Classical Tunneling: Instantons Explained

[scariari]

Im learning about quantum tunneling and read something about that there are classical solutions at imaginary times, so called instantons? Can anyone help me out with this connection?

 

[humanino]

This is not easy, and maybe should be delayed until you have actually gone beyond QM, and learned a good deal of QFT.

When they say "imaginary times" it refers to the fact that those solutions are calculated in Euclidean spacetime : this is just ordinary metric 4-dim space. This is NOT Minkowski space, which is the usual way to think about vacuum (at least in classical terms).

Do you know what a soliton is ?
A soliton typically interpolates between different vacua at infinity.
http://www.maths.surrey.ac.uk/research/Geom/sg.gif
they can go through each other.
Now imagine there is only one of those around : at infinity on the left, there is not the same winding number than at infinity on the right.
An instanton is just a soliton in 4-dim space.
Actually, instantons change the winding number of the vacuum too. It is just less easy to picture in the case.

The reason we call this a tunnel effect, is that there is an energy barrier between the two vacua states.

 

[humanino]

and by the way : the avatar I am using is the formula which gives the winding number of the F field

 

[jasonc65]

When they say "imaginary times" it refers to the fact that those solutions are calculated in Euclidean spacetime : this is just ordinary metric 4-dim space. This is NOT Minkowski space, which is the usual way to think about vacuum (at least in classical terms).

You mean where ds^2 = dx^2 + dy^2 + dz^2 + dt^2?