Is mean field theory self-consistency analogous to string theory consistency?

Question

My question is vague, so I'm hoping the answers will help me ask more concrete questions and maybe produce some interesting discussion.

In mean field theory, say for the Ising model, we treat the magnetization $m$ as a parameter and derive the equation $$ \langle\sigma_i\rangle = \tanh (\beta J dm), $$ where $J$ is the neighbor coupling, $\beta$ is the inverse temperature, and $d$ is the spatial dimension. Self-consistency requires $$ \langle\sigma_i \rangle = m, $$ from which can be derived all the ordinary mean field theory conclusions.

However, we don't have to impose this equation by hand if instead we minimize the mean field free energy with respect to $m$.

In string theory something superficially similar happens. Consistency conditions on the world-sheet theory are equivalent to equations of motion of the target-space theory. I have always wondered why this is the case, and it would be interesting to see a correspondence like mean field consistency $\simeq$ world-sheet consistency, mean field equilibrium for liberated $m$ $\simeq$ target-space vacuum.

An observation: mean field theory maps a physical system with site-to-site interactions to a single site problem, where the other sites are treated as creating the background magnetization $m$.

An analogous observation for the string: to derive Einstein's equations, we consider a string propagating in a background metric (which would be emergent from a sea of strings) and show that the background metric gets renormalized according to the Einstein-Hilbert action.

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Answer 1

Mean field theory is just a useful ad hoc approximation; the world sheet description of string theory is the whole exact thing. Well, the latter is also an approximation – a machinery to produce the Taylor expansions i.e. perturbative series – but the perturbative series are unique and canonical. The mean field is just useful.

So in the case of mean field theory, one simply defines the approximations so that a classical field optimally replaces the actual quantum-fluctuating field, and your self-consistency conditions are examples of that.

The logic in string theory is much tighter and more unavoidable. The deformation of the world sheet action has to be by marginal operators. The equations enforcing the condition that these deformations will remain consistent – the world sheet theory will remain conformal – express the "marginality" of the deformation operators. Marginality means that their dimension $L_0=0$, in some conventions.

These are equivalent to the same spacetime equations (like Einstein's equations) that may be derived from the scattering amplitude of actual strings, perturbations above the background. The reason is that these actual strings also obey $L_0=0$ but $L_0$ acts on the states defined on a string, not operators.

But these two meanings of $L_0=0$ are equivalent due to the so-called state-operator correspondence. By conformal transformations, namely the logarithmic map, a plane (with an operator inserted at $z=0$) may be mapped to an infinite cylinder, the history of a closed string. The condition imposing the $L_0=0$ condition on the operators at $z=0$ is equivalent, via this map, to $L_0=0$ imposed on the state of the closed string. That's why the equivalence holds – why the spacetime equations of motion may be derived in two seemingly different ways.

The equivalence holds at the interacting level, too.

As far as I can say, no state-operator correspondence or conformal mapping is applicable a generic mean field theory. So even though some of the equations may be analogous, the logic determining these equations is different in the two cases.