Next, consider a general two-magnon state of the form

with a two-particle wave-function . The “position space” Schrödinger equation following from then leads to two sets of equations, depending on whether the particles lie next to each other or not: is the eigenvalue of and related to the gauge theory scaling dimensions as . The above equations can be fulfilled by a superposition ansatz with an incoming and outgoing plane wave (Bethe’s ansatz) where denotes the S-matrix of the scattered particles. Note that in the second term describing the scattered contribution, the two particles have simply exchanged their momenta. One easily sees that (55) is fulfilled for an arbitrary yielding the energy as a sum of one-particle energies Equation (56) then determines the S-matrix to be of the form Note that . This solves the infinite length chain. For a finite chain, the momenta are no longer arbitrary continuous quantities, but become discrete through the periodic boundary condition This, in turn, leads to the Bethe equations for the two magnon problem implying with an arbitrary integer . The solutions of the algebraic equations (61) for and then determine the corresponding energies by plugging the resulting quasi-momenta into (58).The magic of integrability now is that this information is all that is needed to solve the general -body problem! This phenomenon is known as factorized scattering: The multi-body scattering process factorizes into a sequence of two-body interactions under which two incoming particles of momenta and scatter off each other elastically with the S-matrix , thereby simply exchanging their momenta. That is, the -body wave-function takes the form [37, 73]

where the sum is over all permutations of the labels and the phase shifts are related to the S-matrix (59) by The -magnon Bethe ansatz then yields the set of Bethe equations with the two-body S-matrix of (59) and the additive energy expression In order to reinstate the cyclicity of the trace condition, one needs to further impose the constraint of a total vanishing momentumAs an example, let us diagonalize the two magnon problem exactly. Due to (66), we have and the Bethe equations (61) reduce to the single equation

The energy eigenvalue then reads which upon reinserting the dropped prefactor of yields the one-loop scaling dimension of the two-magnon operators [10, 92] In the BMN limit with fixed, the scaling dimension takes the famous value , corresponding to the first term in the expansion of the level-two energy spectrum of the plane-wave superstring [33].Hence, from the viewpoint of the spin chain, the plane-wave limit corresponds to a chain of diverging length carrying a finite number of magnons , which are nothing but the gauge duals of the oscillator excitations of the plane-wave superstring.

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