Renormalization theory

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

In this connection there is an important question concerning the infinite selfenergy of a point charge in classical as well as in quantum field theory. The latter uses a renormalization process to solve the problem, namely, by subtracting two infinities to end up with a finite result. Despite the success of such a procedure, a more physically satisfactory way is needed [80]. Possibly the present theory may provide such an alternative, by tackling the divergence problem in a more surveyable manner. The finite result of a difference between two infinities due to renormalization theory would then be replaced by a finite result obtained from the product of an infinity and a zero, as demonstrated by the present analysis. 

We should immediately note that the picture described here in the context. of scaling or renormalization theory leads to the important, concept of concentration blobs. (See Sect. 9.1.)... 

Universality and two-parameter scaling in the general case of finite excluded volume, Be comes about by the much more sophisticated mechanism of renormalization. As will be discussed in later chapters (see Chap. 11, in particular) both the discrete chain model and the continuous chain model can be mapped on the same renormalized theory. The renormalized results superficially look similar to expressions like Eq. (7.13), but the definition of the scaling variables iie, z is more com plica led. Indeed, it is in the definition of R ) and z in terms of the parameters of the original unrenormalized theory, that the difference in microstructure of the continuous or discrete chain models is absorbed. 

This criterion must also hold for the renormalized theory l = = n. ... 

These results of the scaling approach are consistent with the results of renormalization theory and are supported by experiment. We will discuss the experimental situation in later chapters. 

The theorem of renormalizability can be read in two ways. With the renormalized theory taken to be fixed, it implies the existence of a one-parameter class, parameterized by , of bare theories, all equivalent to the given renormalized theory and thus equivalent to each other. This aspect is related to universality a whole class of microscopic models yields the same scaling functions. In the next chapter we will use this aspect to get rid of the technical complications of the discrete chain model. We can however also interpret the theorem as establishing the existence of a one-parameter class of renormalized theories, all equivalent to a given bare theory. This class is parameterized by the length scale r or the scaling parameter... 

This shows that the variables of the renormalized theory depend on temperature, chain length, and segment size via the two parameters z, Ef. For T — 0, i.e. fa T — S - > 0, we find... 

How then can it be used to construct the renormalized theory, finite for d < 4 and valid for all temperatures ... 

Concerning the third point we note that the renormalized theory, expressed in terms of u, does not know whether the underlying bare... 

In this chapter we first show that the continuous chain model is renor-malizable by taking the naive continuous chain limit of the theorem of renor-malizability. We then argue that we can construct renormalization schemes for the continuous or the discrete chain models, equivalent in the sense that they yield the same renormalized theory (Sect. 12.1). In Sect. 12.2 we estab-... 

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the -factors attain finite limits Z(ti), Zu(u), Zn(u). Indeed they are constructed as power series in u, the coefficients for t > 0,6 > 0 taking the form of polynomials in / r) . Thus no problem results from setting t = 0. Since the renormalized theory is finite for d = 4, whereas the bare continuous chain model diverges for d —> 4, showing poles in 5, also the Z-factors must diverge for d —> 4. In the NCL we can therefore formulate the theorem of renormalizability as follows ... 

The renormalization factors can be chosen to absorb all the pole terms of the dimensionally regularized bare theory to yield a renormalized theory finite for d < 4. 

In Chap. 7 we have shown that the bare discrete chain or continuous chain models are naively equivalent only close to the 0-point. We thus might wonder whether the equivalence of the two models, shown above to one loop order, can hold generally. We thus have to show that starting from these different bare theories we nevertheless can construct identical renormalized theories. We consider the renormalized continuous chain limit (RCL), used in the theorem of renormalizability. 

We thus move along a set of bare theories, equivalent in the sense that they yield the same renormalized theory. To relate this way of taking the limit — 0 to the NCL from Eqs. (11.27), (11.33) we note that the set of all equivalent bare theories is defined by fixed values of the parameter sp and of the combination sn n. Now assuming a starting value /o < 1 we note from Eq. (11.28) that S = co-nst implies the asymptotic behavior... 

Thus asymptotically the NCL and the RCL, if applied to the bare functions for d < 4, are identical. In other words, for fo < 1 the set of equivalent theories for i —> 0 reaches the 0-region, where the continuous chain model and the discrete chain model coincide. The same renormalized theory can therefore be constructed from both models. 

Concerning the first question we note that the result of any renormalization scheme based on the continuous chain model via a finite renormalization can be mapped on the renormalized theory derived from the discrete chain model, and vice versa. After renormalization the models are completely equivalent. 

The problem, however, does not ruin our construction of the renormalized theory, and it does not keep us from using the results in some region u = fu > u. Concerning the RG flow we note that we will use the special scheme of minimal subtraction1, where the flow equations depend on d only trivially... 

We thus take the following attitude. For technical reasons we calculate the renormalized theory starting from the continuous chain model. By equivalence of the bare theories for fo < 1 we know that we can derive the same theory from the discrete chain model. Since the renormalized theory in the way we construct it should show no singularity at u, we can use it for u > u. This region however can be interpreted only in terms of the discrete chain model. 

We can scale away hu(l), exploiting the fact that the renormalized theory is a two-parameter scheme. Recall that we may start from the naive continuous chain limit, where all observables are expressed in terms of the variables z,q2i2ofSo Renormalizing these variables according to Eqs, (12.16) we find the variables of renormalized perturbation theory ... 

In all this earlier work the dependence of low order renormalized theory on parameters like no, cq, go has not been considered. Rather some reasonable choice has been made a priori. 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

Within the Standard Model, ae is not sensitive to the short-distance effects. Thus ae is used primarily to test the renormalization theory of QED. Our current goal for ae is to calculate the coefficient of the a4 term to a precision of 0.01. This corresponds to the uncertainty of 0.3 x 10-12 in ae. With the matching improvement in experiment, this will provide a with a precision of 10-9 or better. How far can we go beyond this It is certainly feasible and desirable to improve it further by another factor of 4. This is just a matter of computer time. (It will require about 10 million hours of computing time.) However, improving the a4 term much further will not make sense until the tenth-order term is evaluated or estimated reliably which will be of the order of... 

The difference in the energy of the 2 Sand 2 Pjy2 levels in hydrogenic atoms is a purely electrodynamic effect due to the interaction of the bound electron with the quantized electromagnetic field. The measurement of this splitting was a major stimulus for the development of renormalization theory and still provides an important test of Quantum Electrodynamics. The precise measurement of this split-ting is difficult because of the short radiative lifetime of the 2 P 2 state. 

WloIhIWlo) primitive theory (WloI Plo) renormalized theory... 

The renormalized theory of the effective Hamiltonian implied by the restriction to some subspace S of the full Hilbert space also imposes a requirement for renonnalisation of expectation values of other operators (Freed ). Suppose that we have some operator B and we require its expectation value in a state 0 of the full Schrddinger Eq. (2-2) in complete analogy with the effective Hamiltonian theory described above we define an effective operator B by the requirement that its expectation value in a state A ) in the subspace should equal the exact expectation value (c.f. Eq. (2-4)),... 

On the subject of stars and linear chains, the same authors have employed MC calculations based on Bishop and Clarke s pivot algorithm to study the validity of scaling and group renormalization theories of these interesting molecules. Dimensions and intrinsic viscosities were also calculated. - ... 

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