Path integrals with discarded degrees of freedom

Whenever variables phi = (phi(1), phi(2), ...) are discarded from a system, and the discarded information capacity S(x) depends on the value of an observable x, a quantum correction Delta V-eff(x) appears in the effective potential [1]. Here I examine the origins and implications of Delta V-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 (h) over bar) including the same correction Delta V(eff )as previously derived. This generalises the prior results beyond the limits of the Schrodinger equation; in particular, it allows us to consider discarded variables with a history-dependent information capacity S = S(x, integral(t) f (x(t'))dt'). History dependence does not alter the formula for Delta V-eff. (C) 2018 Elsevier B.V. All rights reserved.