There are many types of conjugate variables, depending on the type of work a certain system is doing. Examples of canonically conjugate variables include the following:
Period and frequency: the longer a musical note is sustained, the more precisely we know its frequency. Conversely, a very short musical note becomes just a click, and so one can't determine its frequency very accurately.
The energy of a particle at a certain event is the negative of the derivative of the action along a trajectory of that particle ending at that event with respect to the time of the event.
The linear momentum of a particle is the derivative of its action with respect to its position.
The angular momentum of a particle is the derivative of its action with respect to its orientation.
The mass-moment of a particle is the derivative of its action with respect to its rapidity.
The electric potential at an event is the negative of the derivative of the action of the electromagnetic field with respect to the density of electric charge at that event.
The magnetic potential at an event is the derivative of the action of the electromagnetic field with respect to the density of electric current at that event.
The electric field at an event is the derivative of the action of the electromagnetic field with respect to the electric polarization density at that event.
The magnetic induction at an event is the derivative of the action of the electromagnetic field with respect to the magnetization at that event.
The Newtonian gravitational potential at an event is the negative of the derivative of the action of the Newtonian gravitation field with respect to the mass density at that event.
Quantum theory
In quantum mechanics, conjugate variables are realized as pairs of observables whose operators do not commute. In conventional terminology, they are said to be incompatible observables. Consider, as an example, the measurable quantities given by position and momentum. In the quantum mechanical formalism, the two observables and correspond to operators and, which necessarily satisfy the canonical commutation relation: For every non-zero commutator of two operators, there exists an "uncertainty principle", which in our present example may be expressed in the form: In this ill-defined notation, and denote "uncertainty" in the simultaneous specification of and. A more precise, and statistically complete, statement involving the standard deviation reads: More generally, for any two observables and corresponding to operators and, the generalized uncertainty principle is given by: Now suppose we were to explicitly define two particular operators, assigning each a specific mathematical form, such that the pair satisfies the aforementioned commutation relation. It's important to remember that our particular "choice" of operators would merely reflect one of many equivalent, or isomorphic, representations of the general algebraic structure that fundamentally characterizes quantum mechanics. The generalization is provided formally by the Heisenberg Lie algebra, with a corresponding group called the Heisenberg group.
