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Not sure this is a complete answer, all I did was basically some data mining + use some sparse domain knowledge + tried to be mature enough to interpretate and correlate information. Anyways, here are my 5 cents... Let us first set the ground here. a) Self-duality and conformal symmetry are closely related in SDYM. b) I'll take "planar limit" as ...


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For rigor, you might try to ask on Math.SE or MO.SE. In this answer, we will give a heuristic derivation via discretization. We will use a slightly different notation to connect to usual time-evolution in QM, but the idea is the same: $$\begin{align} U_{\lambda}(t_f,t_i)~=~&T e^{-\frac{i}{\hbar}\int_{t_i}^{t_f}\! dt~ H_{\lambda}(t)}\cr ~=~&\lim_{N\...


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$T$ is the ordered product of many infinitesimal factors of $$ V(x,t) = \exp\{\epsilon U(x,t)\} \approx 1+ \epsilon U(x,t). $$ Apply Liebniz' product rule to differentiate. If you insist on a rigorous derivation then some properties of the operator $U(x,t)$ need to be adduced.


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Such analytical expressions were derived for special cases by multiple authors. More general results were obtained by Yang and Yang in C. N. Yang and C. P. Yang One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System, Phys. Rev. 150, 327 (1966). For example, the ground ...


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A single autonomous (possibly nonlinear) 2nd-order ODE $$F(x,\dot{x},\ddot{x})~=~0. \tag{1}$$ can in principle be written as a couple of autonomous (possibly nonlinear) 1st-order ODEs of the form $$\dot{x}~=~f(x,y), \qquad \dot{y}~=~g(x,y). \tag{2}$$ One may show that there always exists an integral of motion/first integral for the latter system (2), at ...


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