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===Derivatives of action===
In [[classical physics]], the derivatives of [[action (physics)|action]] are conjugate variables to the quantity with respect to which one is differentiating. In quantum mechanics, these same pairs of variables are related by the Heisenberg [[uncertainty principle]].
* The ''[[energy]]'' of a particle at a certain [[event (relativity)|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 (vector)|position]]''.
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===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 <math> \left (x \right) </math> and momentum <math> \left (p \right) </math>. In the quantum
: <math>[\widehat{x},\widehat{p\,}]=\widehat{x}\widehat{p\,}-\widehat{p\,}\widehat{x}=i \hbar</math>
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: <math> {\sigma_A}^2 {\sigma_B}^2 \geq \left (\frac{1}{2i} \left \langle \left [ \widehat{A},\widehat{B} \right ] \right \rangle \right)^2 </math>
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]] <math>\mathfrak h_3</math>, with a corresponding group called the Heisenberg group <math> H_3 </math>.
===Fluid mechanics===
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