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Suppose I have a system of complex ODE's of the form $$ i\dot{\mathbf{c}}(t)=\mathbf{f}(\mathbf{c}(t))$$ and I can write down a Hamiltonian such that each ODE can be written as $$\dot{c}_j=\frac{\partial\mathcal{H}}{\partial c_j^*}$$ for each $j$ where * denotes complex conjugate. As a very simple example, the Hamiltonian $$ \mathcal{H}=-i\left(|c_0|^2+|c_1|^2\right) $$ leads to the equations $$i\dot{c}_0=c_0, \qquad i\dot{c}_1=c_1. $$ Questions:

What are the conjugate momenta for this system (Are they just the complex conjugates)? Also, is there any way to transform this problem (via action-angle coordinates/madelung transform) to one where the Hamiltonian is purely real or where the system evolves under real dynamics? Is an imaginary Hamiltonian even an issue if I want to analyze a much more complicated non-linear system of this type using canonical perturbation or bifurcation theory?

Cosmas Zachos
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garserdt216
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    (VERY VAGUE) I think that purely imaginary Hamiltonians correspond to "gradient flows" (by which I mean the dynamical systems having the form $\dot{x}=-\nabla V(x)$). My (rough) intuition is that Hamiltonian systems are "conservative" while gradient systems are "dissipative". Putting an imaginary unit in a Hamiltonian should amount to transforming it into a dissipation function. I hope that, searching for the keywords given in this post, you can find something useful. – Giuseppe Negro Jul 27 '17 at 15:10
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    This is not really a "Hamiltonian" in a usual physics sense, as "Hamiltonian" is usually Hermitian (and thus purely real) in the quantum sense, and producing a conserved flow in classical settings. With the imaginary factors you no longer have the same kind of conserved quantity.

    If you want to study the dynamics, one option is to treat $c$ and $c^$ as separate real variables. (Of course you could also do Real(c) and Imag(c), but that usually works out less cleanly.) Then you have 4 real variables, and c_0 / c_0 are conjugate; and c_1 / c_1* are conjugate. But your dimension is doubled.

     – Alex Meiburg Jul 27 '17 at 20:21
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    Something related that you might be interested in is PT-symmetric quantum mechanics. There, one considers non-Hermitian Hamiltonians that nonetheless have an entirely real spectrum. In the latter aspect they are of course different from what you propose, but they might be interesting. – Cyclone Sep 11 '17 at 21:39

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Well, yes, this is imaginary time dynamics, that is, you absorb -i into t, for a new variable $\beta=t/i$ and keep the Hamiltonian real/Hermitean; you run your Poisson brackets, QM, or whatever as usual.

That is to say, a purely imaginary Hamiltonian is essentially a pure real one, once you have analytically continued your time to a pure imaginary parameter, and, as and when opportunities present themselves, all your conventional answers from real time, with care.

Very often, in statistical mechanics, thermodynamics, QFT, β is essentially the inverse temperature and the exponentials of βH are real functions of partition functions, etc... instead of unitary evolution operators, exponentials of -itH.

Cf. Garaschuk 2010.

Cosmas Zachos
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