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There is a class of observables in QFT (event shapes, parton density functions, light-cone distribution amplitudes) whose the renormalization-group (RG) evolution takes the form of an integro-differential equation: $$ \mu\partial_{\mu}f\left( x,\mu\right) =\int\mathrm{d}x^{\prime}\gamma\left( x,x^{\prime},\mu\right) f\left( x^{\prime},\mu\right) . $$ It is well known for such equations that one should distinguish carefully between well-posed and ill-posed problems. A classical example of an ill-posed problem is the backward heat equation: \begin{align*} \partial_{t}u & =\kappa\partial_{x}^{2}u,\qquad x\in\left[ 0,1\right] ,\qquad t\in\left[ 0,T\right] ,\\ u\left( x,T\right) & =f\left( x\right) ,\qquad u\left( 0,t\right) =u\left( l,t\right) =0, \end{align*} while the forward evolution (i.e., the initial-boundary value problem $u\left( x,0\right) =f\left( x\right) $) is well-posed. The fact that the backward evolution is ill-posed (the solution either doesn't exist or doesn't depend continuously on the initial data) models the time irreversibility in the sense of the laws of thermodynamics.

Since the renormalization transformation corresponds to integrating out short-wavelength field modes, the RG transformations are lossy and thus form a semigroup only. My question is — if there is an explicit example (or a demonstration) of an ill-posed problem for RG evolution? I mean, RG evolution equation the solutions (of initial-boundary value problem) of which have some pathological properties like instability under a small perturbation of initial data, thus making a numerical solution either not sensible or requiring to incorporate prior information (like Tikhonov regularization).

Update. Actually, I have two reasons to worry about such ill-posed problems.

The first one: the standard procedure of utilizing the parton density functions at colliders is to parameterize these function for some soft normalization scale $\mu\sim\Lambda_{QCD}$ and then use DGLAP equations to evolve the distributions to the hard scale of the process $Q\gg\mu$. The direction of such evolution is opposite to «normal» RG procedure (from the small resolution scale $Q^{-1}$ to the large one $\mu^{-1}$). Thus I suspect that such procedure is (strictly speaking) ill-posed.

The second: the observables/distributions mentioned above are matrix elements of some nonlocal operators. Using the operator product expansion (OPE), one can reduce the corresponding integro-differential equation to a set of ordinary differential equation for the renormalization constants of local operators. My intuition says that in this case the RG evolution for the distribution will be well-posed at least in one RG direction (thus I think the DGLAP equations are well-posed for the evolution direction $Q\rightarrow\mu$). Therefore, a complete ill-posed RG evolution appears when the OPE fails.

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  • $\begingroup$ Not an expert on this topic, but: Could a possible example be the various hierarchy problems in particle physics? I understood these to be an extreme sensitivity of the IR physics to the UV completion. Would this be similar/related to ill-posedness as you describe it? $\endgroup$ – Michael Brown Jan 21 '13 at 0:12
  • $\begingroup$ @MichaelBrown Actually I don't quite understand what you mean, but I added some clarification to the question. $\endgroup$ – Grisha Kirilin Jan 21 '13 at 20:38
  • $\begingroup$ The standard presentation is that low energy parameters with power-law running, such as the Higgs mass, are very sensitive to loop contributions coming from new physics at higher scales. I see now you're asking about going in the opposite direction. Any irrelevant operators will grow as you go higher in scale and eventually dominate the expansion, so as you say the OPE fails. The usual thing to do in that situation is to rewrite the theory in terms of new degrees of freedom (e.g. Fermi theory -> SU(2)xU(1) at the weak scale), but I'm not really sure how this helps your case. Good question! $\endgroup$ – Michael Brown Jan 22 '13 at 0:24
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Proofs in the mathematical sense are probably hard to come by, but the implicit assumption underlying Wilson's renormalization (semi)group is that the forward direction (integrating out degrees of freedom) is well-posed, while the reverse direction (recovering higher frequency information) is ill-posed.

This is precisely the same situation as for the heat equation, where things can be proved rigorously. Here integration forward in time is well-posed (smoothing, damping high frequencies) while integration backward in time is ill-posed (arbitrarily small perturbations may have arbitrarily large effects in arbitrarily short time). One way to express this is to say that time evolution by the heat equation is a 1-parameter semigroup only, and not a parameter group.

See also https://physics.stackexchange.com/a/373353/7924

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protected by ACuriousMind Apr 5 '17 at 11:11

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