Connection between QFT renormalization and field regularization?

[Jarek 31]

TL;DR Summary

Both renormalization and regularization are used to remove infinities in similar situations (e.g. charged particle), but from different perspectives - is there a connection/correspondence between them?

While in QFT we remove infinite energy problem with renormalization procedure, asking e.g. "what is mean energy density in given distance from charged particle", electric filed alone would say $$\rho \propto |E|^2 \propto 1/r^4 $$
But such energy density would integrate to infinity due to singularity in zero.
In field theories (usually classical, but they can be quantized e.g. https://arxiv.org/pdf/hep-th/0505276.pdf ) we use regularization to remove this kind of infinities, for example vector field with Higgs potential V(u)=(|u|^2-1)^2 - preferring unitary vectors, but allowing for their deformation to prevent infinity like below.

So is there a connection/correspondence between these two approaches to remove infinity of singularity - applied to the same situation like charged particle, but from different perspectives?

 

[A. Neumaier]

Regularization + cutoff to infinity is just a particular method for performing renormalization - the most elementary one.

 

[Jarek 31]

So can we answer "what is mean energy density in given distance from charged particle" question in a way integrating to a finite energy (in contrast to standard rho~1/r^4) like below 511keV for electron?

 

[A. Neumaier]

So can we answer "what is mean energy density in given distance from charged particle" question in a way integrating to a finite energy (in contrast to standard rho~1/r^4) like below 511keV for electron?

Only approximately. Any regularization involves an approximation.

In renormalization, the approximation is removed by redefining the parameters as functions of the cutoff in such a way that one can take the cutoff to infinity, thereby restoring exactness.

 

[Jarek 31]

Approximately would be still great.
Not to exceed 511keV mass of electron, this deformation/regularization of electric field of perfect point charge would need to be femtometer-scale, but what are its details?
Cutoff as in https://en.wikipedia.org/wiki/Classical_electron_radius makes a hole in spacetime - physics rather requires a continuous answer.
I am only aware of Faber's approach to try to answer this fundamental question (fig. 2 in https://iopscience.iop.org/article/10.1088/1742-6596/361/1/012022/pdf ), is there any other?

 

[A. Neumaier]

I cannot give references. In any case you'd look at the quality of the approximation by checking what happens when you change the cutoff by a reasonable amount. If the result changes a lot, it is unreliable.

 

[Jarek 31]

If by cutoff you mean integrating from some minimal radius, it makes hole in spacetime - what happens inside?
Shouldn't there be a continuous rho(r) dependence down to r=0?

 

[A. Neumaier]

If by cutoff you mean integrating from some minimal radius, it makes hole in spacetime - what happens inside?
Shouldn't there be a continuous rho(r) dependence down to r=0?

The cutoff is the point where you stop integration in order to regularize the integral, although formally you should continue. I cannot tell you how to apply it to your particular integral or how to interpret it.

 

[Jarek 31]

Indeed, I understood you would like to calculate total energy of electric field by integrating energy density from some cutoff: nonzero radius - what is brushing infinity under the rug.
Physics requires integration from r=0, complete answer should go there (like in Faber's approach).

 

[A. Neumaier]

If by cutoff you mean integrating from some minimal radius, it makes hole in spacetime - what happens inside?
Shouldn't there be a continuous rho(r) dependence down to r=0?

Indeed, I understood you would like to calculate total energy of electric field by integrating energy density from some cutoff: nonzero radius - what is brushing infinity under the rug.
Physics requires integration from r=0, complete answer should go there (like in Faber's approach).

If the formula can be renormalized you should make the parameters (bare electron mass and charge) dependent on the cutoff $\epsilon$, approximately evaluate the integral with arbitrary finite $\epsilon$ then adjust the $\epsilon$-dependence of bare mass and bare charge such that in the result you can take again the limit $\epsilon\to 0$. With this procedure nothing happens ''inside'' because there is no inside in the limit. This is the recipe that works in QFT.

If you can't get this to work it means you cannot ignore what happens inside, and your ad hoc regularization is just an ugly trick...

 

[Jarek 31]

But the question of concern is: "what is mean energy density in given distance from charged particle".
Regularization e.g. using Higgs potential like in the top post here (or in Faber) allows to resolve this kind of questions down to r=0.

Why answer to such objectively-looking question should depend on our subjective choice of cutoff?

 

[A. Neumaier]

But the question of concern is: "what is mean energy density in given distance from charged particle".
Regularization e.g. using Higgs potential like in the top post here (or in Faber) allows to resolve this kind of questions down to r=0.

Why answer to such objectively-looking question should depend on our subjective choice of cutoff?

If physics at short distance matters then the cutoff must be where this new physics is supposed to appear, and it becomes not a subjective matter to pick it but a parameter to be fitted to data.

If physics at short distance does not matter then the cutoff can be removed by renormalization as described above, and nothing subjective remains.

 

[Jarek 31]

This question requires rho(r) energy density function: asymptotically rho~1/r^4, but regularized: going down to r=0 (not some subjective cutoff) and integrating at most to 511keV mass of electron.
Maybe equal to 511keV (like in Faber) - if below, where is the difference?

At least approximate answer - the complete one should also include angular dependence from energy of magnetic field from electron dipole moment.

Claiming Standard Model is practically complete, shouldn't we know answer to such a basic question?
Maybe e.g. lattice QED could allow to answer it?

 

[A. Neumaier]

Claiming Standard Model is practically complete, shouldn't we know answer to such a basic question?

The standard model is far too complicated to allow handling purely electromagnetic issues. To get to QED one usually simply drops terms and estimates (in the few cases where the accuracy requires it and the computations have been done) the error made based on heuristics.

To answer your question by QED one has to make further approximations. This is nontrivial research work, which you must do yourself. I have neither the time for it nor do I know a source who does it; . Many questions in QED are unanswered, once they go beyond the traditional stuff.

Maybe e.g. lattice QED could allow to answer it?

Lattice QED is completely powerless. It suffers from triviality issues and hasn't produced a single experimentally verifiable result. See the discussion in the PF thread on Lattice QED.

For numerical caclulations one usually works with NRQED. Searching in scholar.google.com for

NRQED "energy density"

(quotation marks needed to get the exact phrase) one finds some references, maybe something there helps you...

 

[Jarek 31]

It is hard to imagine more fundamental question about electron than what is energy density around it.

If it is still too difficult for standard QFT approach, maybe it is worth to support with alternative perspectives for this field regularization problem/question, like using Higgs potential as in the top post?

 

[A. Neumaier]

If it is still too difficult for standard QFT approach

I don't think it is too difficult for standard QFT but that nobody invested the time to work it out, because it is not directly observable. Or maybe someone even did (I don't know the literature completely) and the paper wasn't referenced enough to be easily encountered.

 

[Jarek 31]

Quantum fields also aren't directly observable, but we are interested in understanding them - what simpler question can we ask then of their mean energy density around electron?
Especially when getting ~120 orders of magnitude disagreement of vacuum energy density - problem swept under the rug, which might be worth to finally resolve.

 

[vanhees71]

This question requires rho(r) energy density function: asymptotically rho~1/r^4, but regularized: going down to r=0 (not some subjective cutoff) and integrating at most to 511keV mass of electron.
Maybe equal to 511keV (like in Faber) - if below, where is the difference?

At least approximate answer - the complete one should also include angular dependence from energy of magnetic field from electron dipole moment.

Claiming Standard Model is practically complete, shouldn't we know answer to such a basic question?
Maybe e.g. lattice QED could allow to answer it?

The problem is that the Standard Model (or QED as part of it to keep it simple), based on relativistic QFT, is much less complete than classical electrodynamics with classical point particles, because in fact there is no fully satisfactory description of it, and it's precisely related to the question you address here, i.e., the energy of the electromagnetic field of a point particle, which takes into account the interaction of the particle with its own electromagnetic field.

The best one can come up with still is the Lorentz-Abraham-Dirac equation. This can be done in a modern way using a pretty simple regularization scheme, invented by Lechner et al, which just introduces some length scale $\epsilon$ into the retarded propagator of the D'Alembert operator,
$$D(x)=\frac{1}{2 \pi} \delta(x^2), \quad x^2=x \cdot x=(x^0)^2-\vec{x}^2,$$
defining its regularized form by
$$D_{\epsilon}(x)=\frac{1}{2 \pi} \delta(x^2-\epsilon^2).$$
You calculate the em. field of the charge as a functional of its time-like worldline and then the self-force of this regularized particle along its world line. You end up with an equation, where the piece that is diverging in the physical limit $\epsilon \rightarrow 0$ can be lumped to the left-hand side of the equation of motion, thus providing an inifinite contribution to the particle mass. Assuming also an infinite bare mass for the particle, this diverging part cancels resulting in the physical (renormalized) mass of the particle. Then you can make $\epsilon \rightarrow 0$, and you end up with the LAD equation (working in Heaviside-Lorentz units with $c=1$),
$$m \ddot{x}^{\mu}=K_{\text{ext}}^{\mu}+\frac{q^2}{6 \pi} \left (\dddot{x}^{\mu} + \ddot{x}^2 \dot{x}^{\mu} \right).$$
The trouble with this equation is that it is obviously of 3rd order in the derivatives (the dot stands for derivatives wrt. proper time), and this gives rise to some unphysical solutions, with the acceleration going to infinity at the remote past ("self-acceleration and runaway solutions") which have to be excluded by boundary conditions concerning the motion for $\tau \rightarrow \pm \infty$. Basically one has to assume that the acceleration goes to $0$ in the infinite past and future. Even excluding the runaway solutions there is also preacceleration, i.e., assuming an external force acting only within a limited space-time region, admitting the boundary conditions concerning the acceleration to be fulfilled, the acceleration of the particle sets in before the external force is actually supposed to be acting. For not too large external forces this violation of causality is limited to the time scale given by $q^2/(6 \pi)$ in the LAD equation, but it's nevertheless a violation of causality. So there's strictly speaking no satisfactory self-consistent model for point charges, except for the free particle, where the boundary conditions exclude the self-acceleration and runaway solutions, leading to the "trivial" solution $\dot{x}^{\mu}=0$ as it should be for a free particle with the (boosted) Coulomb field as the particle's field, which is of course having the infinite self-energy of the particle which has to be considered to be absorbed in the particle's finite physical mass.

It's also clear, however, that the time scale $q^2/(6 \pi)$ related to the socalled classical electron (if we consider electrons as the charged particles) radius up to some factor, is way smaller than the Compton wavelength (even for the lightest known charged particle, the electron), and this indicates indeed that the classical point-particle limit must break down way before we can resolve these small length and/or time scales such as the classical electron radius. That justifies the approximation of the LAD equation by the socalled Landau-Lifshitz equation, which has no preacceleration behavior.

I've started to write all this down in my SRT FAQ article (but it's not finished yet):

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

For the complete discussion see the quoted textbook by Lechner and the also quoted paper as well as the paper by Nakhleh.

The more classical regularization is to consider charged particles of finite extent, including the introduction of Poincare stresses to keep this "particle" stable, leading to a differential-difference equation, which is nicely discussed (in the non-relativistic limit though) in

https://doi.org/10.1119/1.3269900

 

[A. Neumaier]

Quantum fields also aren't directly observable, but we are interested in understanding them - what simpler question can we ask then of their mean energy density around electron?
Especially when getting ~120 orders of magnitude disagreement of vacuum energy density - problem swept under the rug, which might be worth to finally resolve.

go and resolve it instead of complaining that nobody does!

 

[Jarek 31]

Sure - in the top post here I have also presented Higgs potential regularization approach used e.g. by Faber, which I see promising to understand energy density around electron - so do you agree or disagree with it?

ps. Below is fig. 2 from https://iopscience.iop.org/article/10.1088/1742-6596/361/1/012022/pdf (r0=2.21fm): red is standard electric field energy density for electron, to prevent it going to infinity in zero there is activated Higgs potential (blue). Experimental consequence of such regularization/deformation is running coupling - deformation of Coulomb potential in high energy collisions (fig. 4).

 

[A. Neumaier]

I've started to write all this down in my SRT FAQ article (but it's not finished yet):

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

... which seems to be pat of a much bigger project at https://itp.uni-frankfurt.de/~hees/pf-faq/

Don't forget to include the no-go result of

• Currie, Jordan and Sudarshan, Reviews of Modern Physics 35.2 (1963): 350

on classical relativistic multiparticle dynamics and the Einstein–Infeld–Hoffmann Lagrangian; cf.

• https://arxiv.org/pdf/1906.00308.pdf

 

[vanhees71]

I guess you mean this conclusion from Currie et al:

In Sec. IV
we show that if one assumes these equations and
assumes that the generators II, P, J, and K satisfy
the Poisson bracket equations characteristic of the
Lorentz group, then at least for the case of two parti-
cles (not two particles and a field) one can conclude
that both of the particles must have a constant
velocity so that the theory is unable to describe any
nontrivial interactions between the particles.

That should make indeed a nice addition to Chpt. 2.

The paper about the Einstein-Infeld theory I didn't know yet. That's of course very interesting too. The question is, whether in a manuscript about SR one should take into account models "beyond standard physics"...

 

[A. Neumaier]

I guess you mean this conclusion from Currie et al:

yes

The paper about the Einstein-Infeld theory I didn't know yet. That's of course very interesting too. The question is, whether in a manuscript about SR one should take into account models "beyond standard physics"...

EHI is very old, pre renormalization, and only uses classical relativistic views.

 

[vanhees71]

Yes, I know. What I meant is, it's not in the usual "canon" taught in standard physics textbooks. I'm not so sure, whether one should include it in a manuscript about special relativity meant to summarize the standard theories used today.

 

[A. Neumaier]

Yes, I know. What I meant is, it's not in the usual "canon" taught in standard physics textbooks. I'm not so sure, whether one should include it in a manuscript about special relativity meant to summarize the standard theories used today.

Well, it gives the first post-Newton approximation mediating between special and general relativity. See the derivation in the much cited 1991 paper https://journals.aps.org/prd/abstract/10.1103/PhysRevD.43.3273

Se also the discussion in the thread https://www.physicsforums.com/threads/einstein-infeld-hoffman-paper.997784/

 

[Jarek 31]

Just found https://www.researchgate.net/publication/338980602_Something_is_rotten_in_the_state_of_QED (how accurate it is?) with quotes from Dirac and Feynman about removing infinities in QFT:

All these techniques are illegitimate from a mathemati-
cal perspective, as demonstrated by Dirac: “I must say that I
am very dissatisfied with the situation because this so-called
’good theory’ does involve neglecting infinities which ap-
pear in its equations, ignoring them in an arbitrary way.
This is just not sensible mathematics. Sensible mathematics
involves disregarding a quantity when it is small – not ne-
glecting it just because it is infinitely great and you do not
want it!.” [33]
This technique of ignoring infinities is called renormal-
ization. Feynman also recognized that this technique was
not mathematically legitimate: “The shell game that we play
is technically called ’renormalization’. But no matter how
clever the word, it is still what I would call a dippy process!
Having to resort to such hocus-pocus has prevented us from
proving that the theory of quantum electrodynamics is
mathematically self-consistent. It’s surprising that the
theory still hasn’t been proved self-consistent one way or the
other by now; I suspect that renormalization is not mathe-
matically legitimate.” [1]

 

[vanhees71]

Well, yes. A lot of physicists' "robust use of mathematics" was not mathematically rigorous. Sometimes it lead however to the development of entire subbranches of (applied as well as pure) mathematics. A prime example is Dirac's (in)famous "$\delta$ function" which triggered the devlopment of modern functional analysis and the theory of "generalized functions" or "distributions".

 

[Jarek 31]

Indeed Dirac delta (generalized) function/distribution (e.g. for electron charge) is very nice from mathematics perspective, but in physics would usually lead to infinite energy - requires regularization of this idealization, we should search for - beside removing infinities by hand, or cutting holes in spacetime with cutoffs.

 

[A. Neumaier]

involve neglecting infinities which appear in its equations, ignoring them in an arbitrary way.
I suspect that renormalization is not mathematically legitimate.

This is a very shallow, invalid view of renormalization. Infinities are neither neglected nor ignored in renormalization, but avoided by taking careful limits. Moreover, by using distributions, renormalization can be set up in a way that everything is mathematically rigorous (and hence legitimate).

Well, yes. A lot of physicists' "robust use of mathematics" was not mathematically rigorous. Sometimes it lead however to the development of entire subbranches of (applied as well as pure) mathematics. A prime example is Dirac's (in)famous "$\delta$ function" which triggered the development of modern functional analysis and the theory of "generalized functions" or "distributions".

And the infinity-ridden renormalization recipes of Feynman, Tomonaga, and Schwinger triggered the development of modern [URL='https://www.physicsforums.com/insights/causal-perturbation-theory/']causal perturbation theory[/URL], which is as rigorous as the theory of distributions.