Question

What is the physical significance of 2 operators to anti commute?

Answer 1

Actually if two operators commute, there is a clear physical interpretation - they are both elements of a complete set of commuting observables - i.e. you can make a measurement on the system of both observables in any order and also simultaneously - e.g. L - the total angular momentum and Lz the z component. If two operators do not commute any measurement made of one of these with a certain accuracy will result in an uncertainty of the expectation value of the second operator - e.g. position and momentum