Re: Renormalization

Chris Oakley wrote:

[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?

There are many ways of doing it, and for Phi^4 theory the details are
almost irrelevant. The simplest is to cut off all p with p_mu^2>Lambda^2. This preserves the O(4)-symmetry of the Euclidean
field theory. One could also cutoff all p not in Omega, where Omega
is any set that contains a sphere of radius Lambda and is contained
in a sphere of radius O(Lambda).


Arnold Neumaier


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