.$$

The Brownian law therefore assigns probability zero to these finite-kinetic-action paths. Writing a Euclidean weight with $\int\dot q^2$ cannot mean evaluating that classical expression on almost every Brownian trajectory. The Gaussian reference law already encodes the kinetic term.

The free real-time kernel has infinite total variation

In one spatial dimension, the free kernel at $t>0$ is

$$K_t(y,x)=\left(\frac{m}{2\pi i\hbar t}\right)^{1/2} \exp\left[\frac{im(y-x)^2}{2\hbar t}\right].$$

Its modulus is a nonzero constant in $y$, so

$$\int_{\mathbb R}|K_t(y,x)|\,dy=\infty.$$

Suppose a finite complex measure on paths starting at $x$ had this kernel as its endpoint marginal, even just in the distributional sense. Pushforward cannot increase total variation, so its endpoint marginal would be a finite complex measure. A measure with distributional density $K_t(\cdot,x)$ would instead have the infinite variation computed above. This is a contradiction.

The conclusion excludes this specific ordinary finite-measure interpretation. It does not exclude distribution-valued constructions, oscillatory time-slicing limits, analytic continuation, or more specialized integration theories. Oscillatory Integrals develops one such interpretation at finite dimension.

Brownian bridges condition endpoints without losing positivity

For a fixed final point $y$, the Brownian bridge from $x$ to $y$ over time $T$ has mean $x+(y-x)t/T$ and covariance

$$2D\left(\min(s,t)-\frac{st}{T}\right).$$

It is the conditional Gaussian law used in heat-kernel formulas. The conditioning is defined through regular conditional distributions or the explicit Gaussian bridge construction; it is not division by the probability of the zero-probability event $X_T=y$.

The Feynman–Kac Formula multiplies this reference law by a potential weight. The free density $p_T(y-x)$ remains outside the bridge expectation and carries the endpoint normalization.

Exercise: conditioning does not make paths smooth

Represent a centred bridge by $B_t=W_t-(t/T)W_T$. Determine its dyadic quadratic variation.

Solution

On a uniform partition, direct expansion gives $\sum_j(\Delta B_j)^2=Q_n-W_T^2/n$. The second term tends to zero almost surely, while $Q_{2^k}\to T$ almost surely. Thus the bridge has the same quadratic variation $T$ as Brownian motion, or $2DT$ after diffusion scaling. Fixing both endpoints does not turn typical paths into finite-action classical trajectories.

References

• Simon, Barry, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, 2005, doi:10.1090/chel/351 — Wiener measure, bridges, and quantum semigroups.
• Wiener, Norbert, “Differential-Space,” Journal of Mathematics and Physics 2, 131–174 (1923), doi:10.1002/sapm192321131 — the original construction of integration on continuous path space.