Quantum Renormalisation Made Easy
Table of Contents
What Is The Issue With Renormalisation
If you have an interest in physics you have likely come across renormalisation before, although what it really is would probably have not been explained. Instead, you would read Dirac never accepted it, Feynman called it a ‘dippy process’, and if you look under a quantum theorists rug you will find all these infinities.
I read it all and thought – surely it can’t be that bad, and decided to have a look at some papers on it. Typical of what I found was the following:
http://arxiv.org/abs/hep-th/0212049
From the section on conventional renormalisation:
‘In other words, the bare mass has to diverge in such a way that its divergence cancels the divergent loop correction to yield a finite result. It amounts to shuffling the infinities to unobservable quantities like the bare mass. This is the part in renormalization theory which is very difficult to comprehend at the first sight.’
Shuffling infinities around? I was despondent – Feynman was right – it’s dippy. Having a background in math it’s also not legitimate math.
Ok – you do a bit more searching and you find a guy called Wilson sorted this out. But exactly what did he do – how did he resolve it? There doesn’t seem to be much detail at an elementary level explaining it. This is a pity, because once you get past the mind-boggling and mind-numbing manipulations that are done in Quantum Field Theory, it really is pretty simple, as I hopefully will explain.
Perturbation Theory Order Notation
First, we need to get out of the way a standard notation used in perturbation theory. If you have a polynomial in x whose lowest term is of order n then that is written as O(x^n). For example, if you have a polynomial like x^3 + 3*x^4 + 2*x^10, it’s a polynomial of lowest order 3 and is written as O(x^3).
A Look At A Quantum Field Theory Infinity
Ok without further ado just what is the issue. What is your typical QFT infinity? Well, normally you will find these horrid detailed manipulations involving Feynman diagrams with complex improper integrals in 4 dimensions that would make even the most enthusiastic mathematician wince. Do I have to wade through this to get to the nut?
Fortunately, you don’t – you simply need to see the final result. The one I will look at is from meson-meson scattering and you will find the detail on page 145 of Zee – Quantum Field Theory In A Nutshell. I will however write the equation in a simpler and I think even more transparent form than Zee uses.
M(K) = iλ + iλ^2*f(K) + limit Λ →∞ iCλ^2 log(Λ^2) + O(λ^3).
Here M is what’s called the scattering amplitude, K is a momentum, λ the coupling constant in meson theory, f a well-behaved function, C a constant the exact value of which isn’t germane, and limit Λ →∞ iCλ^2 log(Λ^2) the result of evaluating an improper integral over momentum, where the formal definition of such as a limit has been included for reasons that will shortly be obvious.
We can see the problem immediately – the limit blows up.
This is exactly what caused Dirac to despair. If you take the limit you get nonsense. What if you don’t take the limit, but instead assign Λ some value. Such is called introducing a cut-off in the theory. Yes, you get a finite answer – but as Dirac expressed it – the answer depends strongly on the cut-off. So you think about it a bit – what exactly is the cut-off – its a cut-off in momentum. Are we really sure integrating infinity in an improper integral is correct? Large values of momentum mean large values of energy – are we really that sure our theory is correct for energies way beyond what we can reach. It doesn’t seem likely. A cut-off looks a reasonable bet. OK – but what cut-off.
Before seeing how Wilson looks at renormalisation, and how it resolved this, I will for simplicity divide the equation by i, call M(K)/i, λ’, and instead of taking the limit will have Λ fixed at some large, but unknown value. This gives:
λ'(K) = λ + λ^2*f(K) + Cλ^2 log(Λ^2) + O(λ^3). (1)
Now we are ready for the trick of renormalisation. Wilson’s idea was to take the cut-off seriously. We don’t know what cut-off to use – but we think there should be one that gives the correct value of λ’. Suppose Λ is that value and we choose some K denoted by U to give:
λ'(U) = λ + λ^2*f(U) + Cλ^2 log(Λ^2) + O(λ^3). (2)
λ'(U) is given a fancy name – λr the renormalised coupling constant. A fancy name for what really is a simple concept. It’s something that can be measured so using it in a formula is not a problem.
Now (as Zee expresses it in his textbook) behold the wonders of renormalisation. All it is is a simple bit of algebra to replace the coupling constant with λr.
First, its not hard to see, O(λr^n) = O(λ^n) and λ^2 = λr^2 + O(λr^3) so if you subtract (2) from (1) you get
λ'(K) = λr + λr^2*(f(K) -f(U)) + O(λr^3).
The cut-off is gone. Everything is expressed in terms of quantities we know. That we don’t know what cut-off to use now doesn’t matter.
Conclusion
We now have a formula without infinities and the cut-off removed. This is what renormalisation is about. By assuming some cut-off and expressing our formulas in quantities that can be measured, in the above, the renormalised coupling constant, we can actually predict things. There is nothing mathematically dubious happening. It’s a very reasonable and straightforward process.
There is more, indeed a lot more, that can be said. If anyone wants to go further I have found the following good:
http://arxiv.org/abs/hep-th/0212049
Hopefully, this very basic introduction has demystified what is really a simple concept.
My favourite interest is exactly how can we view the world so what science tells us is intuitive.
“That’s interesting, and that answers my question about why, after renormalization, the power series look very much the same as before, except for the replacement of bare constants by renormalized constants. That’s a feature of renormalizable theories, but not in general.”
“First – yes it only works for renormalisable theories.
How non-renormalisable theories are handled – I have that on my to do list.”
Even for non-renormalizable theories, the way to make the series “the same” except with different constants is to include all possible terms consistent with the low energy degrees of freedom and symmetry, ie. include both renormalizable and nonrenormalizable terms. We do this for renormalizable and non-renormalizable theories. As we flow to low energies from high energies, all these terms are automatically generated. Since we included all the terms, obviously we solved the appearance of the traditional counterterms in a “trivial” way. But now we have a “problem”, because we also have non-renormalizable terms even for theories that are traditionally called renormalizable. It turns out that as we flow to low energies, the non-renormalizable terms are suppressed by powers of the cutoff.
A calculation is done in detail by Srednicki in Chapter 29 (p185) of [URL]http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf[/URL]. He writes the generation of all possible terms in Eq (29.9) to (29.11). Then he goes on a digression before returning to the idea in the comments just before Eq (29.24) that we should start in principle by writing down all possible terms where he says “we simply treat this effective action as the fundamental starting point”. This is impractical, but he shows a way to handle it, finally ending on p193 with “The final results, at an energy scale ##E## well below the initial cutoff ##Lambda_{0}##, are the same as we would predict via renormalized perturbation theory, up to small corrections by powers of ##E/ Lambda_{0}##.”
“I remember some other characterization of renormalizable/non-renormalizable theories, and I don’t see the connection with what you’re saying, though. What I’ve heard some people say is that a renormalizable is characterized by a finite number of measurable parameters; in the case of QED, there is just the electron mass and charge. But (this is my paraphrase of what I’ve heard–it’s not my own conclusion) in a non-renormalizable theory, every new order of perturbation theory gives new parameters whose values must be determined experimentally. Do you have any idea what that means? (Whether or it’s true or not, I’m wondering why people would say it)”
First – yes it only works for renormalisable theories.
How non-renormalisable theories are handled – I have that on my to do list. I have seen all sorts of claims I need to look into eg all theories, are in a sense, renormalisable.
“There is yet another aspect of renormalization that I remember dimly, which is the approach of fixing the infinities that arise in the naive approach by introducing counter-terms that cancel the infinities at large momenta. That doesn’t seem exactly the same as introducing a cut-off (although it has the same feature of making larger momenta irrelevant).”
It isn’t exactly the same – in fact after going through the standard approach in the past, such as the link I gave in the paper, I think its utter nonsense.
It’s only taking this cut-off idea literally that makes sense to me.
Thanks
Bill
”
For renormalisable theories it would seem you can shift your energy scale (ie exactly where you stop that limit in getting the field or as Atty says your graining) and it makes no difference to the form of your equations.
This is the interesting renormalisation group flow aspect of the issue.”
That’s interesting, and that answers my question about why, after renormalization, the power series look very much the same as before, except for the replacement of bare constants by renormalized constants. That’s a feature of renormalizable theories, but not in general.
I remember some other characterization of renormalizable/non-renormalizable theories, and I don’t see the connection with what you’re saying, though. What I’ve heard some people say is that a renormalizable is characterized by a finite number of measurable parameters; in the case of QED, there is just the electron mass and charge. But (this is my paraphrase of what I’ve heard–it’s not my own conclusion) in a non-renormalizable theory, every new order of perturbation theory gives new parameters whose values must be determined experimentally. Do you have any idea what that means? (Whether or it’s true or not, I’m wondering why people would say it)
There is yet another aspect of renormalization that I remember dimly, which is the approach of fixing the infinities that arise in the naive approach by introducing counter-terms that cancel the infinities at large momenta. That doesn’t seem exactly the same as introducing a cut-off (although it has the same feature of making larger momenta irrelevant).
“If we go down the energy scale, we average over the lattice. So we go down a bit, we get a coarser lattice. Then we do down again, and we get another even coarser lattice.”
That correct.
If you go right back to the foundations of field theory its a discreetised large number of things that’s assumed to obey some Lagrangian. One then takes the limit to get a continuum, and calls that the field.
The modern view is its the taking of this limit that’s the cause of the problem. That’s why we have a cut-off and take that cut-off seriously.
The interesting thing, as I posted previously, is:
λ’ = λr + λr^2 f(K) – λr^2 f(U) = λr + λr^2 f(K) + λr^2 C*Log (Λ^2) for some cut-off Λ.
That’s exactly the same form as the original equation but with a different λ and cutoff.
For renormalisable theories it would seem you can shift your energy scale (ie exactly where you stop that limit in getting the field or as Atty says your graining) and it makes no difference to the form of your equations.
This is the interesting renormalisation group flow aspect of the issue.
Thanks
Bill
“Very interesting. I look forward to trying to read that one. The discrete scale in-variance part is clear. Love the stack of lattices. Very much what I have been picturing for re-normalization and (way over-simplistically I’m sure) the MERA thing.”
Renormalization as getting a coarse grained lattice from fine-grained one was Kadanoff’s idea. Among his many amazing achievements, Wilson showed that if you do this successively you automatically generate all the mysterious counterterms. But the basic idea of renormalization is simply going down to low energy, which means long wavelength, which means spatial averaging or coarse-graining.
“The last sentence about how that system of lattices obeys the equations of GR. Do you mean the mathematical properties of GR fall out of it (emerges from, or is consistent as a property of such a system of lattices) ? That’s what I am assuming that means, rather than, it’s just “another place where GR goes like GR goes? In other words the idea is see if one can derive the way GR goes from the fundamental properties of such a system of lattices applied to the dimensions of space, time, energy, mass?”
This part about renormalization and GR is special to string theory. In general, renormalization does produce an extra dimension which has the meaning of “energy”, but it has nothing to do with GR. In string theory, when the starting lattice has certain properties, renormalization is conjectured to produce a space that has exactly all the properties of string theory, which has GR as a low energy limit.
“Very interesting. I have to say, I have always felt like LQG and the MERA were somehow so similar. This has helped clarify the difference between them. Or at least provide contrast to what was previously a pure fog of confusion.”
There are intriguing similarities. AdS/MERA was suggested by Brian Swingle (Physics Monkey), and you can see his comments on the relationship between tensor networks (the MERA is a tensor network) and the spin networks of LQG in these posts.
[URL]https://www.physicsforums.com/threads/condensed-matter-physics-area-laws-lqg.376399/#post-2573731[/URL]
[URL]https://www.physicsforums.com/threads/condensed-matter-physics-area-laws-lqg.376399/#post-2576472[/URL]
Very interesting. I look forward to trying to read that one. The discrete scale in-variance part is clear. Love the stack of lattices. Very much what I have been picturing for re-normalization and (way over-simplistically I’m sure) the MERA thing.
The last sentence about how that system of lattices obeys the equations of GR. Do you mean the mathematical properties of GR fall out of it (emerges from, or is consistent as a property of such a system of lattices) ? That’s what I am assuming that means, rather than, it’s just “another place where GR goes like GR goes? In other words the idea is see if one can derive the way GR goes from the fundamental properties of such a system of lattices applied to the dimensions of space, time, energy, mass?
Very interesting. I have to say, I have always felt like LQG and the MERA were somehow so similar. This has helped clarify the difference between them. Or at least provide contrast to what was previously a pure fog of confusion.
[Edit] Sorry to keep asking question. It’s a reflex. You’ve given me a lot to think about. Much appreciated.
“I wouldn’t mind having a reference I could dig into that explains more what the energy scale as spatial dimension means. That’s not connecting to anything I understand at the moment and it sounds like it could.”
OK, let’s take the renormalization as running down the energy scale (don’t worry about up or down here, where the direction is a bit contradictory with what I said above).
If we go down the energy scale, we average over the lattice. So we go down a bit, we get a coarser lattice. Then we do down again, and we get another even coarser lattice. And we do this again and again, getting coarser and coarser lattices. We “stack” the lattices, with the finest lattice at the “top”, and the coarser lattices “below”. For simplicity, let’s start with a 1 dimensional lattice. Then the stack of lattices will be 2 dimension (by the common sense idea of a stack). In ordinary renormalization the dimension created by the stack is “energy”, but maybe it can also mean “space”. In itself that is not special, since we have just renamed the stacking dimension from “energy” to “space”. The special thing is that string theory conjectures that under certain conditions, this space has gravity that obeys the equations of Einstein’s general relativity.
You can see the stack of 1D lattices in Fig. 1 and the emergent 2D space in Fig. 2 of [URL]http://arxiv.org/abs/0905.1317[/URL], where the fine lattice is on top, and the successively coarser lattices are below.
“if the solid model can be extended up the energy scale, that would mean making the lattice spacing finer and finer, corresponding to a continuum.”
I was picturing it having a limit, that stopped short of continuum.
I wouldn’t mind having a reference I could dig into that explains more what the energy scale as spatial dimension means. That’s not connecting to anything I understand at the moment and it sounds like it could.
“So the Wilsonian “solids” lattice (I now have 100 times more context for lattice guage theory that I did 5 minutes ago) could/would extend up the energy scale, if it turned out to be that space-time was discrete, since that’s how it looks at things in the first place.
I have a book on QCD… that I have not started. :frown:”
No, if the solid model can be extended up the energy scale, that would mean making the lattice spacing finer and finer, corresponding to a continuum.
There is an interesting way in which the Wilsonian view of renormalization may “fail” and yet simultaneously succeed his wildest dreams. In the MERA picture, in line with other ideas of renormalization and holography in string theory, the renormalization does not bring you up and down the energy scale. Instead the “energy scale” along which you move is a spatial dimension – it is a way in which space and gravity can be emergent.
So the Wilsonian “solids” lattice (I now have 100 times more context for lattice guage theory that I did 5 minutes ago) could/would extend up the energy scale, if it turned out to be that space-time was discrete, since that’s how it looks at things in the first place.
I have a book on QCD… that I have not started. :frown:
“And I need to revisit that MERA paper to see if I missed a similar natural, rather than introduced, re-normalization thresholding process they were proposing.”
The MERA network is a little special, and it was first introduced to describe systems that (in some sense) don’t require a cutoff. But it is totally within the Wilsonian framework.
Again, the idea is simple. We don’t know the true high energy degrees of freedom like strings or whatever. But at intermediate energies, these degrees of freedom are not needed, and we only need things like electrons and protons. These can (skipping a detail) basically be described as solids, where everything is on a lattice. At high energies, we know the lattice will breakdown, but it is ok at intermediate energies, and we only want to use the lattice at low energies. Renormalization is simply the process of extracting the low energy theory from the lattice.
There we had the lattice and ran the renormalization downwards on the energy scale, making an average lattice that was coarser and coarser.
But could we run the renormalization upwards to high energies, making our lattice spacing finer and finer? In general, our theory will break down because we didn’t put in strings or whatever is needed for consistency at high energies. But in special cases, we can run the renormalization backwards. These special theories make sense even at the highest energies, and the MERA is most suited to dealing with these theories. Although the standard model of particle physics cannot be renormalized upwards, a part of it – QCD – can be (at least in the non-rigourous physics sense). This feature of QCD is called “asymptotic freedom”.
I got a useful lead in terms of my confusion from the first part of the second reference Bhobba gave. From that I have a cartoon under construction (as in it’s a pile of mud and sticks) that the problem has to do with probing (the integral of all the Feynman thingamajigs) inside the plank scale where the energy domain is one that “creates” particles rather than observing them. If we are trying to count a set that our counting is creating, we will have a bit of a feedback loop.
I can imagine this is wildly flawed.
The idea of using a “cutoff” on the “observable” domain seems on the one hand just practical – to get at some useful answers. That paper on the MERA Ansatz is one I’ve tried to understand n times now. It invokes what I have learned about “re-normalization” from Sornette, just in terms of how it looks.
The part I am puzzling about… Today, is whether that threshold fixing process is only invented, or arguably natural. In Sornette’s book “Critical Events in Complex Financial Systems” the idea of critical points was primary (obviously). But in hindsight their naturalness, as introduced, was as much about everyday intuition about the system he was using as an example (investor optimism), rather than a clearly demonstrated fundamental mechanism.
Had a bit of an epiphany diving back into “Evolutionary Dynamics” by Nowak this morning. In sec 7.1 “One Basic Model and One Third”, he shows how critical points form as a pure function of N (size of finite population) under conditions of weak selection. According to the model he describes, the only thing required for real critical points of population “fixing” (where one of two species a and b disappears) would be expansion of the number of a and b, even at the same ratio. Other requirements are: Some non-flat payoff matrix and therefore fitness functions for a and b. That a and b are the best response to each other (strategically stable, or evenly matched for payoff). “Selection intensity” is weak (only some encounters induce selection). Pretty elegant and weird. At least I think that’s what he said.
I need to see if I can find a paper by him, maybe on that chapter. And I need to revisit that MERA paper to see if I missed a similar natural, rather than introduced, re-normalization thresholding process they were proposing.
[Edit] the non-sequitur to Evolutionary Dynamics, goes-like “if space-time is discrete, are there mechanisms that could explain problematic observations, such as probabalistic irreversibility, and the fact that reality doesn’t blow up, even though integrals over QM momenta suggest it should/could/would if we didn’t somewhat arbitrarily re-normalize those integrals”
[Edit] There are a number of papers by M.A. Nowak on arxiv. I’ll have to look to see if there is one on that particular model in his book.
[URL]http://arxiv.org/find/q-bio/1/au:+Nowak_M/0/1/0/all/0/1[/URL]
[Edit] There are also a number of papers by D. Sornette on Arxiv. This one really grabbed me.
[URL]http://arxiv.org/abs/1408.1529[/URL]
[SIZE=6]Self-organization in complex systems as decision making[/SIZE]
[URL=’http://arxiv.org/find/nlin/1/au:+Yukalov_V/0/1/0/all/0/1′]V.I. Yukalov[/URL], [URL=’http://arxiv.org/find/nlin/1/au:+Sornette_D/0/1/0/all/0/1′]D. Sornette[/URL]
(Submitted on 7 Aug 2014)
The idea is advanced that self-organization in complex systems can be treated as decision making (as it is performed by humans) and, vice versa, decision making is nothing but a kind of self-organization in the decision maker nervous systems. A mathematical formulation is suggested based on the definition of probabilities of system states, whose particular cases characterize the probabilities of structures, patterns, scenarios, or prospects. In this general framework, it is shown that the mathematical structures of self-organization and of decision making are identical. This makes it clear how self-organization can be seen as an endogenous decision making process and, reciprocally, decision making occurs via an endogenous self-organization. The approach is illustrated by phase transitions in large statistical systems, crossovers in small statistical systems, evolutions and revolutions in social and biological systems, structural self-organization in dynamical systems, and by the probabilistic formulation of classical and behavioral decision theories. In all these cases, self-organization is described as the process of evaluating the probabilities of macroscopic states or prospects in the search for a state with the largest probability. The general way of deriving the probability measure for classical systems is the principle of minimal information, that is, the conditional entropy maximization under given constraints. Behavioral biases of decision makers can be characterized in the same way as analogous to quantum fluctuations in natural systems.
Bhobba’s statement is “We decide the cuttoff, which had to be there to make the calculation work and we based it on observation”. I think there is actually a serious loop of truth to that way of describing it. The question of how… that decision got made, the whole chain of “fixing”… to me… is more than a little spooky, and vertiginous.
:wideeyed:
Understood, thanks. I liked your presentation – renormalization is pretty intimidating for a layman and you make it seem more approachable : )
The appeal of the kind of argument I mentionned is that it gives an intuitive explanation for why the infinities are innocuous, by suggesting they secretly stand for “large unknown quantities” – even if the actual presentation uses infinity to avoid unnecessary complications that would might make it harder to read.
To me it’s a bit like epsilon delta arguments – they are useless as explanations, but it’s good to know that they could be used to make things rigorous.
“IMy recollection of that argument is very hazy so I am unsure about it, but does this work (or rather, something similar, presumably after some renormalization of the argument :wink: )?”
Yes that would work, but simply assuming the integral has a cut-off is the usual way.
There are also other cut-off schemes depending on the regularisation method used, but I didn’t want to get into that.
Thanks
Bill
“You have to read the reference I gave – its simply the result of calculating the equation.
Very briefly if you have a look at the resulting equation you can simplify it to M(K) = iλ + iλ^2*f(K) + 1/2*λ^2 ∫d^4k 1/k^4 (the integral is from -∞ to ∞) where k is the 4 momentum (its done by breaking the integral into two parts – one for very large k and the other for k less than that). That’s a tricky integral to do but after a lot of mucking about, is i π^2 ∫ 1/k^2 dk^2 where the integral is from 0 to ∞. This is limit Λ→∞ i π^2 log (Λ^2).
But as far as understanding renormalisation is concerned its not germane to it. You can slog through the detail but it wont illuminate anything.
Thanks
Bill”
Isn’t there another way to look at it ? I am not familiar with renormalization but one argument I recall seeing went (very) roughly along the following lines :
Assume the integral is actually finite, because the “true” integrand is not 1/k^2 but some unknown function phi(k) such that phi(k)~1/k^2 for k not too large (or small, wherever it diverges), and phi(k) is summable (i.e. assume we don’t really know the high energy behavior, but whatever it is must give a finite answer, perhaps because spacetime is quantized or whatever reason).
Then the calculation still holds, there are no infinities but the unknown parts cancel off at low energy. The result is the same but the mysterious infinities have been replaced by unknown finite quantities.
My recollection of that argument is very hazy so I am unsure about it, but does this work (or rather, something similar, presumably after some renormalization of the argument :wink: )?
“There are a few things that are a little mysterious about it, still. So if you assume that you understand the low-momentum behavior of a system, but that its high-momentum behavior is unknown, it makes sense not to integrate over all momenta. But why is imposing a cut-off the right way to take into account the unknown high-energy behavior?”
The idea is that below a certain energy, there are “emergent” degrees of freedom that are enough for describing the very low energy behaviour we are interested in. For example, even though the standard model has quarks, for condensed matter physics, we just need electrons, protons and neutrons. The cut-off represents the energy where we will be obliged to consider new degrees of freedom like supersymmetry or strings. If we knew the true degrees of freedom and the Hamiltonian at high energy, we could integrate over the high energies and by an appropriate change of “coordinates” obtain the emergent degrees of freedom and Hamiltonian at the cut-off. However, in practice we do not know the high energy details, so we make a guess about the low energy degrees of freedom and the low energy symmetries (here low means much lower than the high energy, but still much higher than the energy at which we do experiments). So we put in all possible terms into the Hamiltonian with the low energy degrees of freedom that are consistent with any known symmetry. In other words, we must do the integral over the unknown high energy degrees of freedom (as required in the path integral picture), and we attempt to do it by guessing the low energy degrees of freedom and symmetry.
It turns out that even if we use a simple Hamiltonian that is lacking many possible terms, if we have a cut-off that is low enough that we know there are not yet new degrees of freedom, but still much higher than the energy scale we are interested in, then we can show that the low energy effective Hamiltonian will contain all possible terms consistent with the symmetry – these turn out to be the counterterms.
The usual explanation of the counterterms is physically senseless. It is better to think of them as automatically generated by a high cutoff, and a flow to low energy. However, it is incredibly inefficient to start calculations by writing down all possible terms. The counterterm technology is a magically efficient way to get the right answer (like multiplication tables :oldtongue:).
Naturally, our guess about the low energy degrees of freedom may be wrong, and our theory will be falsified by experiment. However, a feature of this way of thinking is that a non-renormalizable theory like QED or gravity, by requiring a cut-off, shows the scale at which new physics must appear. In other words, although experiment can show us new physics way below the cut-off, the theory itself indicates new physics in the absence of experimental disproof.
“There are a few things that are a little mysterious about it, still. So if you assume that you understand the low-momentum behavior of a system, but that its high-momentum behavior is unknown, it makes sense not to integrate over all momenta. But why is imposing a cut-off the right way to take into account the unknown high-energy behavior?”
I think the answer lies in the condensed matter physics these ideas sprung from. I don’t know the detail but the claim I have read is pretty much any theory in the low energy limit will look like that.
Certainly it’s the case here – to second order we have:
λ’ = λr + λr^2 f(K) – λr^2 f(U) = λr + λr^2 f(K) + λr^2 C*Log (Λ^2) for some cut-off Λ.
Its the same form as the un-renormalised equation. This is the self similarity you hear about, that is said to be what low energy equations must be like.
“The second thing that’s a little mysterious is the relationship between renormalization and the use of “dressed” propagators. It’s been a while since I studied this stuff (a LONG while), but as I remember it, it goes something like this:”
Wilson’s view where the cut-off is taken seriously looks at it differently.
If you look at the bare Lagrangian its simply taken as an equation valid at some cut-off – we just don’t know the cut-off. You write it in terms of some renormalised parameters by means of counter-terms where the cut-off is explicit ie the counter terms are cut-off dependant. You then shuffle these counter terms around to try to get rid of the cut-off – similar to what I did.
I am not that conversant with the detailed calculations of this method but I think its along the lines of the following based on the example before.
I will be looking at second order equations. First let λr be some function of λ so λ = λr + aλr^2 for some a ie λ = (1 + aλr)λr where aλr is the first order of the counter term in that approach since we are looking only at second order.
You substitute it into your equation to get the renormalised equation with the counter term:
λ’ = λr + (f(K) + C*log Λ^2 + a) λr^2.
Now we want the cut-off term to go away – so we define a = a’ – C*log Λ^2 and get
λ’ = λr + (f(K) + a’)) λr^2.
We then apply what we say λr is to determine a’ – namely λr = λ'(U) so a’ = -f(U).
My reading of the modern way of looking at renormalisation using counter-terms is its along the lines of the above.
Thanks
Bill
There are a few things that are a little mysterious about it, still. So if you assume that you understand the low-momentum behavior of a system, but that its high-momentum behavior is unknown, it makes sense not to integrate over all momenta. But why is imposing a cut-off the right way to take into account the unknown high-energy behavior?
The second thing that’s a little mysterious is the relationship between renormalization and the use of “dressed” propagators. It’s been a while since I studied this stuff (a LONG while), but as I remember it, it goes something like this:
You start describing some process (such as pair production, or whatever) using Feynman diagrams drawn using “bare” masses and coupling constants. You get loops that would produce infinities if you integrated over all momenta. Then you renormalize, expressing things in terms of the renormalized (measured) masses and coupling constants. This somehow corresponds to a similar set of Feynman diagrams, except that
[LIST=1]
[*]The propagator lines are interpreted as “dressed” or renormalized propagators
[*]You leave out the loops that have no external legs (the ones that would give infinite answers when integrating over all momenta)
[/LIST]
The first part is sort of by definition: The renormalization program is all about rewriting amplitudes in terms of observed masses and coupling constants. But the second is a little mysterious. In general, if we have two power-series:
[itex]A = sum_n A_n lambda_0^n[/itex]
[itex]lambda = sum_n L_n lambda_0^n[/itex]
we can rewrite [itex]A[/itex] in terms of [itex]lambda[/itex] instead of [itex]lambda_0[/itex] to get something like:
[itex]A = sum_n B_n lambda^n[/itex]
For a general power series, you wouldn’t expect the series in terms of [itex]B_n[/itex] to be anything like the series in terms of [itex]A_n[/itex]. But for QFT, it seems that they are basically the same, except that the [itex]B_n[/itex] series skips over the divergent terms.
I’m guessing that the fact that the renormalized Feynman diagrams look so much like the unrenormalized ones is a special feature of propagators, rather than power series in general.
“Can you explain more why the improper integral over momentum blows up?”
You have to read the reference I gave – its simply the result of calculating the equation.
Very briefly if you have a look at the resulting equation you can simplify it to M(K) = iλ + iλ^2*f(K) + 1/2*λ^2 ∫d^4k 1/k^4 (the integral is from -∞ to ∞) where k is the 4 momentum (its done by breaking the integral into two parts – one for very large k and the other for k less than that). That’s a tricky integral to do but after a lot of mucking about, is i π^2 ∫ 1/k^2 dk^2 where the integral is from 0 to ∞. This is limit Λ→∞ i π^2 log (Λ^2).
But as far as understanding renormalisation is concerned its not germane to it. You can slog through the detail but it wont illuminate anything.
Thanks
Bill
“Wow, that’s all renormalization is? Just saying, “Let’s not use our theory at absurdly high energies”?”
Yes.
Hopefully the post I did above helped understand why it remained a mystery for so long.
Stuff we use in our equations such as the electron mass and charge secretly depended on the cut-off. We did experiments at low energy from which we obtained small values, which when you look at the equations, means in effect its a low cut-off – otherwise you would get large values. This fooled people for yonks.
The way it works, in QED for example, is by means of what’s called counter-terms:
[URL]http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-5-RenormalizedPerturbationTheory.pdf[/URL]
You start out with a Lagrangian expressed in terms of electron mass and charge. But we know those values actually depend on the cut-off. We want our equations to be written in terms of what we actually measure which pretty much means measure at low energies. This is called the renormalised mass and charge. We write our Lagrangian in terms of those and you end up with equations in those quantities plus what are called counter terms that are cut-off dependant. You do your calculations and of course it still blows up. But now you have these counter terms that can be adjusted to cancel the divergence. Basically this is subtracting equation (2) from (1) in my paper. The cut-off dependant quantities cancel and you get finite answers.
Tricky – but neat.
Thanks
Bill
Hi Guys
Thanks so much for your likes – its appreciated.
One thing I wanted to mention, but didn’t since I wanted to keep it short, is why, since the solution is actually quite straightforward, it took so long to sort out.
To see the issue, we defined λr in a certain way to be a function of λ and the cutoff. To second order, once we invert the function, λ = λr + aλr^2.
λ’ = λ + (f(K) + Clog(Λ^2))λ^2 = λr + aλr^2 + (f(K) + Clog(Λ^2))λr^2.
Now since λr = λ'(U), λr = λr + aλr^2 + (f(U) + Clog(Λ^2))λr^2 to second order, so
a = -(f(U) + Clog(Λ^2))
Since λ = λr + aλr^2 this means, to second order at least, λ secretly depends on the cutoff, and as the cutoff goes to infinity, so does the coupling constant. Normally when pertubation theory is used, you want what you perturb about to be a lot less than one. If not the higher orders do not get progressively smaller. If, for some reason, that’s not the case, it will likely fail, and you have made a lousy choice of perturbation variable. At the energies we normally access this corresponds to a cutoff that’s not large, and you get a λ that’s small, so you think its ok to perturb about. But, as we have seen, it secretly depends on the cutoff, in fact going to infinity with the cutoff. This makes it a downright lousy choice of what to perturb about. Not understanding this is what caused all these problems and took so long to sort out.
Thanks
Bill
Nice first entry [USER=366323]@bhobba[/USER]
Can you explain more why the improper integral over momentum blows up?
Wow, that’s all renormalization is? Just saying, “Let’s not use our theory at absurdly high energies”?