  Whenever variables $\phi=(\phi^1,\phi^2,\ldots)$ are discarded from a system,
and the discarded information capacity $\mathcal{S}(x)$ depends on the value of
an observable $x$, a quantum correction $\Delta V_\mathrm{eff}(x)$ appears in
the effective potential [arXiv:1707.05789]. Here I examine the origins and
implications of $\Delta V_\mathrm{eff}$ within the path integral, which I
construct using Synge's world function. I show that the $\phi$ variables can be
`integrated out' of the path integral, reducing the propagator to a sum of
integrals over observable paths $x(t)$ alone. The phase of each path is equal
to the semiclassical action (divided by $\hbar$) including the same correction
$\Delta V_\mathrm{eff}$ as previously derived. This generalises the prior
results beyond the limits of the Schr\"odinger equation; in particular, it
allows us to consider discarded variables with a history-dependent information
capacity $\mathcal{S}=\mathcal{S}(x,\int^t f(x(t'))\mathrm{d} t')$. History
dependence does not alter the formula for $\Delta V_\mathrm{eff}$.
