The Many Mysterious Coincidences of Particle Physics

Let us look at the fundamental particles of nature, and consider the question: how many, if any coincidences can we find there? At this time I will not be considering what are sometimes called anthropic coincidences, meaning an agreement between some value of nature and the value that is required for beings like us to exist (those interested in that topic can see this post). Here I will merely be considering numerical coincidences, meaning a case of one fundamental number in nature matching another in nature without any obvious explanation, or one number being the exact opposite (or a simple integer multiple) of another number in nature, without any obvious explanation.

The Stable Particles in Nature

To start out looking for possible coincidences, we can create a table listing all of the massive stable particles in nature. The list will include a particle called the positron (which is known as the antiparticle of the electron), and a particle called the antiproton (known as the antiparticle of the proton). Note that both positrons and antiprotons are completely stable particles which will stay around forever when they exist by themselves undisturbed in nature. In actual practice, however, positrons and antiprotons tend to be quickly destroyed in our universe because a positron is converted to energy whenever it comes in contact with an electron, and an antiproton is converted to energy whenever it comes into contact with a proton. (Conversely, an electron and a positron can be produced by a collision of two energy particles called photons, and a proton and antiproton can be produced by a collision of two photons.)

So here is the table listing the stable massive particles in nature that can exist by themselves (throughout this discussion I am going to ignore the “ghost particles” called neutrinos):

Particle Name	Mass	Charge
Electron	9.10938291 X 10-31 kg	-1.60217657 × 10-19 coulombs
Positron	9.10938291 X 10-31 kg	1.60217657 × 10-19 coulombs
Neutron	1.674927351 X 10-27 kg
Proton	1.672621777 X 10-27 kg	1.60217657 × 10-19 coulombs
Antiproton	1.672621777 X 10-27 kg	-1.60217657 × 10-19 coulombs

I list these figures to ten decimal places, because the observational studies described here and here  verify that these masses and charges have been determined to ten decimal places.

Looking at the table above, your first impression will be that there are a number of coincidences. The coincidences can be listed as follows:

1. The absolute magnitude of the charge of the electron, the proton, the positron, and the antiproton are all exactly the same. (An absolute magnitude is the number you are left with after you remove any negative sign. For example, the absolute magnitude of -22 is 22, and the absolute magnitude of 23 is 23).
2. The mass of the electron is exactly the same as the mass of the positron.
3. The mass of the the proton is exactly the same as the mass of the antiproton.

There is one way (at least in theory) that we might be able to explain some of the coincidences in the list given above. If we can prove that these coincidences are due to a “simple integer combination” situation, then it might make one or more of the coincidences seem not very coincidental.

Let me give an example of what I mean by a “simple integer combination” situation. Imagine if two men equipped with empty bags go up to a barrel of tiny gold nuggets. The men both fill the bags with as much gold as they can carry, and then go home and weigh how much is in each bag. They find that both bags contain exactly the same weight of gold, to a tenth of an ounce. That would be a huge coincidence, with a very low likelihood. But imagine that instead of little nuggets the barrel contains only big 50 kilogram gold bars. Such bars would be so heavy that each of the men could carry no more than 2 or 3 of them. It would therefore be not much of a coincidence at all if the men then came home, weighed their bags, and found that both bags weighed the same. Since we would expect that both bags would be a simple integer combination of 50 kilograms (50, 100, or at most 150) then the likelihood of both men coincidentally carrying the same weight would be relatively high, around 1 in 3. This example shows how a “simple integer combination” situation can get make a coincidence seem not very unlikely.

Looking at the situation in regard to fundamental particles, we can imagine a universe in which a simple integer combination might explain away the coincidence of the proton mass matching the antiproton mass. Protons are believed to be made up of particles called quarks. The two main types of quarks are called the up quark and the down quark.

The following table (omitting antiquarks) shows the relation between the charges in the fundamental particles that have charges (in this table e refers to a charge of 1.60217657 × 10^-19 coulombs):

Particle	Charge
Proton	1e
Antiproton	-1e
Up Quark	2/3 e
Down Quark	-1/3e
Electron	-1e
Positron	1e

Protons are believed to be made of two Up quarks and one Down quark. This gives them a charge of 2/3e +2/3e + -1/3e, which equals 1e.

Now we can imagine a universe in which a simple integer combination might explain away the coincidence of the proton charge equaling the antiproton charge. It would be a universe in which antiprotons were made up of three Down quarks. Then we would explain the -1e charge of the antiproton without having to believe in any big coincidence. Since there are 4 ways in which you can make 3 combinations of an Up quark and a Down quark, we would then have a minimal coincidence with a likelihood of only 1 in 4 – not very unlikely at all.

The only problem is: that is not an accurate description of the antiproton. When we look at what antiprotons are made of, the coincidences don't become smaller and more likely – instead they become greater and more unlikely.

Scientists say that an antiproton is made up not of quarks but particles called antiquarks. Rather than being made of three Down quarks, an antiproton is made up of two Up antiquarks and one Down antiquark. So there is not at all any “simple integer combination” which explains why the proton charge is exactly the opposite of the antiproton charge.

When we list all of the particles and component particles, we have quite a list of coincidences to explain. Here is the expanded table (in this table e refers to a charge of 1.60217657 × 10^-19 coulombs):

Particle	Charge	Composition
Proton	1e	2 Up quarks, 1 Down quark
Antiproton	-1e	2 Up antiquarks, I Down antiquark
Up Quark	2/3 e	N/A
Down Quark	-1/3e	N/A
Up Antiquark	-2/3 e	N/A
Down Antiquark	+1/3e	N/A
Electron	-1e	N/A
Positron	1e	N/A

So now we have 8 fundamental particles, all of which have charges that are simple integer multiples (either negative or positive) of the charge 1/3e ( one third of 1.60217657 × 10^-19 coulombs).

It is not an unlikely coincidence that the proton charge has a simple numerical relation to the Up quark charge and the Down quark charge, because the proton is made of two Up quarks and one Down quark.

It is not an unlikely coincidence that the antiproton charge has a simple numerical relation to the Up antiquark charge and the Down antiquark charge, because the antiproton is made of two Up antiquarks and one Down antiquark.

However, all of the following are coincidences incredibly unlikely to occur if there is no underlying principle that explains them:

1. The coincidence that the electron charge is the exact opposite of the positron charge.
2. The coincidence that the absolute magnitude of the charge of the Up quark is exactly twice the absolute magnitude of the charge of the Down quark.
3. The coincidence that the absolute magnitude of the charge of the Up antiquark is exactly twice the absolute magnitude of the charge of the Down antiquark.
4. The coincidence that the absolute magnitude of the charge of the Up quark is exactly the same as the absolute magnitude of the charge of the Up antiquark.
5. The coincidence that the absolute magnitude of the charge of the Down quark is exactly the same as the absolute magnitude of the charge of the Down antiquark.
6. The coincidence that the absolute magnitude of the proton charge is exactly the same as the absolute magnitude of the electron charge.
7. The coincidence that the electron mass exactly equals the positron mass.
8. The coincidence that the proton mass exactly equals the antiproton mass.

So that leaves us with a total of eight fundamental numerical coincidences in nature – cases where two numbers match exactly, even though the chance of them matching in a random universe would seem to be very, very low. When I say “exactly match” in the above list, I mean to ten decimal places.

Do We Know of Some Explanation for the Match Between Antiparticles and Particles?

I'm sure that some people will try to explain most of these coincidences simply by evoking an “every particle has an antiparticle” principle. But “every particle has an antiparticle” is an empirical generalization, not an explanation.

Some may claim that there are theoretical reasons why there need to be antiparticles. They may claim that physicist Paul Dirac predicted the existence of an antiparticle (the positron) before it was discovered. So doesn't that show that there is some theoretical reason why antiparticles must exist?

Not really. The situation in regard to Dirac and the positron is more complicated than it is usually described. Dirac's “prediction” of the positron came in 1931, one year before the official discovery of the positron in 1932. But he did not plainly say that the particle exists – he basically just said that it might exist. So it wasn't really a prediction. Also, at the very time that Dirac made this supposed prediction, another scientist named Patrick Blackett was accumulating photographic evidence that the positron exists – evidence he had not yet published, but which Dirac was familiar with. This calls into question whether Dirac deduced the positron's existence on purely theoretical grounds. Another interesting fact is that Dirac's “prediction” of the positron was made within the context of a larger theory that is not widely accepted today – ideas such as negative mass, negative energy, infinite charge density, the theory that the universe is filled with an infinitely dense sea of negative energy particles, and that there are “holes” in this sea. So the theoretical basis that Dirac advanced for suspecting the existence of an antiparticle doesn't seem to have been an explanation that still holds up today.

In 1986 physicist Richard Feynman gave some extremely complicated lecture called The Reason for Antiparticles. However, according to this analysis by a physicist, Feynman's explanation for the existence of antiparticles was very different from Dirac's. Feynman's explanation for antiparticles is based on the idea that an antiparticle is a regular particle traveling backward in time, but that curious idea is subject to criticism and is doubted by quite a few. Commenting on his lecture, one physics buff says, “Funny that Feynman, who is normally perceived as a master of exposition, is not able to come up with a convincing story here.”

In his book Symmetry and the Beautiful Universe (page 234), physicist Leon M. Lederman offers this attempt to give a reason for antiparticles:

Quantum theory forces electrons to have both positive and negative energy values for any given value of the momentum. We would say that the negative-energy electron is just another allowed quantum state of the electron. But this would be a disaster as well, since it would mean that ordinary atoms, even simple hydrogen atoms, could not be stable. The positive-energy electron could emit photons, adding up to an energy of 2mc^2, and become a negative-energy electron and begin its descent into the abyss of infinite negative energy. Evidently the whole universe could not be stable if negative-energy states truly existed.

But this does not seem to be an explanation for why antiparticles have to exist. It's basically just a statement saying that it is convenient for antiparticles to exist.

It is often claimed that special relativity implies the existence of antiparticles (or that the combination of special relativity and quantum mechanics implies the existence of antiparticles). One sees statements such as this: antiparticles are needed because of very subtle reasons buried deep in the fabric of special relativity and quantum mechanics. But virtually no one who makes such a claim explains what such reasons are, and as a general principle one should perhaps be suspicious about reasons described as deeply buried very subtle theoretical reasons.

In the book Mathematical Quantization (page 159), Professor Nik Weaver at the very prestigious Washington University at St. Louis makes these statements: "A related argument claims that relativity implies the existence of antiparticles.. This argument is wrong...Thus the most that one can say is that relativity implies the existence of phenomena that are reminiscent of multiple particles or antiparticles. But even this seems misleading."

On this  page a physics PhD admits that “Quantum mechanics plus special relativity does not necessarily require antiparticles: although it naturally accommodates them.”

We can't explain the coincidences between the masses and charges of antiparticles and particles just by saying that they are required by the Standard Model or predicted by the Standard Model. The Standard Model evolved after the discovery of different types of antiparticles, so in this regard it simply reflects the coincidences that had already been discovered.

We also cannot explain any of the above coincidences by evoking some principle of symmetry, whether it be a CPT theorem or any other principle of symmetry. Such reasoning would be circular. Certain types of symmetry in nature may be possible because of the near-exact correspondence of the features of particles and antiparticles, but that does not explain such a near-exact correspondence. If X makes Y possible, we generally don't explain X by mentioning Y. For example, a student's high SAT scores may explain his admission to Harvard, but his admission to Harvard doesn't explain his high SAT scores.

We might have some explanation for why the masses and charges of antiparticles match the masses and charges of their corresponding particles if we knew of some mechanism for charge flipping, the changing of a negative charge into a positive charge. Then a scientist could say, “Why, of course the absolute magnitude of the charge of the positron is the same as the absolute magnitude of the charge of the electron, and of course their masses are identical – it's because the charge of one particle just flips, and then you have the antiparticle.” But no such thing happens. Scientists do not believe that particles ever change into antiparticles, nor do they believe that antiparticles change into particles. Scientists believe that what happens is that both a particle and an antiparticle are created from a collision of high-energy photons.

It seems that there are really no theoretical a priori reasons why a universe has to contain any antiparticles. If one were to cite some formula or equation that implies the existence of antiparticles, that really would not be equivalent to showing any necessity behind the existence of antiparticles. Such an equation would merely reflect that we happen to live in a universe in which antiparticles exist. Given a million external random universes with a million random characteristics and a million random set of laws, it would seem there is no reason to think even one of them would be a universe in which there are only a few stable particles with every particle having an antiparticle that is exactly the same in mass but with an exactly opposite charge.

I may conclude this section by noting that even if one were to have some completely convincing explanation for antiparticle/particle coincidences, we would still be left with no explanation for items 2 and 6 on my previous list of coincidences, both of which are not really explained by any widely accepted theory.

Other Coincidences

In this post I have merely discussed the coincidences involving stable particles, particles with a very long lifetime. There are also many similar coincidences involving unstable particles which I have not even mentioned. I may also note another coincidence involving the apparent net electric neutrality of the universe. Protons are 1836 times more massive than electrons. Judging from the intuitive principle that smaller things tend to be more common than more massive things, we might expect that electrons would be perhaps one or two thousand times more common than protons. But instead the number of protons and electrons in the universe seem to be equal, a fact referred to as the net electric neutrality of the universe.

All in all, there are too many mysterious exact numerical coincidences here, which may suggest that a deeper or more complete explanation of things is needed.