# Dressed electrons are not defined as point masses...

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In @A. Neumaier 's excellent Physics FAQ, he notes under "Are electrons pointlike/structureless?" that

"Physical, measurable particles are not points but have extension. By definition, an electron without extension would be described exactly by the 1-particle Dirac equation, which has a degenerate spectrum. But the real electron is described by a modified Dirac equation, in which the so-called form factors figure. These are computable from QED, resulting in an anomalous magnetic moment and a nonzero Lamb shift removing the degeneracy of the spectrum. Both are measurable to high accuracy, and are not present for point particles, which by definition satisfy the Dirac equation exactly."

I presume having extension means that the mass of a physical electron is not considered to be concentrated at a single point?

However, Wikipedia (hmm) says this:

"Nevertheless, there is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket can be represented as a quantum superposition of quantum states wherein the particle is exactly localized. Moreover, the interactions of the particle can be represented as a superposition of interactions of individual states which are localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero."

So I'm confused, are these talking about the same thing?

vanhees71
Gold Member
In quantum theory no particle can be exactly localized since the Heisenberg uncerstainty relation ##\Delta x \Delta p_x \geq \hbar/2## tells you that the position vector has always a finite uncertainty. Thus the argument of Wikipedia is flawed by not considering the foundations of quantum theory.

The notion of "point particle" is also completely irrelevant for understanding what an electron or any other quantum is. All we can observe is described by relativistic QFT, and there the quanta are described by quantum fields and the observables finally by correlation functions (N-point Green's functions) or S-matrix elements to evaluate cross sections of scattering or decay processes. Thus there's no necessity to introduce an idea of "point particles".

One should also note that in classical physics "point particles" (at least charge "point particles") lead to contradictions that are only resolvable approximately. This has to do with the notrious problem of "radiation reactions" in the classical electrodynamics of "point particles". The solution working FAPP is either to use continuum-mechanical descriptions (i.e., the electrons in an accelerator are described as a fluid/plasma) or a model for "particles" of finite extension, e.g., as Born rigid bodies as in the excellent paper

http://stacks.iop.org/JPhysA/39/3801
https://arxiv.org/abs/physics/0508031

Thanks @vanhees71 - I'm still a bit confused about @A. Neumaier 's FAQ answer...presumably you mean cross-sections in scattering are always finite?

My main reason for asking about the non-existence of true point particles - or of the mass not being concentrated exactly at one point, is that it then sidesteps the singularity issue... (e.g., electrons would not be infinitely dense)... but @A. Neumaier 's FAQ doesn't mention mass specifically (although I assume this is in part what he means by spatial extent).

Just looking at the paper above for the "classical" regime - I'm wondering what the QFT answer is currently? Finite extent as @A. Neumaier notes?

Ok here's perhaps a better question: using @A. Neumaier FAQ, if an electron, for example, is an extended particle (still understanding that ultimately it is a field quanta), then at the scales probed to date you get an incredibly small size in scattering experiments, which means an enormous density.

Singularity or not, should this not result in tremendous warping of spacetime near an electron? There is a very small mass, but it is incredibly concentrated.

This, I would assume, would have some quite significant effects experimentally - and yet electrons seem to happily absorb and radiate photons, etc., - so what gives?

vanhees71
Gold Member
As I already said, the idea to think about elementary quanta as if they were classical objects is misleading. It is pointless to think of them as points or of objects with a certain shape and extension. An elementary particle is, according to our current best understanding, the Standard Model of elementary particle physics, which is a local microcausal relativistic QFT, elementary if it behaves up to the highest available energies as described by a quantum field in the Lagrangian of the Standard Model. This has to be taken, however, again with a grain of salt since the strongly interacting fields, describing quarks and gluons, do not have asymptotic free states and thus cannot be observed directly as free quanta. This is, however, possible in the deep inelastic scattering regime, where the strong interaction becomes weak and perturbative evaluations of cross sections become applicable.

Of course, there's also no such thing as a "bare electron" in Nature. It's just a mathematical fiction to start a systematic approximation procedure called "renormalized perturbation theory" to describe the true electron, which is always "dressed". We cannot prepare an electron without electroweak interactions. Particularly it always carries its electric Coulomb field as well as its magnetic moment with it. So it is also pointless to say it's an extended object, only because it doesn't behave as at leading order perturbation theory, which is described by the lowest-order Feynman graphs with no loops ("tree diagrams"). This is just the lowest-order approximation of the true electron, as is reflected in the leading order "form factors" and "gyromagnetic factor". To associate something like a "shape" with the formfactor comes also from the leading-order semi-classical approximation, where the form factor for extended objects (like baryons) occurs as the Fourier transform of the charge distribution in a semi-classical approximation.

@vanhees71 Thanks - but this still doesn't answer the question about the electron scattering cross section or the notion of infinite mass density. I am less concerned about 'shape' and more concerned about what @A. Neumaier asserted - that electrons are at least at some sense extended' since they do not satisfy the Dirac equation.

I am of course talking about dressed (physical) electrons... how, specifically, is the problem of a singularity avoided?

DarMM
Gold Member
I presume having extension means that the mass of a physical electron is not considered to be concentrated at a single point?
Nonperturbatively the electron does not possess a sharp mass and its two point function does not have a pole. The book "Perturbative Quantum Electrodynamics and Axiomatic Field Theory" by Othmar Steinmann has a long discussion on this in the later chapters.

Hi @DarMM - thanks - I have the text, but I must say it's a bit beyond me at the moment.

How about in the perturbative case? I mainly would like to know how the singular mass / infinite charge situation is avoided (by dressing), and what exactly this means for scattering? (since I keep reading a lot of things which assert that the electron is a true point).

Also, in Steinmann's book he makes it very clear that electrons are "point like" but certainly not points, which is leading to some confusion on my part (granted I'm still a beginner trying to jump in the deep end).

DarMM
Gold Member
Basically typical perturbation theory QED doesn't really deal with the electron directly. It computes n-point functions which include the electron field. From those functions you can extract the scattering of electrons.

A. Neumaier
in Steinmann's book he makes it very clear that electrons are "point like" but certainly not points,
This is indeed the proper designation, which the wikipedia article you cited ignores by confusing ''point particle'' and ''pointlike particle''. Wikipedia is not very trustworthy when talking about that part of physics where the story for lay people is very loose only and often more misleading than helpful when trying to move from the laymen's level to a deeper level.

A. Neumaier
which means an enormous density.
In solid matter, electrons are quite diluted, occupying all the space between the nuclei. You should think of them as a fluid, not as tiny, very dense points. The fluid picture remains intuitively valid down to the smallest scales - even reactions involving a few elementary particles only are often modeled by hydrodynamics (with equations derived from the standard model), but never by point particle collisions.

vanhees71
Gold Member
@vanhees71 Thanks - but this still doesn't answer the question about the electron scattering cross section or the notion of infinite mass density. I am less concerned about 'shape' and more concerned about what @A. Neumaier asserted - that electrons are at least at some sense extended' since they do not satisfy the Dirac equation.

I am of course talking about dressed (physical) electrons... how, specifically, is the problem of a singularity avoided?
The electron scattering cross sections are calculated perturbatively (sometimes one has to resum infinitely many diagrams, but that's a technical detail). This is what's well defined by QFT, and today we have nothing better to understand what electrons are than relativistic QFT.

You cannot define a "mass density" nor a shape nor an extension or something similar. It's just not defined to make sense, if you cannot tell what you measure. It makes very well sense to define form factors, which deviate from the tree-level form factors and thus from the naively interpreted Dirac equation as describing a first-quantization-like wave function. I still don't think that it is very useful to define the form factors in terms of classical charge distributions.

@A. Neumaier when you say dilute, are you referring to the electron wave function being spread out in space? (which seems fine)

@vanhees71 Right - but then as @DarMM suggested, if i'm calculating the the scattering matrix for an electron-electron collision, what would I obtain?

I still see sources (even in physics journals - I'll look for one) claiming that the electron can be no larger than 10e-18 m or similar, which, as above, folks have indicated makes no sense. But in even coming up with a number like this, what is being considered - some bound on the wave function?

A. Neumaier
when you say dilute,
I am referring to the ensemble expectation value of the mass density field.
claiming that the electron can be no larger than 10e-18 m or similar
Such sizes most likely refer to semiclassical sizes as computed from form factors, which are the sizes relevant for scattering experiments.

Ultimately (and sorry I don't mean to keep going on) I asked the original question (which may be muddled a bit now) because, from a GR perspective, I'd like to know two things:

1. Just how much a lowly dressed electron out there warps spacetime around it, and
2. If you were to excite said electron with a photon of reasonable energy, how the warping changes.

Is the warping at all significant (I know QFT is background dependent, so not sure how one even goes about the above)?

I unsure how to approach this without have some description of energy density - I'd love to hear the community's answer as it would be very helpful.

PeterDonis
Mentor
2020 Award
Just how much a lowly dressed electron out there warps spacetime around it
Theoretically, we don't know, because we don't have a good theory of quantum gravity. There are various ways of trying to calculate an "energy density" for a single electron, but none of them give useful results.

Experimentally, single electrons don't warp spacetime enough for us to detect.

If you were to excite said electron with a photon of reasonable energy, how the warping changes.
Theoretically, again we don't know, because we don't have a good theory of quantum gravity.

Experimentally, since "a photon of reasonable energy" presumably means something like "a photon we can actually produce in an experiment", the answer is again that there will not be enough spacetime curvature for us to detect.

DarMM
Gold Member
I unsure how to approach this without have some description of energy density - I'd love to hear the community's answer as it would be very helpful.
The problem is for an electron you can compute its density for various quantities, e.g. charge density, magnetic moment density, energy density and let's say define their "shape/size" by where that density drops to roughly zero. The problem is that those densities don't produce the same shapes or sizes, so you just can't obtain a consistent idea of the size of the electron.

And then there are the more fundamental problems mentioned by PeterDonis above. Even when you calculate the energy density (properly Stress-Energy density) we don't know a valid way to couple it to gravity.

Thanks @PeterDonis (very helpful as usual) and @DarMM. @DarMM are you saying " define their "shape/size" by where that density drops to roughly zero" to simply make an arbitrary choice (because you have to choose some boundary)?

Experimentally, single electrons don't warp spacetime enough for us to detect.

Experimentally, since "a photon of reasonable energy" presumably means something like "a photon we can actually produce in an experiment", the answer is again that there will not be enough spacetime curvature for us to detect.
@PeterDonis, last question (promise ;-) How would something like this be tested experimentally?

PeterDonis
Mentor
2020 Award
How would something like this be tested experimentally?
You would have to try to detect tidal gravity on very small distance scales. We don't currently have a way to do that. The best we can do is to say that, on the smallest distance scales at which we can detect tidal gravity (roughly millimeters, if memory serves), we can't detect any from electrons--or any other elementary particles, or indeed any microscopic objects. I believe the smallest objects we have detected any gravitational effects from are on the order of 1 kg balls a few centimeters in diameter.

@PeterDonis - if the electron or any other subatomic particle were doing something more exotic (e.g., black hole ilke - not saying that they are) - shouldn't the effects of curvature show up in other ways?

PeterDonis
Mentor
2020 Award
if the electron or any other subatomic particle were doing something more exotic (e.g., black hole ilke - not saying that they are) - shouldn't the effects of curvature show up in other ways?
What other ways? Spacetime curvature is tidal gravity. That's how it shows up.

vanhees71