Re: Renormalization

> Igor Khavkine wrote:
> > Yes, you do go back and change the initial axioms and relationships.
> > However, you do it in such a way that repeating the same exact steps you
> > did before gets you to an integral that is no longer divergent but is
> > regulated in a specific way. However, physicists often skip the explicit
> > steps of going back and changing the Lagrangian and simply perform
> > manipulations directly on the integral expressions. Just because these
> > steps are often skipped, does not mean that they are not there
> > implicitly.
> >
> > Igor
>
> If this is the case then you and I are in argreement. This was the
> point I was trying to make all along - you should go back and change
> the formal structure (in this case the Lagrangian) so that the end
> results are derived from the initial formal structure.
>
> Dan Solomon

Actually this is not true at all. For example, I have never seen anyone
develop a quantum field theory for an arbitrary complex number of spacetime
dimensions in order to justify dimensional regularization later on - it is
simply an assumption that this is (a) possible and (b) leads to similar
formal expressions as that for 3+1 dimensions. This is why I was interested
in Dr. Neumaier's claim that he could develop QFT in 3+1 dimensions with an
extra parameter (the cutoff) being present in the theory right from the
beginning. In a thread where I was inviting him to develop his argument, I
posted the following, but it never reached the newsgroup, but at the same
time, I received no message from the moderator to say that the message was
unacceptable. Let me therefore try again:

[Quote from earlier post by AN, with corrections & equation numbers]

> Consider
> V:= integral V(t) dt [1]
> with
> V(t,\x):= gamma integral d\x Phi(t,\x)^4. [2]
> which is the interaction defining Phi^4 theory in the
> interaction picture.
>
> Introduce x=(t,\x) and rewrite V in 4-momentum space.
>
> Then regularize by introducing a cutoff Lambda, to get a
> regularized V_Lambda.
>
> Then go back to spacetime coordinates and convince yourself
> that the regularized theory corresponds to
> V_Lambda(t,\x):= gamma integral d\x Phi_Lambda(t,\x)^4, [3]
> where Phi_Lambda(t,\x) can be given explicitly in terms of
> its Fourier transform. Phi_Lambda(t,\x) is simply a smeared
> version of Phi. It satisfies almost standard CCR, except that
> the scalar function arising from the commutator is smeared, too.
> But this makes it nonsingular, and hence amenable to a canonical
> treatment.

It is certainly clear that if I write

\Phi(x) = \int d^4p e^{ip.x} \Phi(p)

and then put some kind of limiting on the four-momentum integral with
parameter Lambda to get \Phi_\Lambda then equation [3] follows from
equation [2].

However, we have just one Lambda and four dimensions of momentum. Are
you using Lambda to limit all the momentum integrals or just the spatial
ones? If so, how, exactly?

.

Relevant Pages

  • Re: Renormalization
    ... my concerns about regularization. ... a theory of physics should be a mathematical theory. ... On pages 238-242 they discuss the Ward identity. ... Equation 7.67 is the difference between two integrals. ...
    (sci.physics.research)
  • Re: Renormalization
    ... theories that use integrals at all, and why we don't look for ones that ... > trying to come up with a mathematical model to describe some system. ... > In answer to that I say - why not use regularization. ... The original Rcalc is recovered when d=1. ...
    (sci.physics.research)
  • Re: Renormalization
    ... The integrals are beta functions ... I remember puzzling over dimensional regularization as a graduate student, ... and you cannot make an analytic continuation from that. ...
    (sci.physics.research)
  • Re: Renormalization
    ... which is the interaction defining Phi^4 theory in the interaction picture. ... Then regularize by introducing a cutoff Lambda, ... you using Lambda to limit all the momentum integrals or just the spatial ...
    (sci.physics.research)
  • Re: Renormalization
    ... I think that you are saying that we must incorporate ... > the process of dimensional regularization as a postulate of the theory. ... cutoff theories, unless they're conformal. ... integrals. ...
    (sci.physics.research)