Path-dependent Time Transformations

Main

In Lorentzian signature:

(1)
\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

(2)
\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$\hat Hcan be written as (3) \begin{align} \left( \frac{\partial}{\partial t} - 12 \pi^{2} a^{-1+2(j+2k)} \left[ \hbar^{2} a^{-(j+2k)} \partial_{a} a^{-(j+2k)} \partial_{a} +a^{1-2(j+2k)}V_{QP}(a) \right] \right) u(a,t) = 0; \end{align} we are looking for the propagatorK(a, a', t)$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$\tilde K (a, a', s)for (4) \begin{align} \left( \frac{\partial}{\partial s} - 12 \pi^{2} \left[ \hbar^{2} a^{-(j+2k)} \partial_{a} a^{-(j+2k)} \partial_{a} +a^{1-2(j+2k)}V_{QP}(a) \right] \right) u(a,t) = 0 \end{align} which can now be done using a Wick-rotated version of Parker's curved-space path integral. However the problem is now how to relate a path integral expression for the propagator of 3 with that of 4, because the time transformationt \rightarrow sis path-dependent: (5) \begin{align} s = \int_{0}^{t} a^{-1+2(j+2k))} \left( t' \right) \ dt' \end{align} so that if we naively transformK \left( a, a', t \right) \rightarrow \tilde K \left( a, a', t \right)$, 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.

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

page revision: 34, last edited: 10 Sep 2009 18:10