Systematically Accelerated Convergence of Path Integrals

We present a new analytical method that systematically improves the convergence of path integrals of a generic $N$-fold discretized theory. Using it we calculate the effective actions $S^{(p)}$ for $p\le 9$, which lead to the same continuum amplitudes as the starting action, but that converge to that continuum limit as $1/N^p$. We checked this derived speedup in convergence by performing Monte Carlo simulations on several different models.

Figure 1

Deviations from the continuum limit $|A_N^{(p)}-A|$ as functions of $N$ for $p=1$, 2, 4, and 6 for an anharmonic oscillator with quadratic coupling $\lambda =10$, time of propagation $T=1$ from $a=0$ to $b=1$. $N_{\mathrm{MC}}$ was $9.2\times {10}^9$ for $p=1$, 2, $9.2\times {10}^{10}$ for $p=4$, and $3.68\times {10}^{11}$ for $p=6$. Dashed lines correspond to appropriate $1/N$ polynomial fits to the data. Solid lines give the leading $1/N$ behavior. The level $p$ curve has a $1/N^p$ leading behavior.