In quantum field theory, e.g. in quantum electrodynamics, renormalisation is used to make sense of an infinite number of virtual particles. This, crudely, involves the subtraction of infinities. But which cardinality of infinities, given that Cantor has provided us with more than one member of the set of infinities? Again, can we just go back to the pre-Cantor days of one simple infinity to make things easier for ourselves.
Quantum Field Theory – Cardinality of Infinities in Renormalization Explained
mathematicsquantum-field-theoryregularizationrenormalization
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A good way to do dimension analysis in computing amplitudes relies on a good power-counting of the action. Let me explain how it works before answering your question. For simplicity, in the following, I will consider a QFT in $d=4$.
Usually, it is very useful to distinguish between couplings, decay constant and mass scales. In order to do so, we restore the dimensions of $\hbar$. What I mean is that the action of the system has to have dimension of $\hbar$, namely
$$ [\mathcal{S}] = [\hbar].\qquad\qquad\qquad (1) $$
Let us consider the very simple example of a One Coupling One Scales scalar theory (1C1S scalar theory), characterized by one coupling $g_*$ and one scale $\Lambda$ (that you interpret as a cut-off which may drive the derivative expansion of the EFT in powers of $\partial/\Lambda$).
Having restored the powers of $\hbar$, the dimensions of the fields are not just those of powers of energy. Indeed, from the kinetic term
$$ [\hbar]=[\mathcal{S}] = [\int d^4x \, (\partial\phi)^2] = M^{-4} M^2 [\phi]^2 $$
where $M$ just means "dimension of a mass". From this dimensional analysis we see that $[\phi] = \hbar^{1/2} M $, and analogously for spinorial fields $[\Psi] = \hbar^{1/2} M^{3/2}$.
We define then a coupling $g_*$ whose dimension is $[g_*] = [\hbar]^{-1/2}$. For the scalar theory, you can see this as an equivalent definition of $g_*$ as the square of the coefficient of the marginal $\phi^4$ interaction. Indeed, the $\phi^4$ interaction has the correct dimension as in Eq.(1)
$$ \int d^4 x\, [g_*^2 \phi]^4 = M^{-4} [\hbar]^{-1} M^4 [\hbar]^{2} = [\hbar]. $$
You see then the Lagrangian can be written as
$$ \mathcal{L}(\phi,\partial\phi) = \frac{\Lambda^4}{g_*^2}\hat{\mathcal{L}}\left(g_* \frac{\phi}{\Lambda},\frac{\partial}{\Lambda}\right) $$ where $\hat{\mathcal{L}}$ is a dimensionless function of its dimensionless arguments. In general, any insertion of the field can carry a $O(1)$-coefficient that is not fixed by the power-counting (these coefficients will be called $a_1,a_2,a_3....$; see below). So, for example, you can write the most general terms compatible with the symmetries of the theory. In this case, we do not have any symmetry, and we can write
\begin{align} \mathcal{L} &\supset \frac{\Lambda^4}{g_*^2}\left( a_1 g_*^2 \frac{(\partial\phi)^2}{\Lambda^4} + a_2 \,g_*^2 \Lambda^{-2} \phi^2 + a_3\, g_*^3 \Lambda^{-3}\phi^3+a_4\, g_*^4 \Lambda^{-4} \phi^4 + a_5\, g_*^4 \Lambda^{-6}(\partial\phi)^2\phi^2\right)\\ &= a_1(\partial\phi)^2 + a_2 \, \Lambda^{2} \phi^2 + a_3\, g_* \Lambda\phi^3+a_4\, g_*^2 \phi^4 + a_5\, g_*^2 \frac{(\partial\phi)^2\phi^2}{\Lambda^2}\,.\qquad \qquad \qquad(2) \end{align}
To have a canonically normalized scalar field we can set $a_1=1/2$.
Let's come to your question. First of all, you have to identify the correct dimension of the scatting amplitude (and this depends on the definition of the S-matrix and the normalization of the states of the Hilbert space). I use the same relativistic normalization of states and definition of $S-$matrix as in the Schwartz's book of QFT. For a $2\rightarrow 2$ scattering in $d=4$, the amplitude has dimension
$$ [\mathcal{A}_{2\rightarrow 2}] = [\hbar]^{-1} = [g_*]^2\,\qquad \qquad \qquad (3) $$ Indeed, my scattering amplitude contribution from $g_*^2\phi^4$ goes like $\sim g_*^2$
Please, note the dimension of the amplitude depends on the space-time dimensions and the number of scattered particles. In general $d$ dimension the $n-$point amplitude has dimension $d-n\,d/2+n$.
Eq.(3) means that terms like $g_*^2 (E^2/\Lambda^2 + E^4/\Lambda^4)$ are generically allowed if there are irrelevant operators which contribute at these orders in the energy; here $E$ is the energy of the center of mass $E= \sqrt{s}$ with $s$ the Mandelstam variable.
In the example of 4-Fermi theory, you can identify (in my normalization)
$$ G\simeq \frac{g_*^2}{\Lambda^2} $$
If you have the operator $ \frac{g_*^2}{\Lambda^2} (\bar{\psi}\gamma^\mu \psi)^2$, the only contribution at tree-level is $g_*^2\, s/\Lambda^2 $. The contribution proportional to $G^2$ comes from a loop. This is because loops carry a power of $\hbar$ which cancels the dimension of the extra $G$ insertion. When dealing with loops and renormalization, you have to fix a renormalization scheme.
$$ \text{loop contribution}\simeq \frac{g_*^4}{\Lambda^4} (\#) $$
What do we put in $(\#)$ ?
If you use cut-off regularization (i.e. you cut-off the momenta integral of the loop) you can have a combination of powers of $s$ and $\Lambda$; However, these contributions are local and can be eliminated by choosing counter-terms; they are not physical. The only physical contribution from the loop-integral is a logarithm, namely $s^2 log(s/\Lambda^2)$.
I believe that Zee is discussing two different scenarios in a very general way, without entering in details and without discussing renormalization of fields
Very high cut-off (but you know the theory has a physical cut-off), small probing energies. The loop integral $I = \int d^4 k \frac{1}{k}\frac{1}{k+p}$ where $p$ is an external momenta, has dimension $2$. Then, the result can be $s$ or $\Lambda^2$. In the assumption $s\ll \Lambda$ you take $\Lambda^2$ possibly multiplying a log.
You might believe your theory is UV-completed and that it can be probed at any energy; namely, we do not have a cut-off and the only dimensionful quantity is the center of mass energy (and maybe the renormalization scale or particle masses which are however assumed to be small); then, the result of the loop can be $s$ possibly multiplying a log.
Note that in a massive (UV-completed) QFT the amplitude is constrained by the Froissart-Bound to be bounded from above, namely $|\mathcal{A}(s)| < s\cdot log^2(s)$.
Best Answer
Definitions differ, but cardinal numbers are either specific sets (e.g. the ordinals $\omega$ where any ordinal $\alpha$ isomorphic to an initial segment of $\omega$ cannot be bijectively mapped to $\omega$), or (proper) equivalence classes (e.g. the cardinality of a set $X$ is the proper equivalence class of all and exactly those sets $Y$ such that there is a bijective map from $X$ to $Y$). In neither case is there strictly speaking a set of infinities, at least not in ZFC (the overwhelmingly popular formalization of modern mathematics). A set with more than finitely many things in it has an infinite cardinality, but that's about a set, and one set has one cardinality and there just is no set of all the cardinalities (for no interesting reason, it is just inconsistent with the most popular things mathematician like to assume).
I don't bring that up to be pedantic. The purpose is to point out that (in ZFC) there isn't a set of all the infinite cardinals, and that cardinal numbers are associated with sets, and only with sets. In set theory the cardinals only apply to sets and a cardinal number itself may (or may not) be a set (the word proper class is used to say that something isn't a set), but (in ZFC) if you have a set of cardinalities, then you don't have them all. This will turn out to have nothing whatsoever to do with any infinities in physics, which are all about real parameters and their limits (getting larger, getting farther away from each other, etcetera). In mathematics they want to talk about sets with lots of things in them. In physics we want to make actual predictions so an infinity is something you can approach (approaching an infinite distance means going farther and farther away, you don't actually get there).
So now let's turn to quantum field theory. The idea is to make a theory that can be compared to experiment. This means you have room for some parameters (that can be set by experimental measurements) and you have some predictions you want to get from the theory (that can depend on the parameters). Often it is hard to find the an expression or algorithm for finding an exact answer with a finite amount of effort. But often you can setup a series of calculations designed to converge to a prediction where if you want more precision in your answer you need to do more steps in your calculation. particualr examples of this are called perturbation theory. But the thing you are trying to predict is a regular ordinary (and finite) real number (e.g. a lifetime for an unstable particle, an energy level, a portion of an S matrix, etcetera). The closest to an infinity you might see is that you might do an integral over all space, and this kind of infinity is really a limit (imagine doing an integral in spherical coordinates and doing it over a sphere where the radius gets larger and larger). It is 100% not an infinity like in set theory, and strictly speaking we don't truly need infinite sets to do it since we want a finite precision answer and we want to get it after a finite amount of work and you can interpret everything we write down as "merely" (or actually) describing actual implementable finite algorithms.
To see where and how renormalization comes up we need to get at the concept of an effective theory. The idea is that you can make a theory with some parameters that might look like they relate in a simple way to measurements we've done. But then when you include all the interactions you realize that all real measurements happen in a world with interactions, so the measured numbers should be the outcomes of your predictions from the actual theory not from a toy idealization of it. So you adjust the parameters in your theory so that the predictions of your theory match what we actually see in experiments. This is called renormalization. It happens in many theories even ones without any divergences. When we say that at theory is renormalizable what the term actual means is there a finite number of measurements can fix the undetermined parameters in the model. Strange as it may seem, when interactions are complex, many theories do not have that property. But any theory that first requires we do an infinite number of measurements before they can make any predictions are technically useless to scientists (even if they were true in some sense of the word).
So let's talk about the divergences. Remember when we had a theory whose parameters originally looked like they might bear a simple relationship to measurements? Sometimes when you ask for more precise predictions the differences between those parameters and the measurements grows larger. But they never were the same, it's a psychological thing to think they should be related. For instance people talk about bare masses and effective masses for the electron. The bare mass is a parameter in the theory. An effective mass is based on the fact that when very isolated, the energy and momentum of an electron seem to bear a certain relationship to each other characterised by a scalar. But electrons aren't really isolated and the effective mass is a idealized construct. But by measuring it to some level of precision we can constrain the bare mass in the theory, which is the actual parameter. The fact that these numbers are so different is simply because the electron doesn't exist by itself, the electron we say is isolated is actually interacting, so the thing we measure is really the electron and the net effect of all the interactions. Since interactions make an effect, the parameter value and the measurement value turn out to be very different are very different. That's fine, interactions are real.
So we've talked about cardinality, renormalization, and a bit about effective theories. But not really about divergences yet. So firstly the original theory was hard, so we worked out ways to make approximations. And the theories we called renormalizable were actually the ones whose approximations are renormalizable. We want a whole family of approximations that can get arbitrarily close to the right answers. For these approximations we need to group enough stuff together to get an unbiased answer to avoid being way off, and we also need to control how off the approximation is from the original (hard) theory so that our family can get closer and closer (to the original, hard) theory. But remember that in renormalization we adjust the parameters so that some of the predictions agree with actual measurements, and that after we have some small number of those measurements we've fixed all the parameters.
But those actual measurements are about the full interaction, not the approximation. So when we take an approximate theory about make the measurements fit the approximate theory we get wrong parameters (but good predictions). These values of the parameters work for the approximation (because we made them to do that), but they wouldn't work well for the original theory. Thus when we switch to another member of the family we get different values of the parameters from the renormalizations.
So there are different infinities coming up. 1) There are cases of the parameters getting extreme in value as the approximations get better, I'm not even sure if this an actual issue to worry about at all, you could for instance use $1/r$ instead of $r$ when doing spherical or polar coordinates, but a choice of parameters going to infinity could just be all about your choices. If enough research money was put on the table for this nonissue people might investigate whether different parametrizations avoid this problem entirely. This is not a set theoretical infinity since it is just a limit things are approaching as we go to better and better approximations. 2) There are limits of integration involved in computing integrals, this is also not a set theoretical infinity since it is just a limit used for taking an integral, by choosing an appropriately symmetric shape that was very large you can often just get some small and finite error in your answer by cutting off your integral and this is often all that is meant by saying that you have infinite limits of integration 3) Some of the integrals might themselves diverge (so we can't just cut off a distance away like above). But these integrals are meant to be part of an approximation, all you have to do is accept that they are meant to approximate the real predictions and use the whole framework as a tool to do that. If parts are meant to cancel, make them cancel, pair off symmetries that cancel each other out by noting exactly what the approximations are trying to approximate. You have to focus on the approximation itself and pay attention to what it is doing. You must pay attention to the fact that if you want the approximation's predictions to agree with measurements you must adjust the approximation's parameters to fit (as long as a finite number of measurements can be used to fully fit all of the parameters this is OK). For renormalizable theories this can eliminate any problems in the approximation, and the theories where this can be done are called renormalizable. The price you pay is you might see these parameters getting larger and larger in magnitude for the better and better approximations. This is just #1 above which might not mean anything whatsoever. Which is the important thing. There simply isn't a thing to worry about. The divergent integrals are not set theoretical infinities, because they are just integrals that produce finite numbers for large spheres and that just don't give finite numbers if each sphere gets infinitely large by itself, you have to group them together and make the spheres you integrate on get larger together. Why? Because that's the thing we are calling the approximation. It's a human desire to have each integral involved give a finite number when the sphere gets large, because then we can work one them one at a time. Human desires are just that. The approximation itself involves making the spheres we integrate over get large together, because that's the method that we found approximates the original theory that was hard for us humans to compute with directly.
I originally wanted to give more details about effective theories (which is just more details about how we decide how to group things, our motivates for that, and how we get out approximations in a way that does also leave room for different theories to make different predictions than our current theories), but now I'm thinking the above actually fully addresses your question about set theoretical infinities.