Magnus Expansion · [Apr 29, 05:28 AM]
I have been occupied with the Magnus expansion for quite a while now, and I thought I might write something about it here and the way I think it describes a transition from classical to semi-classical to quantum.
The Magnus expansion is an example of a geometric integrator for differential equations on Lie groups. It means that using the Magnus expansion you get approximations that honor the geometry of the Lie group, while they are made of linear elements elevated later by the exponential map. In fact, they are somewhat natural extension of “integral” to Lie algebras but they are very useful.
That was just a bunch of big words. Take a time-driven differential equation: $$\frac{dy(t)}{dt}=-iA(t)y(t).$$ And let us think of $A$ as an operator or just a matrix. There billions of ways of approximating the solution, depending on the circumstances [of $A$]. Magnus expansion is about getting a solution in the form $$y(t)=e^{-i\Omega(t)}y(0)$$. If you think of $y$ as a quantum state and $A(t)$ as a time-dependent Hamiltonian, $\Omega(t)$ is like an effective Hamiltonian and $e^{-i\Omega(t)}$ is the propagator which will belong to some special unitary group. The operator $\Omega(t)$ satisfies a non-linear differential equation (due to Hausdorff) that I will not quote here but can be integrated approximately, in order of increasing powers of $A$ using the Magnus expansion:
$$\Omega(t)=-\int_{0}^t A(s)ds + i\frac{1}{2}\int_0^{s_1}\int_0^t [A(s_2),A(s_1)]ds_2 ds_1+\cdots$$
The stuff in [...] is very interesting and I like to call it third and higher order terms. Let’s just say that it is composed of all possible time-ordered commutators that there exists at that order.
What do I see here? The first order term is classical or semi-classical. The second term starts looking more like a quantum object as it will involve commutators and it gets more and more quantum as it progresses. The time-ordering also becomes more and more important, another quantum feature. Of course there is nothing specially quantum about the differential equation that we started with and it is our own choice to interpret it as the Schrodinger equation…
I will elaborate on a couple of applications of this expansion and how it might play with information correlations later.
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