What does "conjugate variable" encompass, and how does it do this?

Question

Context for the question: I am studying Legendre transforms in statistical mechanics. I have read that the Legendre transform allows us to transform between conjugate variables in expressions. For example, if I begin with $E(S,V,N)$, a Legendre transform can get me to $E(T,V,N)$.

The confusion come in when the term "conjugate variables" is used. I have seen three examples of conjugate variables:

1. In statistical mechanics, entities related by Legendre transforms are conjugate variables (e.g. T & S, P & V, and $\mu$ & N).
2. In quantum mechanics, canonical conjugate variables are pairs a pair of variables which generate the displacement of each other (e.g. x and p).
3. In classical mechanics the derivatives of action are conjugate variables to the quantity with respect to which one is differentiating.

My question is how are all of these different relations encompassed by the term "conjugate variable"? I am wondering how I can know if I can Legendre transform between two given variables in statistical mechanics, but this would require identifying terms of conjugate variables, and I am unsure how to do this. Thus, I am seeking clarity on what this relation "being conjugate variables" truly means and why it seems to manifest across so many different forms (e.g. differentiation, commutation, legendre transform, etc.).

There are some discussions here and here, but I seek a more physically motivated understanding than is currently presented in the answers. I am not able to make sense of the existing answers to the second link, and the first link evades the specific question I am asking, as it does not touch the topic of statistical mechanics. Existing answers also do not discuss the unification of all uses of "conjugate variable" under this single term.

Answer 1

The most simple-minded answer is that $(X, Y)$ are conjugate if some fundamental quantity $\mathcal{Q}$ in your theory obeys $d \mathcal{Q} = X \, dY + \ldots$, or in other words, if $X = \partial \mathcal{Q} / \partial Y$. That's what gives you all of the other properties in each case.

Statistical mechanics: the fundamental quantity is $U$. If $dU \supset X \, dY$, then the function $U' = U - XY$ will obey $dU' \supset - Y \, dX$, i.e. we can Legendre transform between $X$ and $Y$.

Classical mechanics: the fundamental quantity is the on-shell action $S(q, t)$, which obeys $dS \supset - H \, dt + p \, dq$. By considering second derivatives of $S$, this implies that $(q, p)$ satisfy Hamilton's equations of motion.

Quantum mechanics: the action becomes the phase $e^{iS/\hbar}$, and the fact that $dS \supset p \, dq$ means that $p$ is the rate of change of phase as $q$ varies, which means that on wavefunctions $p$ acts like $- i \hbar \partial_q$, which means that $[q, p] = i \hbar$, which implies that $q$ and $p$ generate displacements of each other and obey an uncertainty relation.

Answer 2

What you are encountering is a more general notion of duality in the mathematical formalism of physical theory. The term "conjugate variable" is probably used more loosely than some would like, but fundamentally it is used to describe duality between two variables. In principle variables that are duals of each other can be used to describe the same object and are thus relatable and transformable from one basis to another.

The easiest way to understand duality is to consider any function that is plotted on a standard x-y plot where x is the "domain" and y is the "range". We generally consider the function as taking the value of the domain and mapping it to a value in the range, however there is a "dual" representation that maps the range to the domain with the use of an "inverse function".

We are not used to thinking this way because we are often taught without any consideration of "duality" being a fundamental attribute of most of mathematics. The use of "conjugate" is a little more descriptive than "dual" but still just means that there is a pairing between two variables. The details of the pairing depends on specific mathematical context and use which can be confusing when there is sloppy descriptions.

If you think of complex numbers, every complex number has a conjugate which simply describes the change in sign of the imaginary component. The importance becomes apparent when you multiply a complex number by its conjugate and you get a real number that is the sum of the squares of the real and imaginary parts.

The point of this is that just because people use similar terminology to decribe a concept doesn't mean the concepts are immediately relatable but there may be a justification/motivation for the use of the terminology at a slighly deeper level.