B What are the predicted sizes of elementary particles?

[Nathan Warford]

I understand that the Standard Model of QFT treats elementary particles like the electron, quark, photon, muon, etc. as point-like objects. But I've also heard that a "point-like particle" is nothing more than an idealization of a particle. Elementary particles can be treated as point-like objects in experiments because they have no detectable size, but just because we can't detect their size doesn't mean that they have some finite size smaller than what we can detect (I've heard that this scale is about 10^-18 meters).

Some string theorists have said that elementary particles have a size on the scale of the Planck Length. But "on the scale of" means nothing more than a size that can be expressed in Planck units, as opposed to a specific, well-defined number of Planck units. Are there any theoretical models that predict the sizes of the various elementary particles?

 

[Demystifier]

But "on the scale of" means nothing more than a size that can be expressed in Planck units

No, that's not what "on the scale of" means. It means that string (that is an "elementary" particle according to string theory) has size of the same order of magnitude as Planck length.

 

[Nathan Warford]

It means that string (that is an "elementary" particle according to string theory) has size of the same order of magnitude as Planck length.

The way I understand it, an order of magnitude is the same as a factor on ten. So saying that a string is of the same order of magnitude means it could range anywhere from ¹/₁₀ of a Planck length to 10 Planck lengths. Am I understanding the definition of "order of magnitude" correctly?

 

[Demystifier]

Yes, except that order-of-magnitude is not always defined that precisely. For instance, a factor 1/20 or 20 can also sometimes count as order-of-magnitude. Rarely even 1/100 or 100 can be counted so. People use that expression when they only have a rough idea how big a number is.

 

[Vanadium 50]

In the muon decay, the terms of order 1 end up being 192π³.

 

[Nathan Warford]

terms of order 1

I'm not familiar with that terminology. Could you explain it to me?

 

[ohwilleke]

"Terms of order 1" are parameters expected to have a value of the same order of magnitude as 1, which in a narrow sense means an expectation of about 0.1 < x < 10, but only approximately. In the situations where you talk about a "term of order 1" (often in the sense of a Bayesian expected value of a parameter in a beyond the Standard Model theory motivated by "naturalness") a value of 0.02 or 50, for example, wouldn't be thought of as strongly inconsistent with the original expectation.

"order of magnitude" when used in the context of "on the order of" is almost always an approximate concept rather than a more narrow and precise one, in much the same way that in some contexts the word "couple" means exactly two (e.g. "a couple is going to the dance"), while in other contexts "couple" (e.g. "I got a couple of things at the store") it means "a small number of".

The Bayesian expectation itself is often derived from a bit of physics theory folk mythology that assumes that all unknown fundamental constants should be of order 1 unless there is some know reason that they shouldn't be. When the actual value of a parameter violates that folk myth for no obvious reason, it is called an "unsolved problem" in physics.

 

[ohwilleke]

I understand that the Standard Model of QFT treats elementary particles like the electron, quark, photon, muon, etc. as point-like objects. But I've also heard that a "point-like particle" is nothing more than an idealization of a particle. Elementary particles can be treated as point-like objects in experiments because they have no detectable size, but just because we can't detect their size doesn't mean that they have some finite size smaller than what we can detect (I've heard that this scale is about 10^-18 meters).

Some string theorists have said that elementary particles have a size on the scale of the Planck Length. But "on the scale of" means nothing more than a size that can be expressed in Planck units, as opposed to a specific, well-defined number of Planck units. Are there any theoretical models that predict the sizes of the various elementary particles?

There is more than one sensible definition of the term "size" with respect to an elementary particle.

For example, if all particles had truly point-like properties for all purposes, in the limit, nothing would ever interact with anything else, because all Standard Model forces involve carrier bosons interacting with fermions (and sometimes with other carrier bosons as well) in a contact-like manner. But, for purposes of figuring out if a particle moving in the general direction of another particle will "scatter" due to one of the forces in the Standard Model, a particle has a finite size, often within a few orders of magnitude of femtometers, that is part of the equation governing the likelihood that the particles will interact. A description of some of these concepts (there are several similar such concepts) is found in the Wikipedia article on Compton wavelength.

Another important concept to keep in mind when trying to make sense of the size of a fundamental particle is the uncertainty principle, which provides that one can't know both a particle's location and momentum at the same time at a combined precision greater than that permitted by an equation proportional to Planck's constant. Basically, the harder you look, the more something gets out of focus, not just from a practical perspective, but fundamentally as a intrinsic feature of Nature itself.

Also, special relativity further messes with our normal conception of even basic concepts like length, which turns out to be a function of velocity relative to the speed of light (in a way that nonetheless is not frame dependent).

And, the point-like nature of a fundamental particle is also messed with by the quantum mechanical concept that an unobserved particle is in some sense not in any single place at one time, instead, it is smeared as a probability distribution over all places it could possibly be at once.

Further, while it is contrary to our ordinary intuition, point-like fundamental particles that are not scalars (i.e. spin-0) have directional properties that our ordinary life experience can't make sense of outside a non-point-like topology in an object. (And, indeed, most physics textbooks illustrate these particles as non-point-like in order to explain these properties for heuristic reasons.)

Thus, while to say that fundamental particle is point-like isn't really a category error (there are lots of mathematical purposes for which treating it as point-like is the correct treatment), being point-like doesn't have all of the intuitive implications that it would for an object in your daily life that you could observe unaided.

Somewhat related is the trichotomy of reality, causality and locality in quantum mechanics. The intuition in your question hinges heavily upon the "reality" of a fundamental particle, but in quantum mechanics we know that there are circumstances where all three of those concepts can't be simultaneously true, which is basically a problem with the concepts themselves, as much as it is a statement about what properties Nature has or lacks.

 

[Demystifier]

"Terms of order 1" are parameters expected to have a value of the same order of magnitude as 1, which in a narrow sense means an expectation of about 0.1 < x < 10, but only approximately. In the situations where you talk about a "term of order 1" (often in the sense of a Bayesian expected value of a parameter in a beyond the Standard Model theory motivated by "naturalness") a value of 0.02 or 50, for example, wouldn't be thought of as strongly inconsistent with the original expectation.

"order of magnitude" when used in the context of "on the order of" is almost always an approximate concept rather than a more narrow and precise one, in much the same way that in some contexts the word "couple" means exactly two (e.g. "a couple is going to the dance"), while in other contexts "couple" (e.g. "I got a couple of things at the store") it means "a small number of".

The Bayesian expectation itself is often derived from a bit of physics theory folk mythology that assumes that all unknown fundamental constants should be of order 1 unless there is some know reason that they shouldn't be. When the actual value of a parameter violates that folk myth for no obvious reason, it is called an "unsolved problem" in physics.

It could be made precise by the following definition. The unknown positive number $x$ is of the order of $a$ if the probability of given value of $x$ is given by
$$p(x)={\rm Gauss}({\rm log}_{10}x-{\rm log}_{10}a;1)$$
where ${\rm Gauss}(x;\sigma)$ is the Gaussian distribution with width $\sigma$.

 

[ohwilleke]

It could be made precise by the following definition. The unknown positive number $x$ is of the order of $a$ if the probability of given value of $x$ is given by
$$p(x)={\rm Gauss}({\rm log}_{10}x-{\rm log}_{10}a;1)$$
where ${\rm Gauss}(x;\sigma)$ is the Gaussian distribution with width $\sigma$.

Clever.

 

[no-ir]

It could be made precise by the following definition. The unknown positive number $x$ is of the order of $a$ if the probability of given value of $x$ is given by
$$p(x)={\rm Gauss}({\rm log}_{10}x-{\rm log}_{10}a;1)$$
where ${\rm Gauss}(x;\sigma)$ is the Gaussian distribution with width $\sigma$.

Clever formalization of one's intuition, indeed. :) However, it contains a slight error. Namely, if we want $\log_{10}x$ to follow a normal (Gaussian) distribution, then $x$ itself should follow a log-normal distribution, not a normal distribution. The probability densities of $\log_{10}x$ and $x$, respectively, should thus be
$$\begin{align}\frac{dP}{d(\log_{10}x)}&={\rm Gauss}(\log_{10}x-\log_{10}a;1),\\
\frac{dP}{dx}=\frac{dP}{d(\log_{10}x)}\frac{d(\log_{10}x)}{dx}&=\frac{{\rm Gauss}(\log_{10}x-\log_{10}a;1)}{x\cdot\ln 10}.\end{align}$$
Notice the extra $1/x$ in the probability density of $x$.

Of course, in practice, the width of the distribution (chosen to be 1 above) is entirely subjective and should really depend on the situation, as already alluded to by Vanadium 50. ;)

 

[Nathan Warford]

We're starting to get into some complex mathematics that I'm not entirely familiar with (my math training only goes up to freshman-level calculus). I was hoping that someone knew of some theoretical models (highly speculative) that would predict more constrained dimensions of the various fundamental particles ("on the order of" is too ambiguous for my liking).

I don't think that the Compton Wavelength is a very good value to assign to particle size. If we assume the Compton Wavelength to be the size of a quark, then that quark would be a thousand times larger than the proton that it's contained within.

 

[ohwilleke]

I don't think that the Compton Wavelength is a very good value to assign to particle size. If we assume the Compton Wavelength to be the size of a quark, then that quark would be a thousand times larger than the proton that it's contained within.

The point of discussing Compton wavelength is not to say that a quark IS the size of its Compton Wavelength.

Instead the point of mentioning it is to illustrate the point that "the size of a quark" isn't a coherent single concept that is the same for all purposes as it is in the case of classical objects. The "size of a quark" in ordinary English is definitionally incoherent in quantum physics. It doesn't have a well defined meaning. The comprehensive classical meaning of "the size of an object" does not have a close equivalent at the quantum level which is why efforts to answer the question are generally misleading no matter how you answer them.

For some purposes it is appropriate to think of a quark as truly point-like. For other purposes, it is useful to think of a quark as having a non-zero length dimension. Even then, there is not one length dimension that is appropriate for all purposes. In some contexts one concept about the "size of a quark" in one or more length dimensions is appropriate, and in others a different conceptualization of the "size of quark" is appropriate. For example, the volume of a quark for determining its likelihood of having an electromagnetic interaction is not the same the the volume of a quark measured on a probability distribution basis.

Also, there is something to be said for not entirely disregarding the wave side of the wave-particle duality, even though the modern trend is to emphasize the particle interpretation as more fundamental than the particle one. There are plenty of circumstances where one's intuition predicts physical behavior more naturally in a wave-like conception than than a point particle-like one, which is currently emphasized in lay simplifications of physics. The point-like particle description often leads to bad intuitive expectations about how fundamental particles behave.

 

[Nathan Warford]

I came across an article on arXiv today concerning the nature of the electron.
<link removed>

From what I gathered by skimming the article, the author posits that an electron's diameter is effected by its velocity. The more energy the electron has, the smaller its diameter becomes. On page 32 of the article, he writes: "when an electron is accelerated to the energy of 100 GeV or 200 GeV, its diameter has diminished to 1.97×10^-16 cm or 0.99×10^-16 cm respectively." Does this interpretation of the electron have any merit? If it does have merit, can the equations be used or modified to find values for other particles like the muon, the various flavors of quark, the photon, the Higgs, etc?

 

[DrClaude]

I came across an article on arXiv today concerning the nature of the electron.
<link removed>

This is not an acceptable reference. It is pseudoscience and won't be discuss at PhysicsForums.

 

[jtbell]

People use [order of magnitude] when they only have a rough idea how big a number is.

In more colloquial American English, one might say "in the ballpark of..."