When it comes to explaining antiparticles: Dirac or Feynman?

I am working with my director, Nic Stacey, and exec producer, Paul Sen, on a way of visualising on television the way combining quantum mechanics and special relativity necessitates the existence of anti-particles. I know how the textbooks do it, but not very visual. Here is a discussion of two of the ways this stuff can be explained.

By 1927, quantum mechanics was developing along two separate lines. On the one hand, Erwin Schrödinger had come up with a wave theory to describe quantum entities like electrons, describing them not as tiny particles but as waves of pure energy. At the same time, Werner Heisenberg was working with abstract mathematics and had developed a theory based on matrices (basically arrays of numbers). But it was Paul Dirac who saw how to combine these two approaches: Schrödinger’s waves and Heisenberg’s mathematical matrices. But his real motives were different.

You see all this was just a couple of decades after Einstein’s work on relativity theory in which he had shown than when a body moves at close to the speed of light, the rules of the game change. And the only way to describe such a fast moving object properly was to unify the three dimensions of space with the one dimension of time into what is called 4-D space-time.

So, we have the Schrödinger and Heisenberg who could to describe how subatomic entities like electrons move and behave and interact, and we have Einstein’s relativity theory to describe what happens when objects move very fast. BUT… what if it is an electron that is moving very fast? Dirac knew he needed to combine quantum mechanics with relativity. What we say is that he was looking for a relativistic quantum theory. In early 1928, Dirac published his relativistically invariant equation for the electron. What he did was unify If we ignore relativity, then quantum mechanics is described by the Schrödinger equation. Here the energy (E) of a free particle (one just plodding along minding its own business and not under the influence of any force) is related to its momentum via the relation: energy equals the square of momentum divided by twice the mass (E=p2/2m). But Dirac wanted to describe what happens if a particle like an electron moved very fast, close to the speed of light. It is here that we have to take Special Relativity into account. According to Einstein, the total energy of the electron is now related to its momentum in a more complicated way, because we also have to consider the energy frozen in the electron as its mass (the famous mc2 bit). And the awkwardness is that the total energy is not just the sum of the old expression and this mc2, but comes in as the sum of their squares, namely: E2 = (mc2)2 + (pc)2. If you can remember school geometry then this is similar to Pythagoras’ theorem whereby the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides. So, why is this so awkward?

Well, with Pythagoras, to work out the length of the longest side of a right-angled triangle we square each of the other two sides, add them, then take the square root of the answer. Fine. All pocket calculators do this for us. But if we are being mathematically rigourous we have to remember that there will be two answers. Consider a triangle with the two shorter sides of length 3 and 4 cms. The sum of their squares (9+16) is 25. So we known the hypotenuse must be of length 5 cm, because 5 × 5 = 25. BUT, 5 × −5 = 25 too (negative times a negative is a positive). Naturally, we can ignore this answer since we are unlikely to find a triangle with a a side of length −5 cm!

But in the energy equation of Einstein’s we also get two values for the total energy of the electron. One positive and one negative. Of course you can also say that we can quite reasonably throw away the negative energy solution because it doesn’t make sense. But Dirac was worried: if you apply this equation in quantum mechanics to describe an electron it is not obvious we can do this. Consider an electron moving freely. We know it cannot have an energy less than zero, so we are safe. But if it is sitting in an electromagnetic field then the way it interacts with the field (according to quantum mechanics) is that it gives off light, as photons. Every time the electron emits a photon, it drops to a lower energy state. But what is stopping it from continuously interacting and emitting photons? Dirac said the electron would eventually be left with negative energy!!!

Other physicists hated this idea, particularly Heisenberg and Pauli. So it is quite OK for the rest of us mere mortals to be uncomfortable with it too! But Dirac insisted that such negative energy states are required and were exactly what his equation was telling him if he kept the ‘other’ solution from his square root formula. How can this make sense? How can a particle have less than nothing energy? Dirac got round this problem by calling upon one of the rules of quantum mechanics, called the Exclusion Principle. This said that no two electrons can have the same energy (well, more correctly, they cannot be in the same quantum state – which is a bit like saying they cannot be in same place at same time with same energy, spinning in same direction etc). He then said, that all possible negative energies the electron might want to have are already assigned to an infinite number of virtual (not real) electrons. So an electron was forbidden from emitting a photon that would reduce its energy to less than zero (taking it to one of the basement levels of the multi-storey) not because having less than nothing energy was silly, but because any such state was already ‘occupied’ by another ‘virtual’ electron. This became known as the Dirac sea and you can understand why it only made other quantum physicists even more upset.
The upshot is that this idea has not survived in its original form, but it turned out to nevertheless be correct that we cannot ignore these negative energy solutions. They didn’t mean that there could be electrons with negative energy, but rather that there are antimatter electrons (positrons) with positive energy! Dirac predicted the existence of these new particles and they were very soon discovered in cosmic ray experiments.
Fast-forward to 1986 and the Dirac Memorial Lecture at Cambridge given by the great Richard Feynman. He gave another explanation for the need for antiparticles. In special relativity, if two events are close enough in time yet far enough apart in space such that the only way the earlier one could have influenced/affected/caused the later is by some faster than light signal, then we say they are space-like separated and the second event lies outside the light cone of first. It also means that there are frames of reference in which the two events would look reversed in time. So, if first event is starting point of a particle and second event is it reaching destination, then in our original frame it is travelling faster than light (ftl). But in second frame it looks like it is also moving backwards in time. I use this explanation in my undergraduate lectures to prove impossibility of ftl.

Feynman said that if a particle only has positive energies (as it propagates from the first point to the second) then the QM formalism insists on the possibility of ftl travel. Of course we cannot really talk about a particle moving ftl. Instead, as Feynman does, we say there is a non-zero prob that the particle could find itself at a space-like separated point. I must say I don’t like this violation of relativity explanation in the least.

OK, the reason Feynman does this is because he can use this business about another frame of reference in which it looks as though the second event happens first. But rather than a particle arriving at this point then moving on, it seems like a particle and antiparticle spontaneously appear at this point (pair creation). The antiparticle is basically the same as a particle travelling backwards in time. The anti-particle moves forward in time until it reaches the later time (that was the earlier time in first frame), whereby it annihilates the original particle. So, with this argument he shows necessity of existence of antiparticles: they are just particles moving faster than light as viewed from different inertial reference frames.

All this is fine, but the connection with the Dirac equation is lost. Basically, Feynman does use the relativistic energy equation E2 = (mc2)2 + (pc)2, but he then says he insists on positive energies only and that this is what gives rise to antiparticles. Whereas, for Dirac it was the insistence on retaining negative energies that suggested need for antiparticles. I find this a little puzzling to reconcile and welcome comments.

About Jim Al-Khalili

Professor of theoretical physics at the University of Surrey, author and broadcaster.
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13 Responses to When it comes to explaining antiparticles: Dirac or Feynman?

  1. Bob Dickinson says:

    I understood the connection to be the fermion field term in the QED Lagrangian… you can either treat the fermion field either (a) as a single field with positive and negative energy solutions, or (b) as two fields with positive solutions
    If you take the path integral for any physical event, the only paths that contribute to the probability of an event are:
    (a) in the single-field version, paths which are connected over space-like or time-like intervals, and
    (b) in the two-field version, paths which are connected over time-like intervals, and allowing particle-antiparticle pair production and annihilation.
    Do correct me if I’m oversimplifying or if it doesn’t really work like that. It’s not a tv show explanation, but it might not be a bad place to start thinking from…

    • Jim Al-Khalili says:

      Thanks Bob. My understand is that Feynman allows for only positive energy solutions in his plane wave propagator (that connects to space-like separated vertices). And space-like separation between two events in one frame means a reversal of the time ordering in another. I do not understand where (in this particular approach of Feynman’s) he ever includes the negative energy solutions. But I have to admit I need to look at this more carefully. Any QED textbooks that do this?

  2. Geoff says:

    Utterly glorious, thank you 🙂

    I’m one of, I suspect, a big group of people who have a background in physics (I’m a non-graduating under-grad) and also have an utter fascination for the subject. We’ve read most of the (usually very, very good) popular science books and now we’re looking for a little bit more, and this exactly what you’re providing, it’s brilliant 🙂 (I’m a science buyer for a UK bookshop, I base my buying decisions on that hypothesis and is seems to be paying off handsomely.)

    Aside from all round congratulations, a slightly more philosophical query:

    From the style of your writing you appear to subscribe to the “maths is real” school of thought, the idea that if something can exist in a working mathematical theory, then it must have an equivalent in our own real life universe. (Wigner’s “Unreasonable Effectiveness Of Mathematics… puts it far more eloquently than I can in a comment). Is this an idea you would subscribe to? Does the very existence of the myriad string theory solutions tie in to the idea of a multiverse for example?

  3. Vandan Parmar says:

    Surely you can apply Dirac’s equation to the frame of reference… In that you can say that in this it looks as if the momentum is this, which is negative… But it will come out positive…. But then when you sqrt it then you can make it negative…? Or not… Am I just confusing everything…

  4. Nick Day says:

    Google returns 4.5 million versus 32 thousand that it’s Pythagoras, not Pythagorus. Or do you know something I don’t, there? Good article, anyway.

  5. Jim Spinner says:

    Can’t you just say the negative solutions to the energy equation correspond to antimatter electrons? I always found that more pleasing than Feynman’s argument, much as I admire him!

  6. Dean Rickles says:

    Hi Jim,

    You mentioned the Dirac sea in your post, but you never mentioned Dirac’s original idea that electrons are holes in such a sea. If it’s visualisability/pedagogy you’re after, then this might be something to discuss (at least as a warm up)?

    On the matter of interpreting Feynman’s argument for the existence of anti-particles (given just positive energies and relativity), I think the answer is quite simple, and just boils down to symmetry. So: the goal is to get a Lorentz invariant quantum theory; and the claim is that this combo somehow enforces antiparticles. To see why requires some CPT ideas (C = charge conjugation; P = parity; T = time-reflection). We know that physics is invariant under the combined operation of these. And clearly, by definition, charge conjugation turns particles into anti-particles. Also, we know that C = PT (Wu parity non-conservation experiment shows this relation in a nice visual way – you should think about using that setup). Physically, this relation means that we can ‘construct’ an antiparticle by performing spatial and temporal inversions (x,y,z,t goes to -x,-y,-z,-t). This makes sense because if we were to then do the C-operation following PT (to give full CPT), we get the original particle back out. The next step is to invoke spacelike separated events so that one can consider their temporal ordering to be switched depending on the frame (the reason for the spacelike separation itself follows from the non-zero amplitude given to paths connecting such events in Feynman’s path-integral approach). One then has the necessary resources to get the antiparticles (one can set up the necessary inversions that will duplicate C = charge conjugation). If one couldn’t do this, then the symmetry would be broken: the theory would not be Lorentz invariant. Hence, the difference between a particle and an anti-particle is not absolute: it is frame-dependent (positive charge flowing backwards in time is negative charge flowing forwards in time).

    This is an explanation in the true sense: we derive antiparticles from the combination of symmetry and QM. Dirac’s hole theory ‘would have’ amounted to an explanation (if it had been right!). I think initially Dirac was playing with equations (as he was fond of saying), so his insistence on keeping the negative energy solutions was due to his characteristic mathematical methods (think of his work on the magnetic monopole for comparison). I think Feynman’s insistence on positive energies stems directly from his path-integral approach to QM and the considerations that go into the weighting of the paths.

    This all seems visualisable, but perhaps you had a different problem in mind?


  7. John says:

    Hi Jim

    Nice blog.

    I’d just like to offer my thoughts here – even though they may be a bit off target.

    Firstly, I think the monikor “antiparticle” is a bit misleading since they are just as much particles as “particles.” It’s just that combining them leads to “pure energy” – all other quantities cancelling out except mass.

    When a gamma photon (near a nucleus) creates an electron-positron pair then its energy become mass. On the other hand (as I think of it), the virtual pair created near a black hole horizon leading to Hawking radiation is one of the pair getting way while the black hole has to cough up the energy deficit by shrinking a little. Surely this is mass (the one that got away) & anti-mass in a sense? (Bit like the bankers getting away with a big bonus & the national debt – “black hole” paying for it? Maybe not.)

    Anyway, it would be good to clarify these disticntions in your program.

    The quantum exclusion approach could be useful because, after all, the hydrogen atom is stable (to a very large timescale anyway) whilst positronium is unstable as the little blighters move in on each other until its game over.

    Funny that Feynman, who is normally perceived as a master of exposition, is not able to come up with a convincing story here. Didn’t he & John Wheeler come up with the wave going forwards & backwards in time when he was doing his PhD?


  8. Barrie Moore says:

    Physicists don’t really accept that time can travel backwards and this is the source of the conceptual problem with antiparticles. However, if time has only travelled one way since the big bang and the second law of thermodynamics obtains, everything we see in the universe has reached its observed state by finding an equilibrium. That equilibrium is disturbed if in a wholly artificial way, you add energy to already “stable” particles. What you will observe is particles trying to revert to an earlier state of equilibrium and every stable particle in the universe will be contributing to the apparent time reversal. Powerful stuff. Light is the least disruptive way to redistribute the energy to the rest of the universe.

  9. This is certainly out of the books point of view your article is. Thanks for such a great read tonight

  10. Paul Secular says:

    I’ve been kind of wondering the same thing, although I’m just an undergraduate at the moment and so understand the topic even less than yourself! Would love to find out more though so please keep us updated.

  11. Scott Gallagher says:

    “The Theory of Positrons:” Richard P. Feynman (1949)

    Traveling in “aunform velocity in a straight line” means one observer’s anti-particle is another persons particle. Therefore, ftl is virtualspace-time process.

    did not Einstein agree that anti-particles could, in theory go faster than the speed of light?

    Wonderful Blog
    Richard Feynman, the legendary physicist from the California Institute of Technology, has been named the seventh greatest physicist of all time in a poll taken by the British journal Physics World. The poll surveyed 130 leading physicists worldwide.
    Feynman, who died in 1988 after four decades on the Caltech faculty, is the only American to appear on the top 10 list, and the only one who did his most important work in the second half of this century. The others are (1) Albert Einstein, (2) Isaac Newton, (3) James Clerk Maxwell, (4) Niels Bohr, (5) Werner Heisenberg, (6) Galileo Galilei, (7) Richard Feynman (8) Paul Dirac, (9) Erwin Schrödinger, and (10) Ernest Rutherford.

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