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arXiv · 1807.07007

Path integrals with discarded degrees of freedom

Luke M. Butcher
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Abstract

Whenever variables $ϕ=(ϕ^1,ϕ^2,\ldots)$ are discarded from a system, and the discarded information capacity $\mathcal{S}(x)$ depends on the value of an observable $x$, a quantum correction $ΔV_\mathrm{eff}(x)$ appears in the effective potential [arXiv:1707.05789]. Here I examine the origins and implications of $ΔV_\mathrm{eff}$ within the path integral, which I construct using Synge's world function. I show that the $ϕ$ 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 $ΔV_\mathrm{eff}$ as previously derived. This generalises the prior results beyond the limits of the Schrödinger 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 $ΔV_\mathrm{eff}$.

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Luke M. Butcher. 2019-02-14. Path integrals with discarded degrees of freedom. https://doi.org/10.1016/j.physleta.2018.12.002

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