Insights Struggles with the Continuum - Part 5 - Comments

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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
 
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A. Neumaier

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Nice to see you again!
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.
 
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?
 
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However, one of many puzzlements: if the integrals diverge, then why not change the measure?
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
 

A. Neumaier

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However, one of many puzzlements: if the integrals diverge, then why not change the measure?
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.
 
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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’m a graduate student at UMass Boston and I really enjoyed your "Struggles with the Continuum” paper. I was wondering if you could give me a little more explanation for one part. I’m by no means a particle physicist, which is probably why I didn’t get what was probably a simple point. When we want to compute the magnetic dipole moment of an electron as predicted by QED, why should we consider a process where an electron exchanges a photon with a “massive charged particle”? Basically I want to know what intuition leads one to realize this is the calculation we want to do.
I replied:

I'm no experimentalist, but to measure the magnetic dipole moment of an electron basically amounts to measuring its magnetic field. Like its electric field, the magnetic field of an electron consists of virtual photons. So when you "measure" its electric or magnetic field, it's exchanging a virtual photon with some charges in your detector apparatus.

The easiest way to model this is to imagine your detector apparatus is simply a single charged particle and compute the force on this particle as a function of its position and velocity, due to the virtual photon(s) being exchanged. (The magnetic force is velocity-dependent.)

But if that charged particle is light, quantum mechanics becomes important in describing its behavior: the uncertainty principle will make it impossible to specify both its position and velocity very accurately. So, it's better to take the limit of a very massive particle. In that limit, you can specify both position and momentum exactly.
 

john baez

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Steve Wenner:

However, one of many puzzlements: if the integrals diverge, then why not change the measure?
"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.
 
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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.
: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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