Editing Instanton (section)

=== Results ===
Results obtained from the mathematically well-defined Euclidean [[line integral|path integral]] may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the (potentially divergent) Minkowskian path integral. As can be seen from this example, calculating the transition probability for the particle to tunnel through a classically forbidden region (<math>V(x)</math>) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential &minus;''V''(''X'')) in the Euclidean path integral (pictorially speaking &ndash; in the Euclidean picture &ndash; this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an ''instanton''. In this example, the two "vacua" (i.e. ground states) of the [[double-well potential]], turn into hills in the Euclideanized version of the problem.

Thus, the ''instanton'' field solution of the (Euclidean, i. e., with imaginary time) (1&nbsp;+&nbsp;1)-dimensional field theory &ndash; first quantized quantum mechanical description &ndash; allows to be interpreted as a tunneling effect between the two vacua (ground states &ndash; higher states require periodic instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written

:<math> V(\phi) = \frac{m^4}{2g^2}\left(1 - \frac{g^2\phi^2}{m^2}\right)^2 </math>

the instanton, i.e. solution of

:<math> \frac{d^2\phi}{d\tau^2} = V'(\phi), </math>

(i.e. with energy <math>E_{cl} = 0</math>), is

:<math> \phi_c(\tau) = \frac{m}{g}\tanh\left[m(\tau - \tau_0)\right],</math>

where <math>\tau = it</math> is the Euclidean time.

''Note'' that a naïve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this ''non-perturbative tunneling effect'', dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential (cf. [[Mathieu function]]) or other periodic potentials (cf. e.g. [[Lamé function]] and [[spheroidal wave function]]) and irrespective of whether one uses the Schrödinger equation or the [[Functional integration|path integral]].<ref>H.J.W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific, 2012, {{ISBN|978-981-4397-73-5}}.</ref>

Therefore, the perturbative approach may not completely describe the vacuum structure of a physical system. This may have important consequences, for example, in the theory of [[axion|"axions"]] where the non-trivial QCD vacuum effects (like the ''instantons'') spoil the [[Peccei–Quinn theory|Peccei–Quinn symmetry]] explicitly and transform massless [[Nambu–Goldstone boson]]s into massive [[Chiral symmetry breaking|pseudo-Nambu–Goldstone ones]].