---------------------------------------- S8d. Renormalization and coarse graining ---------------------------------------- In QFT, there are two different scales, one on the bare level and one on the renormalized level, and the meaning of the renormalization group is slightly different from that in statistical mechanics. On the statistical mechanics level, there is the cutoff beyond which one cannot (or does not want to) observe anything. This effective cutoff is a parameter Lambda in an effective theory defined by coarse graining. The effective theory depends on E: For different values of E you get a _different_ effective theory, though their low energy predictions are essentially the same. This is expressed by the Wilson flow, described by renormalization group equations that relate the parameters g(Lambda,mu) in the different effective theories such that some key low energy observables mu keep the same values. The number of such key observables (i.e, the dimension of mu) equals the number of parameters in the effective theory (i.e, the dimension of g); most other observables are different at different cutoffs (though only slightly if they are observable at low energy), because of the coarse graining done when lowering the cutoff scale Lambda. In QFT, the above is mimicked on the _bare_ level. The cutoff is a large energy Lambda beyond which the bare interaction is modified to be able to get a meaningful limit; this corresponds to coarse-graining. The resulting bare theory with cutoff Lambda is a well-defined effective theory and behaves precisely as described above. To define the renormalized theory, one needs, in addition to the cutoff, renormalization conditions defining the bare parameters in terms of renormalized parameters q. These conditions depend on a renormalization scale E figuring in the equations defining the renormalization conditions. Because of the dimensional nature of momentum, there always has to be such a parameter E, no matter which renormalization procedure is followed. In QFT, one usually refers to a mass scale M, which is the same as E=Mc^2 in units such that c=1. Then M is the constant needed in the renormalization conditions to relate certain computable expressions to the renormalized parameters. This is discussed at length in the QFT book by Peskin and Schroeder, Section 12.2, for a massless Phi^4 theory, and in Section 12.5 for the general case. (For an online source, see, e.g., equations (90-(11) of hep-th/9804079. M is introduced there without comment, the role of M is described later, after (20).) In the following, I continue to use E in place of M. Thus the bare parameters are functions g(Lambda,q,E) of the cutoff Lambda, the renormalized parameters q, and the renormalization scale E. The renormalization group equations in the statistical mechanics sense (the Wilson flow) would describe how g(Lambda,q,E) changes as the cutoff Lambda is altered. However, in QFT, this is of no physical interest. Indeed, Lambda is completely eliminated from considerations: The renormalized theory is obtained at fixed E by letting the cutoff Lambda go to infinity. This has the effect that the bare parameters become meaningless, since the limit lim_{Lambda to inf} g(Lambda,q,E) does not exist. At this stage it becomes obvious that all bare objects are unphysical. Although nonphysical, the renormalization group equations in Lambda are an important tool in the _construction_ of QFTs, where the limit of all correlation functions must be shown to exist in a suitable topology, and the absence of divergences shown. In the weakest topology, based on the ultrametric norm and corresponding to perturbation theory at all orders, this is shown rigorously in a nice book M. Salmhofer, Renormalization: An Introduction, Springer, Berlin 1999. Unfortunately, this topology is too weak to give the existence of the correlation functions as functions; they are only shown to exist as formal power series. All expressions of the theory that survive the limit, in particular all n-point correlation functions, n=1,2,3,..., describe observable physics. They can therefore be expressed as functions of q and E only, whose detailed form comes from the standard theory. However, there is a little twist since the scale E can be chosen arbitrarily, hence cannot be measurable. In terms of a fixed set of physical parameters mu (measurable under well-defined experimental conditions), we can predict mu by some function of q and E, mu=mu(q,E). Solving for q, we can express q in terms of mu and E, q=q_ren(mu,E). But the exact renormalized result of a physical prediction P(q,E) must be completely independent of E, uniquely determined by the physical parameters mu. Thus we get the so-called Callan-Symanzik equations d/dE P(q_ren(mu,E),E) = 0. They are the renormalization group equations of interest in quantum field theory. In contrast to the Wilson flow, however, the sliding scale in the Callan-Symanzik flow is the renormalization scale E and _not_ the cutoff Lambda (which at this stage is already infinite). Moreover, since observable physics is completely independent of the renormalization scale E, the latter has no intuitive 'physical' interpretation. There is no relation between the two flows, except by analogy. The Wilson flow is needed to _get_ the renormalized theory at fixed renormalization conditions, the Callan-Symanzik flow describes what happens when you _change_ these conditions.