Path-dependent Time Transformations


In Lorentzian signature:

\begin{align} H = 12 \pi^{2} \left[ -a \dot a^{2} + V(a)] \end{align}

where $V(a) = - a + \frac{\Lambda}{3}a^{3} \right$. For the Steigl/Hinterleitner factor orderings, this promotes to the quantum operator

\begin{align} \begin{array}{l c l} $\hat H$ & = & $12 \pi^{2} \left[ \hbar^{2} a^{-i} \partial_{a} a^{-j} \partial_{a} a^{-k} + V(a) \right]$ \\ \ & = & $12 \pi^{2} \left\{ \hbar^{2} \left[ a^{-1} \partial_{a}^{2} - (j+2k)a^{-2} \partial_{a} +k(j+k+1)a^{-3} \right] + V(a) \right\} \\ \ & = & $12 \pi^{2} \left\{ \hbar^{2} \left[ a^{-1} \partial_{a}^{2} - (j+2k)a^{-2} \partial_{a} \left] \ + \ \right[ \hbar^{2}k(j+k+1)a^{-3} + V(a) \right\} \right] $ \end{align}

To denote the fact that we have divided the terms into a new effective kinetic term and an effective potential including a quantum potential term, define $\hat T_{m} = 12 \pi^{2} \hbar^{2} \left[ a^{-1} \partial_{a}^{2} - (j+2k)a^{-2} \partial_{a} \left]$ and $V_{QP}(a) = 12 \pi^{2} \left[ \hbar^{2}k(j+k+1)a^{-3} + V(a) \right]$. Note that $\hat T_{m}$ can be rewritten as $a^{-1+2(j+2k)} 12 \pi^{2} \hbar^{2} \left[ a^{-(j+2k)} \partial_{a} a^{-(j+2k)} \partial_{a} \left]$, so that the portion within brackets is the Laplace-Beltrami operator for the 1-dimensional metric $g(a) = \left( a^{2(j+2k)} \right)$.

Thus the Euclidean Schrödinger equation corresponding to $\hat H$ can be written as

\begin{align} \left( \frac{\partial}{\partial t} - 12 \pi^{2} a^{-1+2(j+2k)} \left[ \hbar^{2} a^{-(j+2k)} \partial_{a} a^{-(j+2k)} \partial_{a} +a^{1-2(j+2k)}V_{QP}(a) \right] \right) u(a,t) = 0; \end{align}

we are looking for the propagator $K(a, a', t)$ for this parabolic partial differential equation.

If we were to transform to a time variable $s$ satisfying $\frac{ds}{dt} = a^{-1 +2(j+2k)}$ so that $\frac{\partial}{\partial t} = a^{-1+2(j+2k)} \frac{\partial}{\partial s}$, then we would be in effect looking for the propagator $\tilde K (a, a', s)$ for

\begin{align} \left( \frac{\partial}{\partial s} - 12 \pi^{2} \left[ \hbar^{2} a^{-(j+2k)} \partial_{a} a^{-(j+2k)} \partial_{a} +a^{1-2(j+2k)}V_{QP}(a) \right] \right) u(a,t) = 0 \end{align}

which can now be done using a Wick-rotated version of Parker's curved-space path integral.

However the problem is now how to relate a path integral expression for the propagator of 3 with that of 4, because the time transformation $t \rightarrow s$ is path-dependent:

\begin{align} s = \int_{0}^{t} a^{-1+2(j+2k))} \left( t' \right) \ dt' \end{align}

so that if we naively transform $K \left( a, a', t \right) \rightarrow \tilde K \left( a, a', t \right)$, the $s$ appearing as a dependent variable in $\tilde K$ would not be a single value: for each path $a(t),$a different end-time $s$ corresponds to the original end-time $t.$

For propagators that operate w.r.t. the Lebesgue measure, we can relate the original path integral expression to a time-transformed one, as described in Fischer, Leschke, and Muller.

Path-dependent time transformations as done in "Changing dimension and time: two well-founded and practical techniques for path integration in quantum physics," Fischer et al

Outline for applying time transformations to path integrals for the Steigl/Hinterleitner operator orderings

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