- #36

maline

- 436

- 69

Well, I think I just provided a derivation where this is not necessary...In particular, there must be a factor before the derivative term.

This is indeed an issue that also bothers me quite a bit. The ##Z## factors are defined in terms of a real inner product between normalized states, so ##0\le Z\le 1## is a physical restriction, but in fact ##(1-Z)## is found to diverge to each order in perturbation theory. I have been told that this is the same issue as the existence of the Landau pole, that it indeed shows that theories such as QED and ##\phi^4## are incomplete at high energies, and that for asymptotically free theories such as QCD there is a geometric series over the orders in perturbation theory that shows that the factor goes to zero. I don't know whether any of this is correct.The restriction 0≤Z≤10≤Z≤10\le Z\le 1 plays no role in the renormalization procedure and is never checked since ZZZ is a bare quantity without physical meaning.

You seem to be saying that the values of the counterterms in fact have nothing to do with the inner-product-based definition of ##Z##. But than leaves the question: why do we require that the vertices get a counterterm with the same value (to the appropriate power) as the propagator kinetic counterterm?

Of course I mean ##\partial^\mu \phi \partial_\mu \phi +m^2##. I wrote ##p^2+m^2##, being the corresponding vertex contribution.ppp is not a variable in the Lagrangian, so I don't understand your proposal about adding an interaction term proportional to p2+m2p2+m2p^2+m^2.

Good point, my mistake. The CCR have ##[\phi(x),\pi(y)]=i\delta(x-y)##, where ##\pi(x)=\frac{\delta L}{\delta(\partial_t \phi)}##. So changing the scale of the Lagrangian changes the CCR, just as rescaling the fields would.This holds for a classical theory but not for a quantum theory when, as usual, ℏℏ\hbar is fixed at 1, since scaling the Lagrangian changes the expression in the functional integral.

But this also settles the issue: The terms in the Lagrangian that include time derivatives of the field cannot be arbitrarily rescaled, because their scale encodes the commutation structure of the field operators. So the only way to get a ##p^2 +m^2## counterterm is to use the the ##Z## factors from the LSZ formula, either the way I described or with some other derivation involving a "renormalized field" (which I still don't understand).