Path integrals and the WKB approximation in loop quantum cosmology

We follow the Feynman procedure to obtain a path integral formulation of loop quantum cosmology starting from the Hilbert space framework. Quantum geometry effects modify the weight associated with each path so that the effective measure on the space of paths is different from that used in the Wheeler-DeWitt theory. These differences introduce some conceptual subtleties in arriving at the WKB approximation. But the approximation is well defined and provides intuition for the differences between loop quantum cosmology and the Wheeler-DeWitt theory from a path integral perspective.

Figure 1

For fixed $({\nu }_i,{\varphi }_i)$, the (dashed) curves ${\nu }_f={\nu }_i{e}^{\pm \sqrt{12\pi G}({\varphi }_f-{\varphi }_i)}$ divide the $({\nu }_f,{\varphi }_f)$ plane into four regions. For a final point in the upper or lower quarter, there always exists a real trajectory joining the given initial and final points (as exemplified by the thick line). If the final point lies on the left or right quarter, there is no real solution matching the two points. The action becomes imaginary and one gets an exponentially suppressed amplitude.

Figure 2

Real parts of the exact and WKB amplitudes are plotted as a function of the final volume $n_f≔{\nu }_f/4{\ell }_o\hslash$. The exact amplitude (dots) has support on the “lattice” $({\nu }_f-{\nu }_i)/4{\ell }_o\hslash \in \mathbb{Z}$. At $n_f=n_i{e}^{\sqrt{12\pi G}({\varphi }_f-{\varphi }_i)}=50$ there is the transition from oscillatory to exponential behavior, and the WKB amplitude (solid line) diverges. (It also diverges at $n_f=n_i{e}^{-\sqrt{12\pi G}({\varphi }_f-{\varphi }_i)}=2$.) Here, ${\varphi }_i$, ${\nu }_i$, and ${\varphi }_f$ are kept fixed: ${\varphi }_i=0$, $n_i≔{\nu }_i/4{\ell }_o\hslash =10$, and $\sqrt{12\pi G}{\varphi }_f=\log 5$.

Figure 3

Real parts of the exact and WKB amplitudes are plotted as a function of the final scalar field ${\varphi }_f$ (dots and solid lines, respectively). Here ${\varphi }_i$, ${\nu }_i$, and ${\nu }_f$ are kept fixed: ${\varphi }_i=0$, $n_i≔{\nu }_i/4{\ell }_o\hslash =10$, and $n_f≔{\nu }_f/4{\ell }_o\hslash =20$. The WKB solution is the curve diverging at $\sqrt{12\pi G}{\varphi }_f=\log ({\nu }_f/{\nu }_i)=\log 2$.

Figure 4

Comparison of the exact and WKB amplitudes for initial and final configurations with ${\nu }_f=2{\nu }_i$ as a function of $n_i≔{\nu }_i/4{\ell }_o\hslash$. As $n_i$ increases, the WKB solution (continuous line) becomes closer to the exact solution (dots). (The distance between the amplitudes oscillates, but overall it decreases.) This calculation was done for ${\varphi }_i=0$ and ${\varphi }_f=1/\sqrt{12\pi G}$.