Weak Localization: Explaining the Argument

[aaaa202]

A very handwaving argument for weak localization is the following:
In a conductor the electron can take many paths leading back to its origin. Consider two of these, being the time reversed of each other and denote them +,-. The classical return probability is:
P_classical = A₊² + A_²
While the quantum mechanical (due to constructive interference) is twice this:
P_quantum = lA₊exp(iθ)+A_-exp(iθ)l²

The problem I see with this argument is this: Would a path in general not also interfere with other paths besides its own time reversed path? Am I misunderstanding or how exactly is one to interpret this argument.

 

[Jilang]

Are you asking if it interferes with the paths of other electrons?
Or are you asking if the path interferes with other possible paths of the same electron?

 

[aaaa202]

The last one. Why is only the time reversed path for the same electron relevant.

 

[Jilang]

All possible paths of the electron will interfere, some constructively and some destructively. If there were no loops on average the constructive intereferences would be equal to the destructive intereferences and the classical result would be achieved. However if you have possible loops the each way paths round the loops will always interfere constructively. This means the electrons have slightly more chance of staying in the same place than moving somewhere else. The incidence of these loops is higher in lower dimensions so the effect is more noticeable in thin wires and films.

 

[aaaa202]

But my question was. Will the paths from different loops not interfere and average out the contribution from the loops?

 

[Jilang]

Sure, but the paths from two different loops with have an equal chance of interfering constructively or destructively.