Description of tunneling phenomena with Euclidean methods

In this work we investigate quantum mechanical tunneling phenomenon with Euclidean methods. Quantum tunneling was developed from the study of radioactivity. Friedrich Hund was the first to take notice of tunneling in 1927. Its first application was a mathematical explanation for alpha decay, which was done in 1928 by George Gamow. Tunneling refers to the quantum mechanical phenomenon where a lower energy particle tunnels through a higher potential barrier that it classically could not surmount. Alpha decay is example of this activity. A standard application method for tunneling is Schrodinger's equation. In modern works same problems are explained by functional integral. tunneling amplitude is defined by finctional integral and it's calculation is possible by method of saddle in so-called Euclidean time. For transfering in Euclidean time we need complex transformation ($t\rightarrow -i\tau$) and it calls Wick rotation. Tunneling amplitude is defined by Euclidean action $S_E$, which is integral over classical solution (instanton). Euclidean methods are convenient because it makes easier to generalize in field theory. Euclidean method is described by instanton, which has a great use in quantum field theory. We will consider S.Coleman's method \cite{Coleman:1977py,Coleman:1977,Coleman:1978ae} of approach about tunneling in quantum mechanics.

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