What's smaller: a neutrino, or a string from string theory

Question

I've recently read an article that stated "If an atom were as big as the solar system, a neutrino would be the size of a golf ball". I watch the science channel, and on (I believe) the show How the Universe Works they mentioned that if a hydrogen atom was the size of the solar system a string would be the size of a tree (approx 50 ft tall). Could someone explain this to me, please?

Answer 1

While this question has a lot of issues, I understand where the confusion arose. I originally answered this question wrong, for instance! And thinking about the scale at which the different objects we're aware of operate in is frankly, awe-inspiring. We study both galactic clusters and subatomic particles, and yet we will never be able to completely comprehend the scale of either, for precisely opposite reasons. So I'll try to briefly go over some of these ideas of scale, because I think that's really what your question is getting at here, and then go more into why your question has issues.

First, let's familiarize ourselves with units. We'll go with metric, because it isn't such a pain to convert. The diagram below shows units down to the Planck Scale with a few objects.

Okay, now that we've got some idea, let's expand this chart out a bit. A honeybee is about 2 centimeters long. And it is about one hundredth the length of the average adult male. Now, this really isn't beyond our reckoning. It isn't very exotic for us; we've seen honeybees. Now, then, let's think about a dust mite. (Gross, I know; bear with me.) A dust mite is 50 times smaller than a honeybee. And, again, bear with me, but here's a picture of a dust mite.

And these guys are less than $\frac{1}{2}$ of a millimeter in length, or (if you want to go in micrometers) it's about 400 micrometers in length. Now, let's go smaller. The diameter of a human hair (on average) is 100 micrometers. And here's a picture of a zoomed in human hair. And this, obviously, is about $\frac{1}{4}$ the length of a dust mite.

Then, at about 6 to 8 micrometers in diameter is the human cell. This is so, so tiny. And a diagram of a cell is below. While we certainly can't see a cell, we can sort of see spider silk, which is about 3 to 8 micrometers in diameter. This is the limit of human perception.

On the same sort of scale as the cell, mostly smaller (1 to 10 micrometers) are bacteria. Tons of bacteria. And I said this was the limit of human perception, but really, spider silk is the exception, not the rule here. And, to continue with the picture theme, below is a diagram of a bacteria cell.

Let's get even smaller. Let's go to the scale of nanometers. Just to point out here, a nanometer is $\frac{1}{1000}$ of a micrometer. This is the scale of a virus, which can be anywhere from 20 to 400 nanometers. An "artist's rendering" of a virus is below.

Okay, now to DNA. The double helix of a DNA molecule is about 2 nanometers wide. Just to point out again, how small we are, you can put 500 million DNA molecules side by side (remember, in width) to get a meter. This is tiny. And a diagram of DNA is shown below.

Okay, now to introduce angstroms. An angstrom is not really a part of the conventional metric system, but 0.1 nanometers is an angstrom. Now, 2 angstroms is equal to the width of a water molecule. A diagram of a water molecule is shown below.

Okay, now the diameter of a hydrogen atom is roughly one angstrom. This is the scale of the stuff we are made up of...and we aren't really near neutrinos, and we're definitely not near strings. I think this is helpful though, to get a sense of scale. Okay, here's a diagram of a hydrogen atom.

Okay, now the nucleus of a hydrogen atom has the radius of $\frac{1}{10000}$ of an angstrom. This is so tiny it's ridiculous, really. But there are a few issues we're starting to run into.

First, we've been talking about length, but we've not been talking about how we're measuring it! You could measure with a ruler, or if it's something smaller but still visible to the naked eye, calipers might be more precise. If we get even smaller, and need a microscope, one way to do it is measuring the size when magnified, and then scaling that quantity by how much we magnified the object.

But as we keep going, we're expecting more and more precision out of our measurements. If you've heard of significant figures, this is what they're all about! If you hold a ruler marked into millimeters up to an unmagnified cell (neglecting the difficulty of lining it all up), it's just going to look like it's 0 mm (or '0 thousandths' of a meter) in length. We need precision out to the sixth decimal place for a cell! And we're dealing with something much, much smaller. So your question assumes that we have tools which are good enough to measure this!

Second, you may have heard of the Heisenberg uncertainty principle, which (to oversimplify a bit) limits how much we can know about any object on the subatomic scale. Your question assumes not only that we have tools which can measure these lengths, but that we can collect the information needed to measure them. This may seem like the same thing as the first point, but it's not!

Finally, and most deeply, your question assumes that length has meaning for neutrinos and strings. What if I asked you which was more red, happiness or sadness? Or if I inquired as to the relative spatial volume of energy in two different car engines? These questions make no sense because they ask for information about properties that the things in question do not have.

It turns out that size is something that's meaningful for composite particles - that is, things that are made up of other things - but not elementary particles. Elementary particles are sometimes called point particles for this reason: just like happiness isn't really something which has a color, elementary particles don't have size.

We think neutrinos are elementary particles. (By the way, there are three types, or flavors of neutrinos, which is another aspect in which this question is ill-defined - though differentiating between their masses is somewhere where problem one is an issue - measuring tiny things is hard!) Because of this, they don't have a size, as far as we know. We do know they have a mass - a very, very small one. (An "upper bound" of sorts - I am once again oversimplifying - is one millionth the mass of an electron, which is already a very small quantity.)

The strings from string theory, it should be pointed out, are entirely theoretical. We know neutrinos exist. We have spotted them in experiments and measured some of their properties. Strings, however, exist only on blackboards and in papers. Some physicists think they may solve issues with our current theories of how nature works; others don't. (This is not a debate I am anywhere near qualified to weigh in on. But it's important to note that strings are fundamentally different from neutrinos - we know neutrinos exist; we're not sure if strings exist or have meaning beyond an interesting mathematical object!) Note that a fundamental idea in string theory is that particles aren't the most basic way to describe objects - that particles are different 'vibrations' of the string. As dmckee notes in the comments above, in this way, you could say a neutrino is itself a string (and if you mean your question in the sense of 'which object is more fundamental, a neutrino or a string?' you could say a string - but we don't know if this is true, as, again, we don't have experimental proof of string theory).

Strings do have a length scale - the Planck scale, which is marked in the diagram up at the top of this answer, many orders of magnitude below the last thing we could measure the length of. Strings also have a property called tension (see What is tension in string theory?). This tension sort of (if I understand correctly) gives the string an apparent mass, at length scales larger than strings.

But all of this is to say...your question about relative size doesn't make sense, because neutrinos don't have a length/volume the way you're likely picturing, and strings don't have a mass the way you're likely picturing. I know this isn't a satisfying answer, especially since pop science tends to go wild with comparisons like the ones in your question. But hopefully, the questions that this leaves you with (what exactly is going on with the Heisenberg uncertainty principle, and how is it different from a failure of our measurement tools? what are the different neutron flavors? what makes a particle fundamental? and the many other possible questions you may have!) serve as motivation to learn the deeper math and true answers to these questions. It only gets more peculiar, interesting, and exciting the deeper you go. Good luck!

Note: This answer was heavily edited ~6 years after I wrote it, because it was, as dmckee points out in the comments, straight up wrong at the end. Hopefully this is a better attempt.

Answer 2

The mass of the neutrinos are estimated to some tenths of an $\mathrm{eV}$. The masses of atoms are mostly between 1 and 300 $\mathrm{GeV}$. Thus, considering the masses, this golf ball comparison isn't okay in my opinion. In my mind, comparing the mass of the Moon to the mass of the Solar System would be more realistic.

Their sizes can't be easily compared, because the elementary particles don't really have a size. Currently they are considered point-like particles with uncertain position. This uncertainty depends on their impulse (which depends on their speed and their mass) because of Heisenberg's uncertainty principle:

$$\Delta p \cdot \Delta r \ge \frac{\hbar}{2}$$

The masses of strings is a subject that belongs to professionals. I am not sure it is easy to even define their mass. Their sizes are in the order of the Planck length (around $1.6\cdot10^{-35} m$). Compare this to the size of the nuclei ($10^{-15} m$).