Why do we need to renormalize in QFT, really?

[jakob1111]

There are several reasons given in the literature, why UV infinities arise in QFT in the first place. My problem is putting them together, i.e. understand how they are related to each other.

So... UV divergences arise and thus we need to renormalize, because:

1. We have infinite number of degrees of freedom ín a field theory. (From this perspective, the infinites seem inevitable.)
2. We multiply fields to describe interactions, fields are distributions and the product of distributions is ill-defined.
3. We neglect the detailed short-wavelength structure of scattering processes, and the infinites are a result of our approximations with delta potentials. (From this point of view, the UV divergences aren't something fundamental, but merely a result of our approximation method. )
4. We are dealing with non-self-adjoint Hamiltonians. (This is closely related to the 3. bullet point. From this perspective an alternative to the "awkward" renormalization procedure would be the "method of self-adjoint extension".)

Are these reasons different sides of the same coin? And if yes, how can we understand the connection between them?

 

[DrDu]

0. Noninteracting field theories are an incredibly bad starting point to describe interacting field theories.

 

[dextercioby]

0. Noninteracting field theories are an incredibly bad starting point to describe interacting field theories.

In most of the cases the possible interacting field theories are mathematically derivable from the free/noninteracting ones.

 

[DrDu]

In most of the cases the possible interacting field theories are mathematically derivable from the free/noninteracting ones.

I don't know of any realistic interacting field theory which is derivable from free ones.

 

[dextercioby]

I don't know of any realistic interacting field theory which is derivable from free ones.

The self-interactions in QCD (Yang-Mills fields) are derivable from the free theory of electromagnetism (in fancy mathematics language: the only physically relevant deformation of the U(1) gauge algebra is a compact Lie algebra).

 

[DarMM]

2. and 4. are closely related.

Quantum fields are operator valued distributions. Like all distributions there is no natural notion of multiplication for them. By "natural" I mean there are several ways of defining how one should multiply two distributions (the only thing they have in common is that when restricted to functions they agree with the normal notion of multiplying functions). The Colombeau algebra is one example of a multiplication algebra for distributions.

The naive way of multiplying functions, pointwise multiplication, results in nonsense for distributions. This is the mathematical origin of infinities in QFT.

For QFTs it turns out that the correct notion of multiplication is intimately tied to the operator component. One can only correctly define the multiplication of the fields on a specific Hilbert Space. On that Hilbert Space, the correct multiplication will amount to normal ordering. Once that normal ordering is performed the Hamiltonian automatically becomes self-adjoint.

In this, mathematically rigorous, view renormalization is essentially the process of slowly constructing the inner-product which defines the correct Hilbert Space and normal ordering the fields (order by order in the coupling constant).

Short Version:
Renormalization occurs because QFTs live in different Hilbert Spaces. On the wrong Hilbert Space the field multiplications occurring in the Hamiltonian are nonsensical and ill-defined. Only on the correct Hilbert Space is the Hamiltonian self-adjoint after normal ordering. Unfortunately the correct Hilbert Space is different for each Hamiltonian.

So:
local fields = operator valued distributions = problem above.

1. is simply a consequence of using local fields. Although one can have infinite degrees of freedom without running into the problem above.

3. is just an idea for how one might replace local fields, but I don't think its the reason for renormalization.