有限尺寸量子隧穿的瞬时方法
Instantonic Methods for Quantum Tunneling in Finite Size
摘要：
在渐进极限中重新检验了$ϕ^4$模型势的瞬时方法。在半经典近似下，将经典背景下的作用量展开，导出了系统的路径积分。它示出的奇异性的路径积分，所产生的量子涨落谱中的零模式，可以处理只有假设一个{ \it有限}（虽然大）系统的大小。另一方面，标准的瞬变方法假设（反）扭结作为经典背景。但（反）扭结是{ \it无限}尺寸系统的欧拉-拉格朗日方程的解。这种形式上的矛盾可以通过基于Jacobi椭圆函数形式主义的有限尺寸瞬变理论来一致地克服。对于后者，详细推导了求解有限尺寸欧拉-拉格朗日方程的经典背景，得到了有限尺寸下的广义路径积分。问题和解决方案都为数学和物理之间富有成效的相互作用提供了一个有启发性的例子。{ \bf Keywords：路径积分方法，有限尺寸系统，瞬子}
英文摘要：
The instantonic approach for a $ϕ^4$ model potential is reexamined in the asymptotic limit. The path integral of the system is derived in semiclassical approximation expanding the action around the classical background. It is shown that the singularity in the path integral, arising from the zero mode in the quantum fluctuation spectrum, can be tackled only assuming a {\it finite} (although large) system size. On the other hand the standard instantonic method assumes the (anti)kink as classical background. But the (anti)kink is the solution of the Euler-Lagrange equation for the {\it infinite} size system. This formal contradiction can be consistently overcome by the finite size instantonic theory based on the Jacobi elliptic functions formalism. In terms of the latter I derive in detail the classical background which solves the finite size Euler-Lagrange equation and obtain the general path integral in finite size. Both problem and solution offer an instructive example of fruitful interaction between mathematics and physics. {\bf Keywords: Path Integral Methods, Finite Size Systems, Instantons}
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