What is the physical meaning of renormalization in field theories?

What is the physical meaning of renormalization in field theories?

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en.wikipedia.org/wiki/Renormalization#Self-interactions_in_classical_physics
en.wikipedia.org/wiki/Ising_model
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I dunno but I'm gonna ask magnet man.

Basically subtracting off infinities that arise from the likely in-completeness of your theory in order to recover meaningful results.

An older example is the infinite energy in a point charge.

en.wikipedia.org/wiki/Renormalization#Self-interactions_in_classical_physics

So its not anything physical, its a modeling tool?

Physics itself is a modeling tool.

I have a unique view of renormalization. Keep that in mind, because maybe I'm full of shit.

The unifying idea behind all renormalization (in field theories or in statistical mechanics) is that it's a process of finding an effective theory. An effective theory is just the original theory, but stated only in terms of the degrees of freedom you are interested in. In field theory, you are (typically) interested in low energy degrees of freedom (e.g. the low-momentum modes just above the vacuum). Because neither you nor I know anything about the high-energy physics, we attempt to remove those high-energy degrees of freedom that we know nothing about. It may strike you as odd that we can somehow remove things we don't know about and somehow incorporate them into our low-energy theory in a consistent way. The concept of "universality" tells you that this can be done. It says that, regardless of what the UV theory looks like, low-energy theories can only look like a few classes of theories. Those classes of theories are defined by the fact that they contain terms of appropriate mass dimension/symmetries.

We didn't say anything about infinities here, and that's usually the context in which renormalization is discussed. Where do they come in? It turns out that quantum field theories without cutoffs are not well-defined. If you give me a Lagrangian, I can't make any predictions about it if you don't give me a cutoff. The choice of Lagrangian/cutoff isn't unique. You can simultaneously change the cutoff and the Lagrangian (but you have to do so in a special way), and this is what defines the RG flow in parameter space. There is some family of Lagrangians/cutoffs that all describe the same physics (i.e. they will produce the same scattering amplitudes), and the RG flow defines this family of theories. Without a cutoff, you'll get infinite answers that don't really mean anything. It doesn't make sense to increase the cutoff beyond the energy scales you don't know about.

If we knew what it meant it wouldn't be such a problem.

I guess nothing there was that unique/contentious. But I can get more unconventional if I keep going.

So fudging to fit the data

Yes

No one knows what the bare parameters are, so how is that fudging? You first have to identify what the QFT you observe in nature is in the first place. Only after you've done that can you make predictions (and people do make predictions, and those predictions are confirmed by experiment).

If you can't get your result from first principles it's a fudge.

I guess all of science is a fudge

...

>classical mechanics
>special relativity
>general relativity

Yea, but to figure out a scattering amplitude for an electron (even in classical mechanics) you have to measure the mass of an electron (which is essentially the same kind of parameters fitting you do in QFT). Since you didn't get the mass from first principles, classical mechanics is also a fudge.

using data to get values from your theory isn't a fudge but using data to force your theory to fit the values you are supposed to get is a fudge.

Obviously we can't get everything from first principles (yet) but there is an awful lot more fudging in some theories rather than others.

We need more people like this in sci

Rescaling energies.
Completely wrong. Renormalization is the shifting the poles of the scattering amplitudes to where that of the free theory is. "Cancelling infinities" is merely how this pole shifting manifests at the perturbation level in some theories, namely there are theories in which renormalization does not involve subtracting infinite counterterms at all. Please read Weinberg before embarrassing yourself.
This is one sensible way of looking at it. A prof I know believes that all theories in nature are effective field theories of some kind, and divergence/anomaly cancellations often occur when calculating scattering amplitudes from the Wilson action.

if we had discovered something like zeta function regularization and weird summation methods first you wouldn't ask this question

it only seems weird because of a historical artifact of the progression of algebra, but the algebra doesn't give a fuck about what order you learned things

This is silly. No mention of physical meaning in your post, and that's the whole point of this thread.

>what is the physical meaning of 1
>what is the physical meaning of squaring
>what is the physical meaning of integration
hurrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr

>Completely wrong. Renormalization is the shifting the poles of the scattering amplitudes to where that of the free theory is. "Cancelling infinities" is merely how this pole shifting manifests at the perturbation level in some theories,namely there are theories in which renormalization does not involve subtracting infinite counterterms at all.

So not "completely wrong" then. And a lot of the counter-terms in QFT are formally infinite.

Though you're right it's not really the def. or renormalization in general. I suppose a good example for physical intuition finto what's happening I think looking at how spin-coupling fields' coupling constants change with scale.

en.wikipedia.org/wiki/Ising_model

> Having one of something
>The area of the square with sides of the number
>The area under the curve

It's "completely wrong" in the sense that it's completely wrong to say that renormalization is (defined as) the cancellation of infinities, which is exactly what OP was asking about.
And in terms of the Ising model it's the consequence of conformal symmetry that you are able to renormalize the theory by scaling it. It does not offer the right physical intuition for renormalization in general.

It's a way to integrate out all the contributions of interactions at a particular scale and compensate for it by changing some properties like that mass. If you want intuition it's probably better to look at normalisation in classical mechanics, it's used in fluid dynamics to aggregate the effects of turbulence for example. A very simple example showing the idea which isn't completely correct is as follows, imagine a large ball in a dust cloud with dust of all sizes, you want to calculate the movement of the ball when you apply a force to it, to do this you have to account for all the interactions of the dust with the ball. Since the collisions with dust will make it harder to move you can renormilize it by ignoring all dust smaller than some scale and increasing the mass of the ball (and of the larger dust probably) to compensate for it and get a solution that's simpler but statisticaly the same.

Damn.

Fair enough. OP didn't mention QFTs specifically,

>theory gives infinite energies between the plates
>let's truncate (i.e fudge) the series so it matches real life and call it "regularization"

>Please read Weinberg before embarrassing yourself.
Good post

Well that one kinda makes sense to me.

You have infinite energy outside the plates, but the infinite energy inside the plates is slightly less, so you get internal pressure.

Or that's how I interpret what's going on.

This is a good answer.

Why does everyone mention scaling? You can integrate out degrees of freedom in ways that have nothing to do with scaling. You can have a channel coupled to a device and integrate out the channel degrees of freedom.

You get a self-energy term in the Green's function, etc.

Scaling is the most initiative type of renormilization

Sure, but "scaling" shouldn't be used as a definition of renormalization. It obfuscates the general point.

You have to renormalize the definition.

This is how renormalization is taught by people who just want their students to compute things quickly.
This is the Wilsonian picture of renormalization. It's not really unique.
>A prof I know believes that all theories in nature are effective field theories of some kind
My adviser would agree.

I like that analogy.

you both have good points
Talking about physical scales is a good way to illustrate that some of the degrees of freedom of a model are not as relevant as others. To any feasible degree of accuracy, the physics of b quarks is unimportant to calculating the lamb-shift of Hydrogen. A simple way of arguing to someone why that's true is just to compare the distance scales at which these effects become important.

>"scaling" shouldn't be used as a definition of renormalization
It isn't. It's the physical intuition behind renormalization. The definition is the shifting of poles of the S-matrix elements. On the generating functional level this manifests as integrating out momentum shells of the theory (which can't always be done).