Monday, December 6, 2010

Tunneling without instantons?

This attempts to answer questions raised in a previous post.
Here is one point of view.
Tunneling is always present and as one lowers the temperature (or increases the coupling to the environment) one just has a crossover from transitions dominated by activation over the barrier to tunneling under the barrier. Instantons [or the "bounce solution" which is a solution to the classical equations of motion in an inverted potential] are just a convenient calculational machinery which arises when evaluating a path integral approximately by finding saddle points. There is always a contribution from the trivial solution corresponding to the top of the barrier. Quadratic fluctuations about this saddle point give a "prefactor" which includes quantum corrections due to tunneling and reflection.
Below the crossover temperature T0 this first saddle point becomes unstable and there is  a second saddle point, which is the instanton solution.
I thank Eli Pollak for sharing his thoughts on this subject.
But all this seems against the spirit of the approach to tunneling in dissipative environments, pioneered by Leggett [and reviewed in detail here], which seems to assert that tunneling only exists when instanton solutions are present.
Perhaps, the key distinction is that the instanton captures coherent tunneling whereas the quadratic fluctuations only capture incoherent tunneling. Specifically, if one considers a double well system, the instanton can capture the level splitting associated with tunneling.

I am keen to hear others perspectives.

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