First of all, I am having a hard time finding any good definition of what a conjugate pair actually is in terms of physical variables, and yet I have read a number of different things which use the fact that two variables are a conjugate pair to justify things.
For instance, every once in a while I'll see something which will say something along the line of, "we know the commutator of these variables is non-zero because they are a conjugate pair."
Also, I have gathered that position and momentum are a conjugate pair as are energy and time. I have seen once that the imaginary and real components of an electric field are a conjugate pair.
So, it seems that if I took two variables and their quantum mechanical operators do not commute, they are a conjugate pair. This doesn't feel quite right to me though... Or at least that's a lame definition cause it doesn't speak to classical anything.
So, that brings me to two questions:
1. How does one define a conjugate pair of variables?
2. What important roles do conjugate pairs play in both classical and quantum mechanics?