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A Why do we need to renormalize in QFT, really?

  1. Jun 19, 2017 at 5:53 AM #1
    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?
     
  2. jcsd
  3. Jun 19, 2017 at 8:14 AM #2

    DrDu

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    0. Noninteracting field theories are an incredibly bad starting point to describe interacting field theories.
     
  4. Jun 20, 2017 at 12:21 AM #3

    dextercioby

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    In most of the cases the possible interacting field theories are mathematically derivable from the free/noninteracting ones.
     
  5. Jun 20, 2017 at 1:19 AM #4

    DrDu

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    I don't know of any realistic interacting field theory which is derivable from free ones.
     
  6. Jun 20, 2017 at 2:27 AM #5

    dextercioby

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    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).
     
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