I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce complex coordinates. My question is: why does the below still hold true?

Normally the Lagrange equations are derived using "real partial" derivatives, but of course the partial derivative with respect to a complex variable (like below) is defined as: 1/2( d/dx - i d/dy). I only have a very basic familiarity with the Hamiltonian-Lagrange formalism, but I guess my question is, is it true that the Hamiltonian equations can be extended to complex conjugate coordinates as seen below? If so, why? This seems to be a topic that is essentially nonexistent in the literature, and everywhere I go to investigate this topic of decoupled oscillators takes it as a given.

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  • $\begingroup$ I think actually, as a potential answer to my own question: When you are taking a complex partial derivative, you are technically taking the derivative of both the real and imaginary part. So these are the actual variables that need to be minimized in the Lagrange equation. 71 is essentially two equations in 1. $\endgroup$ – Electric to be May 2 at 3:57

The Hamiltian formalism works with a set of N coordinates. Since complex-conjugate coordinates $Q_s^*$ are independent from the $Q_s$, the Hamiltonian formalism for N complex coordinates is the same as if you had 2N real coordinates. For example, the harmonic oscillator equation (71) can be written as two independent equation for $Q_s$ and $Q_s^*$.

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  • $\begingroup$ I have a little trouble accepting Qs and Qs* to be independent. I know in some sense they are (via Wirtinger derivatives), but this isn't true "independence". After thinking about it, what you CAN take to be independent is the individual real and imaginary components of Qs. It is with respect to these that you extremize the action. However, the above equation 71 are basically two equations for real and imaginary. But now we have 2N free variables instead of N? What gives? The answer is Q(-k) = Q*. Thus this reduces our total independent variables back to N! $\endgroup$ – Electric to be May 2 at 7:02
  • $\begingroup$ Well, if real and immaginary parts of $Q$ are independent, and $Q$ and $Q*$ are two different combination, they must be independent as well right? Regarding the condition $Q(-k) = Q^*(k)$ I think it is related to your particolar problem only. If it bothers you, you can remove the 1/2 and sum on positive $k$ only. This is just a relabelling of the indeces, and you obtain the independence you wanted ($\frac{d Q Q^*}{dQ*} = Q$) $\endgroup$ – Carlo Cepollaro May 2 at 8:47
  • $\begingroup$ Let me call x the real part and y imaginary. I feel like they are independent because I can fix x and y to be whatever (REAL value) I want. Once I do this, it completely determines Q and Q*. But I can't fix Q and Q* seperately, since Q* by definition is related by Q = x + iy --> Q* = x-iy. Thus really my only two independent variables are x and y (which have to be assumed real!) Once this is done Q and Q* are completely fixed and determined. This is what makes the most sense to me, especially in the context of exterminating the action with respect to both x and y. $\endgroup$ – Electric to be May 2 at 19:40
  • $\begingroup$ I have always intended the independence of $z$ and $z^*$ as the independence of (x,y) and (x,-y): it is true that given (x,y) the vector (x,-y) is fixed, nevertheless they are two independent vectors meaning that their linear combination is zero only if all the coefficients are zero. Their independence is on the real vector space with 2N components, and not on the N complex vector space. Can it be the source of the problem? $\endgroup$ – Carlo Cepollaro May 2 at 20:40
  • $\begingroup$ I think this issue of independence of Q and Q* is a bit tangential to my question. Either way, my original question of why I am able to take a complex derivative in the Lagrange-Euler equation is effectively answered by the fact that we are minimizing the action with respect to the real and imaginary parts when we write partial d/ partial Q. Thus we have effectively made a change of coordinates from the original particle positions, to generalized coordinates of real and imaginary parts of Q, that can be compactly written as N complex number. Half of the 2N variables are redundant:Q(-k) = Q*(k) $\endgroup$ – Electric to be May 2 at 21:26

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