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Insights Struggles with the Continuum - Part 5 - Comments

  1. Feb 1, 2016 #1

    john baez

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    Last edited: Sep 6, 2016
  2. jcsd
  3. Feb 1, 2016 #2

    A. Neumaier

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    Nice explanation!

    Those who want to see renormalization in a simpler toy setting not involving fields but just ordinary quantum mechanics might be interested in my tutorial on renormalization.
     
  4. Feb 2, 2016 #3

    bhobba

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    Really nice write up.

    I can see myself linking to it often when discussions of re-normalisation come up.

    Thanks
    Bill
     
    Last edited: Feb 2, 2016
  5. Feb 2, 2016 #4

    john baez

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    Nice to see you again!
     
  6. Feb 2, 2016 #5

    A. Neumaier

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    Well, your former ''this week's find in mathematical physics'' was far more attractive for me than your current blog. Online, I am now mainly active on PhysicsOverflow. But this term I have a sabbatical, which gives me more time to spend on discussions, so I became again a bit active here on PF.
     
  7. Feb 2, 2016 #6
    Wonderful post! This is the first time since I read Feynman's popular little book, QED, that I have felt that I have learned something solid this subject. Can't wait for the next installment!

    However, one of many puzzlements: if the integrals diverge, then why not change the measure?
     
  8. Feb 2, 2016 #7

    bhobba

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    In that regard the following is likely of interest:


    Very nice series of lectures on what divergence really is and even the why of quantitisation. I have viewed them a few times now and enjoy it every time.

    Thanks
    Bill
     
  9. Feb 3, 2016 #8

    A. Neumaier

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    One can avoid the divergences from the outset by using a carefully chosen mathematical setting. This is described in Scharf's book cited in the article, and in more detail in my insight article.
     
    Last edited: Feb 4, 2016
  10. Sep 12, 2016 #9

    john baez

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    I wrote:

    For example, consider the magnetic dipole moment of the electron. An electron, being a charged particle with spin, has a magnetic field. A classical computation says that its magnetic dipole moment is

    $$ \vec{\mu} = -\frac{e}{2m_e} \vec{S} $$

    where ##\vec{S}## is its spin angular momentum. Quantum effects correct this computation, giving

    $$ \vec{\mu} = -g \frac{e}{2m_e} \vec{S} $$

    for some constant ##g## called the gyromagnetic ratio, which can be computed using QED as a sum over Feynman diagrams with an electron exchanging a single photon with a massive charged particle:

    electron_magnetic_moment.png

    Someone asked me a good question about this:

    I replied:

     
  11. Sep 12, 2016 #10

    john baez

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    Steve Wenner:

    "Imposing a cutoff" or "regularization" is one way to change the measure to get a convergent integral. This is an important first step. But in making this step, you are led to inaccurate answers to physics questions. This step amounts to pretending that virtual particles with large momenta are impossible, or less likely than we'd otherwise expect. That's not really true.

    It's like saying "my calculations show that I'll be in debt if I buy a Cadillac. But I don't want to be in debt, so I'll do the calculation differently."

    To get the right answers, you don't want to make false assumptions just in order to get integrals that converge! You want to figure out why the integrals are diverging, understand what conceptual mistake you're making, and fix that conceptual mistake. That's renormalization.

    There are ways to do this, like Scharf's way, where you never get the divergent integrals in the first place. But I believe for most people those are harder to understand than what I explained here. My explanation is more "physical" - or at least, most physicists use this way of thinking.

    So what's the conceptual mistake?

    The conceptual mistake is trying to work with imaginary "bare" particles separated from their virtual particle cloud. There is no such thing as a bare particle.

    It's not an easy mistake to fix, because the particle-with-cloud is a complicated entity. But renormalization is how we fix this mistake. It makes perfect sense when you think about it. I think my explanation should be enough to get the idea. The actual calculations are a lot more work.
     
  12. Sep 13, 2016 #11

    bhobba

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    :biggrin::biggrin::biggrin::biggrin::biggrin::biggrin::biggrin::biggrin:

    I wonder if John, or someone else, could comment on why it works. My limited understanding is its really a mathematical trick to decouple low energy physics we are more certain of from high energy physics that's a bit of a mystery.

    Thanks
    Bill
     
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