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Renormalization and counterterms 
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#1
Oct2313, 02:03 PM

P: 121

Hi all,
Can anyone tell me whether the counterterms introduced in renormalized perturbation theory (see e.g. Chapter 10 of Peskin and Schroeder) have any physical interpretation? In particular, are they taken to model the selfinteractions that take us from a 'bare' to a 'dressed' particle? Thanks a lot! 


#2
Oct2313, 02:58 PM

Thanks
P: 1,948

The counterterms have no physical interpretation. They are a book keeping device introduced to separate the physically significant renormalization effects from the mess of infinities that arise ultimately from our ignorance of the physics at short scales.



#3
Oct2413, 05:49 AM

Sci Advisor
P: 303

In the Lagrangian we have an constant ##\lambda^i_0## multiplying the ##i##th term of the Langrangian, let's say.
It turns out that, unlike classical physics, this constant is not directly interpretable as a physical coupling or mass, but simply as just a number in the Lagrangian. Rather some related constant ##\lambda^i## would be physically meaningful. To make calculations easier we split ##\lambda^i_0## into ##\lambda^i_0 = \lambda^i + \delta\lambda##, this allows the physical coupling to be visible from the beginning, rather than being buried in the poles of some correlation function and involving a complicated calculation to extract. It's also more convenient as ##\lambda^i## is an input to your theory, a value you fix to experiment, so again it is better to have it visible from the beginning. The counterterms are simply the ##\delta\lambda##. Now all of this would have to be done anyway, even if QFT had no divergences. The interesting thing though, is that it turns out that in many theories, like QED or QCD, the ##\delta\lambda## are essentially the negatives of any divergence in the theory and so cancel them out. This has nothing to do with ignorance of physics at short scales. The infinities of QFT do not originate from our ignorance of physics at low scales. 


#4
Oct2413, 07:52 AM

Thanks
P: 1,948

Renormalization and counterterms



#5
Oct2413, 02:40 PM

Sci Advisor
P: 303

I can write down pure YangMills on ##\mathbb{R}^4## and compute its counterterms. There is nothing being swept into them, as I'm only working with pure YangMills, they are simply the computable difference between the bare and physical constants. 


#6
Oct2413, 03:53 PM

Sci Advisor
P: 1,199

Yes, here we have to distinguish asymptotically free theories like QCD (which are believed to exist at arbitrarily small scales) and nonasymptotically free theories (like QED and the full Standard Model) which are believed to require an explicit modification (often referred to as an "ultraviolet completion") at some sufficiently small scale.



#7
Oct2413, 06:36 PM

P: 121

Thanks people. So just to be clear: the counterterms do not in themselves model e.g. the vacuum polarization effects that are taken to 'screen' the electron at finite distances and make the charge welldefined?



#8
Oct2613, 08:03 PM

Sci Advisor
P: 1,199




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