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  How is physics valid at different scales linked in causal perturbation theory?

+ 4 like - 0 dislike

In the Wilsonian EFT framework, there are no infinities either, but at the scales where infinities used to appear because people tried to apply a certain EFT beyond its domain of validity, just different operators become relevant whereas others can be neglected, if I understand this correctly. Is this Wilsonian picture taken over in causal perturbation theory, or does linking physics at different scales differently?

asked Jul 19, 2014 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]

Dilaton, you, probably, know the answer to this question because the "causal" perturbation theory gives the same results as the standard QED. In the standard QED we may safely put the cutoff $\Lambda$ to infinity. As such, it is a self-consistent theory in this limit. There is no need in stuffing it with other operators.

The only place where $\Lambda$ remains is the relationships between "bare" parameters and divergent corrections they "absorb", but who gives a physical meaning to bare parameters in QED?

Being able to send the $\Lambda$ to infinity just means that the theory is (perurbatively?) renormalizable and QED is.

Strictly speaking, I am waiting for @ArnoldNeumaier to put up his answer from the comments elsewhere too, as I probably already have a follow up question :-)

But renormalizability in this sense means there is no special scale in QED, there is no "different" physics to link to each other.

done, in an expanded version. The old remark applied only to Scharf's treatment.

1 Answer

+ 2 like - 0 dislike

Effective field theories are families of non-renormalizable theories whose coefficients depend on multiple parameters, usually including an energy scale $E$ (or equivalent mass scale $M=E/c^2$) that, by construction, approximate the true underlying theory (of which they are an effective theory) at energies close to $E$.

Sometimes, $E$ is a nonphysical parameter, typically an additional scale introduced by a particular renormalization scheme - a necessity in theories like QCD; then $E$ is called a renormalization scale. In this case, all physical results must be (at infinite order) independent of $E$. Therefore the effective theories at different $E$ may be mapped onto each other (up to higher order terms that are neglected) by a renormalization group that also transforms the other parameters and the fields. 

The parameter $E$ must not be confused with the cutoff $\Lambda$ in traditional renormalization schemes. The cutoff is a second energy scale that is moved to infinity to produce the renormalized quantities. The renormalization group describes the effects after this limit has been taken. In causal perturbation theory, there is no cutoff, as all quantities appearing in the theory are renormalized from the start. However, there may still be a renormalization scale, in which case there is also a renormalization group.

QED has a natural intrinsic scale, which is the energy corresponding to the electron mass. However, this scale is fixed by the physical constants defining QED. The cutoff introduces a second scale, which in traditional renormalization schemes is moved to infinity, while in causal perturbation theory it is absent from the start. In Scharf's causal perturbation approach to QED, this leaves no scale to be varied, hence there is no renormalization scale and no renormalization group. 

One could, however, change the renormalization condition used by Scharf such that a renormalization scale is introduced; then a renormalization group would appear (indeed, in the 1995 edition of Scharf's book Finite Quantum Electrodynamics, this is treated in Section 4.8), and Scharf's original results correspond to its infrared fixed point. The latter is the appropriate scale for electromagnetic effects, which are visible at very low (e.g., everyday) energies. 

Similarly for other field theories. For example, in theories with a Higgs sector, causal perturbation theory always produces the broken symmetry phase. Nonabelian gauge theories such as QCD, however, have no natural scale and hence always need a renormalization scale, even in causal perturbation theory. (Though they ultimately get one through dimensional transmutation; then the natural scale is given by the mass gap.)

Which operators are relevant or irrelevant just depends on the renormalization group structure, which in turn depends on the precise renormalization conditions imposed. Hence it appears in the same way in standard renormalized perturbation theory and in causal perturbation theory. 

answered Jul 20, 2014 by Arnold Neumaier (15,787 points) [ revision history ]
edited Jul 21, 2014 by Arnold Neumaier

Thanks Arnold for this enlightening and interesting post !

"Nonabelian gauge theories such as QED", did you mean QCD? By saying electron mass is the natural scale, is "natural" in the sense of "non-renormalization" property of radiactive correction of QED, e.g. as discussed in Weinberg chap 10?

QED $\to$ QCD: corrected. - 'natural" in the sense that it defines a scale, and every quantity in QED with a nontrivial mass or energy dimension is naturally given in powers of the electron mass. In particular, this holds for radiative corrections. 

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