### 4.1 The coordinate Bethe ansatz

We now discuss the ansatz that enabled Bethe to diagonalize the Heisenberg model in 1931 [37]. For this, we will drop the cyclicity constraint imposed on us from the underlying trace structure of the gauge theory operators for the moment, and treat a general non-cyclic, but periodic, spin chain. The vacuum state of the Heisenberg chain is then given by . Let with denote a state of the chain with up-spins (magnons) located at sites , i.e. . It is useful to think of these spin flips as particles located at the sites . Note that the Hamiltonian preserves the magnon or particle number. The one magnon sector is then trivially diagonalized by Fourier transformation
as . The periodic boundary conditions require the one-magnon momenta to be quantized with .

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 [3773]

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 momentum

As 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 [1092]
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.