Recently there has been increasing interest in alternate methods to compute quantum tunneling in field theory. Of particular interest is a stochastic approach which involves (i) sampling from the free theory Gaussian approximation to the Wigner distribution in order to obtain stochastic initial conditions for the field and momentum conjugate, then (ii) evolving under the classical field equations of motion, which leads to random bubble formation. Previous work showed parametric agreement between the logarithm of the tunneling rate in this stochastic approach and the usual instanton approximation. However, recent work claimed excellent agreement between these methods. Here we show that this approach does not in fact match precisely; the stochastic method tends to overpredict the instanton tunneling rate. To quantify this, we parameterize the standard deviations in the initial stochastic fluctuations by $\epsilon \sigma$, where $\sigma$ is the actual standard deviation of the Gaussian distribution and $\epsilon$ is a fudge factor; $\epsilon =1$ is the physical value. We numerically implement the stochastic approach to obtain the bubble formation rate for a range of potentials in 1+1-dimensions, finding that $\epsilon$ always needs to be somewhat smaller than unity to suppress the otherwise much larger stochastic rates towards the instanton rates; for example, in the potential of one needs $\epsilon \approx 1/2$. We find that a mismatch in predictions also occurs when sampling from other Wigner distributions, and in single particle quantum mechanics even when the initial quantum system is prepared in an exact Gaussian state. If the goal is to obtain agreement between the two methods, our results show that the stochastic approach would be useful if a prescription to specify optimal fudge factors for fluctuations can be developed.

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量子隧穿随机方法的定量分析

最近，人们对场论中计算量子隧穿的替代方法越来越感兴趣。特别令人感兴趣的是一种随机方法，该方法包括（i）从自由理论高斯近似采样到Wigner分布，以获得场和动量共轭的随机初始条件，然后（ii）在经典的运动场方程下演化，导致随机气泡的形成。先前的工作表明，在这种随机方法中，隧穿速率的对数与通常的瞬时近似值之间存在参数一致性。但是，最近的工作声称这些方法之间有很好的一致性。在这里，我们表明这种方法实际上并不完全匹配。随机方法往往会过高估计瞬时隧穿率。为了量化这一点，$\epsilon \sigma$，在哪里 $\sigma$ 是高斯分布的实际标准偏差， $\epsilon$ 是一个软糖因素； $\epsilon =\mathrm{1个}$是物理价值。我们用数值方法实现了随机方法，以获取1 + 1维范围内的一系列势能的气泡形成率，发现$\epsilon$总是需要比统一小一些，以将原本较大的随机率抑制为瞬时率；例如，潜在的一种需求$\epsilon \approx \mathrm{1个}/2$。我们发现，即使从其他Wigner分布中进行采样，甚至在初始量子系统准备为精确的高斯态的情况下，预测中也会发生不匹配。如果目标是要在两种方法之间取得一致，则我们的结果表明，如果可以制定出指定波动的最佳软糖因子的处方，则随机方法将很有用。