# Which law/formula proves the existence of antimatter?

Out of curiosity i was watching some physics documentaries. They threw out quantum theory and relativity and i have taken a modern physics course. I cant say i remember antimatter ever coming up in lecture. Im curious about the math.

I get that the double split experiment comfirmed the dual nature of particles but how does that tell me anything about antimatter?

EDIT: no worries ill try to find the relevant article.

I found this paper on google "The Dirac Equation and the Prediction of Antimatter" that answered all my questions. thanks for the lead.

PeterDonis
Mentor
I found this paper on google "The Dirac Equation and the Prediction of Antimatter" that answered all my questions

Do you mean this paper?

http://multimidia.ufrgs.br/conteudo/frontdaciencia/dirac antimatter paper.pdf

If so, you should be aware that, while Dirac did predict the existence of antimatter based on his model of antiparticles being "holes" in an otherwise filled "sea" of negative energy states, that model turned out not to work, and predicting antimatter solely on the basis of the Dirac equation is not a currently accepted explanation of antimatter. The most obvious problem with it is that it only applies to fermions, but bosons also have antiparticles.

plasmon_shmasmon
PeterDonis
Mentor
The best layman's presentation I'm aware of of the reason why there must be antiparticles, from the standpoint of our best current theory (quantum field theory), is in Feynman's 1986 Dirac Memorial Lecture. Unfortunately it doesn't seem to be available online; I have it in book form:

https://www.amazon.com/dp/0521658624/?tag=pfamazon01-20

I'm not aware of any more technical presentation of this, but I would assume there is one somewhere. The basic idea is that antiparticles are an unavoidable consequence of combining quantum mechanics with relativity.

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member 648030
Yes,
he antimatter comes out of the relativistic quantum equation of Dirac, which in particular admits solutions with negative energy. (or with negative time, see Feynman diagrams) In any case, these anti-states were interpreted by Dirac himself as anti-particles: the experiments gave him reason

PeterDonis
Mentor
he antimatter comes out of the relativistic quantum equation of Dirac, which in particular admits solutions with negative energy

See my post #6. This is not a valid explanation of antimatter; Dirac initially thought it was, but it didn't work out.

or with negative time, see Feynman diagrams

If you mean "going backwards in time", this is one interpretation of Feynman diagrams, yes; but Feynman diagrams themselves are tools used in perturbation theory, not fundamental explanations of anything.

In any case, these anti-states were interpreted by Dirac himself as anti-particles: the experiments gave him reason

No, Dirac came up with his anti-particle interpretation before any experimental evidence existed for antiparticles. The positron was discovered based on his prediction that antiparticles should exist.

Staff Emeritus
2021 Award
The existence of antimatter is not proved with a law or formula. It's proved by making it in a laboratory.

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dextercioby
member 648030
See my post #6. This is not a valid explanation of antimatter; Dirac initially thought it was, but it didn't work out.
I'll read it, (I did not understand, in what sense it did not work out)

If you mean "going backwards in time",
right
this is one interpretation of Feynman diagrams, yes; but Feynman diagrams themselves are tools used in perturbation theory, not fundamental explanations of anything.
In the diagrams of feynman, an antiparticle is represented as a particle that goes back in time, we agree? (the arrow is antiparallel to the time axis), in accordance with the initial interpretation of Dirac in which the negative time solutions were just antiparticles

No, Dirac came up with his anti-particle interpretation before any experimental evidence existed for antiparticles. The positron was discovered based on his prediction that antiparticles should exist.
exactly, what I wanted to say, but I do not see where the problem is: a theoretical prediction is then confirmed by the experiments, quite normal in physics, (when physics is right..) like Higgs boson, just to cite an example

PeterDonis
Mentor
(I did not understand, in what sense it did not work out)

See post #6 for the most obvious problem with it. There are other more technical issues that would probably require an "A" level discussion.

In the diagrams of feynman, an antiparticle is represented as a particle that goes back in time, we agree?

If you interpret the arrows that way, yes. But you don't have to interpret the arrows that way. For one thing, Feynman diagrams are much more commonly interpreted in momentum space than in configuration space; in momentum space the arrows can't be interpreted as showing "direction in time".

in accordance with the initial interpretation of Dirac in which the negative time solutions were just antiparticles

No, Dirac's original model did not have antiparticles going backwards in time. His "hole in a sea of negative energy states" model had nothing to do with that at all. (And the antiparticle solutions are "negative energy solutions", not "negative time solutions".)

a theoretical prediction is then confirmed by the experiments

Nope. The fact that Dirac's theory happened to predict the existence of antiparticles does not mean the experimental detection of antiparticles confirmed his theory. It just meant his model happened to get the right answer by luck for this particular case.

You can only count an experiment as a confirmation of a particular theory if there are no other theories that can explain the same experimental result. In the case of antiparticles, quantum field theory explains their existence, and does so without the issues that Dirac's model has (again, the most obvious example is that QFT explains why bosons have antiparticles, whereas Dirac's model does not, and antiparticles of bosons have been experimentally detected). When the positron was first discovered in the early 1930's, it was possible to view this as experimental confirmation of Dirac's model, because QFT had not been developed yet; but that is not a tenable position today, many decades later, when QFT is very well developed and the issues with Dirac's model are very well understood.

king vitamin
Gold Member
From the principles of quantum field theory, there's an argument you can make for antiparticles as follows (this is following the presentation in Weinberg's textbook, but I omit some details and might get some factors of 2 or -1 wrong).

First of all, you ask how you can make a unitary quantum mechanical theory which is relativistic. It turns out that this is possible if every observable of the theory is written in terms of causal quantum fields, which are operators satisfying
$$[\phi_1(x,t),\phi_2(x',t')] = 0 \qquad (x - x')^2 > (t - t')^2$$
for all pairs of fields and any composite you can make out of them. This ensures that the time-ordering of measurements outside of each other's light cones doesn't matter (which is already a nice property to have, considering the no-communication theorem).

Now you expand your fields in terms of creation and annihilation operators, but in order to satisfy the above relation you need to do so in a particular way. You find that the fields need to have the form (schematically)
$$\phi(x,t) \sim \int d\omega \, d^d k \left[ e^{- i \omega t + i p x}c(p,\omega) + e^{i \omega t + i px} c^{\dagger}(p,\omega) \right]$$
where $c$ and $c^{\dagger}$ destroy and create operators respectively. Note the extra sign in front of the time-dependence of the second term: this will eventually convince Feynman that antiparticles are particles moving backwards in time, but it exists even without antiparticles, and is just required by (1) expanding fields linearly in creation/destruction operators and (2) enforcing the commutation relation above.

Ok, now what about if you have a conserved charge, as we do in our universe? In this case we not only want our Hamiltonian to be a Lorentz scalar, we also want it to commute with some operator $Q$. If the particle created by $c^{\dagger}$ has charge q, then
$$[Q,c(p,\omega)] = - q \, c(p,\omega)$$
$$[Q,c^{\dagger}(p,\omega)] = q \, c^{\dagger}(p,\omega)$$
But now the commutation relation of $Q$ with $\phi$ is all messed up! This is a nightmare, since we spent so much time constructing the $\phi$ fields so that we can write our Hamiltonian as a function of them, but now we want $[Q,H] = 0$ which can't be done with the fields as-is.

Here, Weinberg argues that you need to fix this by introducing a new species of particle with charge -q (an "antiparticle"), and writing
$$\phi(x,t) \sim \int d\omega \, d^d k \left[ e^{- i \omega t + i p x}c(p,\omega) + e^{i \omega t + i px} a^{\dagger}(p,\omega) \right]$$
where now $a^{\dagger}(p,\omega)$ creates an antiparticle. Now you have $[Q,\phi] = q\, \phi$ and $[Q,\phi^{\dagger}] = -q\, \phi^{\dagger}$, and enforcing $[Q,H] = 0$ simply boils down to each term having an equal number of $\phi$'s and $\phi^{\dagger}$'s.

(There are a few assumptions here, like that we need (and have) an expansion in terms of creation/annihilation operators.)

protonsarecool and Mentz114
king vitamin
Gold Member
Ok, now let's discuss the issue with the Dirac equation. First of all, it cannot even be fully relativistic because it does not allow the creation/destruction of particles, so we are clearly working in some sort of one-particle limit of an actual relativistic quantum theory (QFT in this case). It also has an infinite energy (and even worse, infinite charge!) "Dirac sea" if you want to describe a stable particle.

With benefit of hindsight, we can derive a one-particle approximation of the QFT of spin-1/2 particles, derive Dirac's original formalism, and see what's going wrong. I'm using the approach of Merzbacher, whose textbook covers this well. As usual in QFT, it's tempting to associate the state
$$\Psi^{\dagger}(\mathbf{r})|0\rangle = |\mathbf{r}\rangle$$
as the position-space wave function of a single electron. Except the state is not normalizable, basically because of the anticommutation relations and the fact that $\Psi(\mathbf{r})|0\rangle \neq 0$ (because it creates a positron! See my last post for why.).

The fix to this which leads to Dirac's original theory is to define an "electron vacuum" $|0\mathbf{e}\rangle$, which is the state satisfying
$$\Psi(\mathbf{r})|0\mathbf{e}\rangle = 0$$
Those familiar with QFT will notice that this is the state where every positron state is filled, and every electron state is not filled. So it has infinite energy and charge compared to the physical vacuum, as advertised. This is an unphysical state - this is the sense in which the limit is pathological. Now you can define one-particle states as
$$| \Psi\_e \rangle = \int d^d \mathbf{r} \ \psi\_e(\mathbf{r}) \Psi^{\dagger} |0\mathbf{e}\rangle$$
where $\psi\_e(\mathbf{r})$ is the usual position-space wave function used by Dirac. You can analyze this and show that it's normalizable and evolves via the Hamiltonian in the correct way, etc.

You'll notice that this state can both create a single electron, or remove one of the infinite number of positrons ("create a hole in the Dirac sea"). The "negative energies" which people wrung their hands over in the 1920s are only "negative" compared to the infinitely positive energy which the "one-electron vacuum" has w.r.t. the physical vacuum anyways, so their appearance is no longer mysterious.

Furthermore, rather than interpreting the "holes" in the Dirac sea as positrons (they actually have the wrong charge to be positrons...), it makes much more sense to treat "one-positron" states by defining a "positron vacuum"
$$\Psi^{\dagger}(\mathbf{r})|0\mathbf{p}\rangle = 0$$
and define a corresponding positron wave function in analogy to the above.

Finally, operators can connect electron states with the hole states, so predictions from the one-electron approximation get worse as time evolves, even at low energies. You need to go back to QFT for more precise answers.

protonsarecool
stevendaryl
Staff Emeritus
You can only count an experiment as a confirmation of a particular theory if there are no other theories that can explain the same experimental result.

I guess it's a matter of opinion, but (1) it's never the case that there is only one theory that can make a given prediction. There might be only one theory under current investigation, but there are always infinitely many alternatives. (2) I really don't think that Dirac's hole theory should count as a wrong guess. Much of the mathematics for dealing with holes in Dirac's theory translate without change to the mathematics for dealing with positrons in QED. Dirac's model was a crucial step toward QED, it seems to me.

In the case of antiparticles, quantum field theory explains their existence, and does so without the issues that Dirac's model has (again, the most obvious example is that QFT explains why bosons have antiparticles, whereas Dirac's model does not, and antiparticles of bosons have been experimentally detected).

Again, this might be a matter of opinion, but it seems to me that Dirac didn't need to describe bosons in order for his theory of fermions to be essentially correct. I don't see QFT as an alternative to Dirac's hole theory, but a cleaning up of it. In sort of the same way that the modern Maxwell's equations are actually much more elegant than Maxwell's original ideas, which he thought required a luminferous aether.

PeterDonis
Mentor
I really don't think that Dirac's hole theory should count as a wrong guess.

I'm not saying it should. I'm saying that today, knowing what we know now, we can't say that the discovery of positrons confirmed Dirac's hole theory, because we now don't think Dirac's hole theory was correct. It was a step on the road, yes, but it's not correct according to our current understanding.

Dirac didn't need to describe bosons in order for his theory of fermions to be essentially correct.

That's not the point I was making. The point I was making was that, if the question is how to explain antimatter, you can't just explain anti-fermions; you also have to explain anti-bosons. So any model that only explains anti-fermions can't explain antimatter by itself.

I don't see QFT as an alternative to Dirac's hole theory, but a cleaning up of it.

Mathematically this might be true. But physically, nobody today believes Dirac's infinite sea of negative energy particles actually exists, whereas people do believe that quantum fields actually exist.

People might find this interesting.

atyy