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I Bare QFT theory with cutoff and multi-particle state...

  1. Jan 10, 2017 #1
    Hi all,

    I was reading Arnold Neumaier's excellent article on the Vacuum Fluctuation Myth, and ran upon one part I have a question about: he notes that "in bare quantum field theory with a cutoff, the vacuum is a complicated multiparticle state depending on the cutoff – though in a way that it diverges when the cutoff is removed, so that nothing physical remains. "

    I'm wondering a) why the introduction of a cutoff leads to a multiparticle state? I thought the ground state even in the bare theory was a state with zero particles, even with a cutoff, and b) since we don't observe this multiparticle state (do we?), is this evidence that there should not be a cutoff?

    Thanks!

    Reference https://www.physicsforums.com/insights/vacuum-fluctuation-myth/
     
  2. jcsd
  3. Jan 10, 2017 #2

    A. Neumaier

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    With cutoff one can work in a Fock space created in terms of bare particles, and with a well-defined normally ordered Hamiltonian that describes the interacting dynamics. The physical vacuum is then the ground state of this Hamiltonian. It contains zero physical particles, but expressed in terms of the particles of the bare theory, it is a very complicated multi-bare-particle state. The only state with zero bare particles is the bare vacuum state, but this is not even an eigenstate of the physical Hamiltonian.
     
  4. Jan 10, 2017 #3
    Ah thanks Arnold - ok, so the observed state of the physical vacuum contains zero physical particles, but if one refers to particles of the bare theory, you have the complex multiparticle state. And the bare vacuum state is not a physical state (since not eigenstate of physical Hamiltonian)?
     
  5. Jan 10, 2017 #4

    A. Neumaier

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    The bare vacuum state is (in the cutoff theory) a very complex physical multiparticle state. In the renormalization procedure it disappears and has no longer a physical interpretation.
     
  6. Jan 10, 2017 #5
    And if one does not apply the cutoff or renormalization, one is left with a divergent, non-physical result, then.

    I guess my final query is when one should expect to see a physical vacuum with zero physical particles - either with an applied cutoff, or in the renormalized case?
     
  7. Jan 10, 2017 #6

    A. Neumaier

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    The only physically correct theory is the renormalized one, after having removed the cutoff. The physical vacuum is of course an idealization, as the real world is filled with lots of fields (not only stuff describable in terms of particles).
     
  8. Jan 10, 2017 #7
    Then I'd suggest not to call it physical.
     
  9. Jan 10, 2017 #8

    A. Neumaier

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    If one is not allowed to make idealizations one cannot do physics until a theory of everything is available....

    Many physical systems are idealizations. There is no isolated system, but systems are often treated as isolated.
     
  10. Jan 10, 2017 #9
    Yes, all this doesn't change the fact that calling physical what is ideal could be more confusing than callint it ideal. Just a practical suggestion.
     
  11. Jan 10, 2017 #10

    A. Neumaier

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    One cannot change this. Physical means tractable in terms of a dynamical model based on physical principles.
     
  12. Jan 10, 2017 #11
    One more question about renormalization: once the cutoff has been removed, if I want a more and more accurate answer, can I sum up to an arbitrary loop order?
     
  13. Jan 11, 2017 #12
    Any takers?
     
  14. Jan 11, 2017 #13

    A. Neumaier

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    The series is only asymptotic, so at first the accuracy improves but at some point it will get worse. For arbitrarily accurate answers you'll need to sum infinite families of terms, using for example the renormalization group. It is unknown whether this is enough.
     
  15. Jan 11, 2017 #14
    Thanks Arnold! So the answers should still remain finite, with the cutoff removed, but of course it's not possible to sum infinite families of terms.
     
  16. Jan 11, 2017 #15

    A. Neumaier

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    Yes, the answers must be finite if the theory is well-defined. But summing infinitely many diagrams of a certain kind (ladder, or rainbow, etc.) is a common activity when evaluating series derived in terns of Feynman diagrams.
     
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