# Classical Tunneling?

1. Dec 6, 2004

### 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?

2. Dec 6, 2004

### humanino

This is not easy, and maybe should be delayed until you have actually gone beyond QM, and learnt 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.
Consider those two solitonic waves
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.

3. Dec 6, 2004

### humanino

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

4. Dec 8, 2004

### jasonc65

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