Wormholes are considered both from the Wheeler deWitt equation, as well as from the field equations in the Euclidean background of Robertson Walker mini-superspace in $R^2$ gravity. Quantum wormhole satisfies Hawking Page wormhole boundary condition in the Euclidean background of mini-superspace, however, in the Lorentzian background wave functional turns to the usual oscillatory function. The Euclidean field equations for $\kappa =0,\pm 1$ lead to the wormhole configuration, as well as oscillating universe in Euclidean time $\tau$. The oscillating universe in Euclidean time $\tau$ transforms to an expanding universe only for $\kappa =1$ in Lorentz time $t$ under analytic continuation $\tau =it$ and asymptotically leads to an exponential solution. An Euclidean wormhole in the very early era evolves to an oscillating universe in $\tau$, thereafter crossing deSitter radius transition to an inflationary era is evident at later epoch only for $\kappa =1$.

虫洞既可以从惠勒德威特方程考虑，也可以从 $R^2$ 重力中罗伯逊沃克迷你超空间欧几里得背景中的场方程考虑。量子虫洞在微型超空间的欧几里得背景下满足霍金佩奇虫洞边界条件，但在洛伦兹背景下波函数转向通常的振荡函数。 $\kappa =0,\pm 1$ 的欧几里得场方程得出了虫洞结构，以及欧几里得时间 $\tau$ 中的振荡宇宙。欧几里得时间 $\tau$ 中的振荡宇宙在解析延拓 $\tau =it$ 下仅在洛伦兹时间 $t$ 中转变为膨胀宇宙 $\kappa =1$ 并渐近导致到指数解。早期的欧几里得虫洞在 $\tau$ 中演化为振荡宇宙，此后跨越德西特半径过渡到暴胀时代仅在 $\kappa =1$ 的后期时期才明显。