Friday, April 23, 2010

LHC and antimatter

The LHC began its routine work at 7 TeV. After the first preprint, ALICE has also submitted its
second preprint and third preprint
Both of them are about the multiplicity of charged particles. But the last one is extracted from the state-of-the-art 7 TeV collisions. As expected, the number of charged particles seems to grow with energy significantly faster than the existing models predicted. It seems unlikely that it's a real problem with the QCD - but it's a problem with the methods used to calculate the "composition" of the protons at high energies.

Meanwhile, another detector, LHCb, has discovered "its" first particle, the bottom quark. Various media somewhat superficially announce that the LHCb is going to investigate the antimatter: they probably mostly mean the CP violation that can be seen in B meson decays etc.

By the way, Anil A. of Nude Socialist wrote about the reasons why the LHC succeeded where the SSC failed. The SSC wasn't ambitious enough when it came to the strength of the magnets - the superconducting magnets that gave the SSC its proud name were not superconducting (and cool) enough - and they didn't manage to use the same magnets for both beams which made the project too expensive.


Because the LHCb's re-discovery of the bottom quark helps them to study the antimatter, it may be a good idea to say a few general things about antiparticles.

In 1928, Paul Dirac theoretically predicted the first known antiparticle, the antiparticle of electron called the positron because of its positive electric charge. How did he do it and how did he guarantee his 1933 Nobel prize, one year after the positron's discovery? Well, he looked at his equation,
(i γμμ - m) Ψ = 0
and he realized that the plane waves, exp(ipx-iEt), multiplied by an appropriate spinor, were solutions. However, not surprisingly, he found that solutions exist whenever the relativistic dispersion relation
E2 - p2 = m2
is satisfied. And because the solutions of the "mass shell" equation above lie on two parts of a hyperboloid, there also exist solutions with negative values of "E". That's bad. Why? Because there should also exist electrons with negative values of "E". In fact, the same electron should be able to switch from a positive-energy state - the kind we know - to a negative energy state - that we don't know. It could emit a photon whose energy exceeds "2mc^2".

Clearly, we don't observe such things. More theoretically, we can't observe such things because the world would be unstable: the energy wouldn't be bounded from below so we could always create new stuff by lowering the energy of the existing electrons. And if we anticipate the fact that we can create new electrons, the possibility to create electrons with a negative energy would make any state even more unstable.

Is there a solution?

You bet. Dirac re-used his solution to a problem about fishermen in which the right number of fish equals minus two: he found this solution as a little boy. That allowed him to manipulate with negative numbers more rationally.

He realized that Nature tries to be thrifty. It wants the vacuum to be the lowest possible energy state She can have. How can you find this state? Well, you surely want to remove all the particles with positive energies. But what about the particles with negative energies?

Dirac realized that in order to lower the energy as much as possible, it is actually economic to add all the particles with negative total energies that you can add. Because he implicitly understood the Pauli exclusion principle, he knew that only one electron can be added for each negative-energy state.

When you occupy all these negative-energy states, you get a state - the physical vacuum state - whose energy can no longer be lowered. The new physical vacuum has an energy that differs by an infinite amount from the unphysical vacuum we started with - but it's reasonable to call the energy of the new physical vacuum "zero" because this vacuum actually matters in physics. (Universal shifts of energy by an additive constant don't affect physics - at least until you turn the gravity on.)

The same comment applies to the electric charge where it is even more important: you need to admit that you didn't know the charge of the unphysical vacuum (which allowed negative-energy excitations) - so that the charge of the physical vacuum including the Dirac sea is equal to zero.

You can still add regular positive-energy electrons to this state. But you can also remove some negative-energy electrons from this state. In this way, you create holes. Such a hole is "minus one electron" relatively to the exact vacuum state. If you remove one particle with a negative charge (of an electron) and a negative energy, it's just like if you add a particle with a positive charge and a positive energy: the positron. The positron is a hole in the otherwise occupied "Dirac sea" of the negative-energy electrons.

You could have started with positrons and discover electrons after an analogous procedure. Even though we usually think that the words "empty" and "full" are very different, the situations are mathematically equivalent. The occupation numbers may only be 0 or +1 for fermions, and there is a pretty nice symmetry "N goes to 1-N" that interchanges them (and it also interchanges creation and annihilation operators, among other things).

It's important to realize that this "hole theory" of the positron works. And it works for all fermions and their antiparticles. On the other hand, it doesn't work for bosons - because you can't ever "maximally occupy" the states for bosons: the number of bosons in one state can grow arbitrarily large. There's no symmetry between "occupied" and "unoccupied" for bosons, either.

However, bosons such as the W+ boson can have their antiparticles, too: in this case, it's the W- boson. So the hole theory is not working for them. You need a different, more general treatment for the bosons: just properly write all the creation and annihilation operators in the Fourier decomposition and claim that there is nothing else to say.

Because you have developed this machinery for the bosons, many people - including Weinberg - say that the hole theory has been "superseded" by the more general methods of quantum field theory. While it's true in some sense, I think that it is not a pedagogically wise statement.


Because pretty much everyone who hears such things - and isn't able to think about them independently - ends up thinking that Dirac's hole theory for the electron just didn't work and it fundamentally differs from the picture we believe to be correct today. Of course, it doesn't differ. The way how we interpret the antifermions is still mathematically equivalent to Dirac's hole arguments.

So I kind of feel that it would be more pedagogic to start with Dirac's original arguments - that won him the Nobel prize - as an explanation for antimatter, and only present the boring decompositions as a generalization of the ideas that also work for bosons. Dirac's original arguments worked for him - and they also lead you to understand why you can find both creation and annihilation operators in the Fourier expansion of a quantum field.

Antiparticles in the path integral

Richard Feynman had his own method to look at quantum field theory. Correlators - two-point functions - between the quantum fields of the same kind are represented as "propagators", lines with two end-points. The energy and momentum are always running through this propagator.

Once again, the energy-momentum in these propagators may correspond both to positive and negative energy. It's clear that the flip from a positive energy to a negative energy must correspond to the replacement of particles by antiparticles again.

But how do we see the "Dirac sea" and the permutation between "occupied" and "unoccupied" in this formalism?

Well, it is replaced by a similar trick: the antiparticles such as positrons are not just ordinary particles such as electrons with negative energy: they're particles with negative energy that move backwards in time! So if you have antiparticles "A+,B+,C+" in the initial state and antiparticles "D+,E+,F+" in the final state, you represent this process by putting particles "D,E,F" with the opposite - negative - energies into the initial state, and "A,B,C" into the final state!

Again, this is not an ad hoc rule. One can see that with this choice, we reproduce the operations we had in the operator formalism. For example, we had the antiparticle "B+" in the initial state. So there was a hole in the Dirac sea at the "B" spot. And this hole disappeared - the state became occupied - because the "B+" particle changed its momentum. Clearly, the negative-energy electron was present in the final state but not the initial state, so you just exchange these two endpoints of time.

Meanwhile, the rest of the Dirac sea - except for the states corresponding to "A,B,C,D,E,F" - remained untouched. You could represent these states by additional (infinitely many) parallel propagators corresponding to all the states. These extra lines would mean that the particles in the Dirac sea just continued from the past to the future. But what's more important is that such an extra "illustration" with infinitely many dull parallel lines will only add a factor of one (or at most a universal phase) to the Feynman diagram, and can therefore be completely ignored.

Nevertheless, it's a fun idea that antiparticles are particles that travel backwards in time. John Wheeler applied a similar idea to his vision that there's only one electron in the world that goes back and forth in time. It sometimes becomes a positron. He would even use this picture to "explain" why all electrons have the same properties: they're the same electron "who" has just traveled to the past with "his" time machine. ;-)

Now, this is cool. And in some reactions, it's even locally correct: for example, if there's a vertex with an electron, positron, and a photon, it's really true that the electron "continues" somewhere - possibly backwards in time - and Wheeler's picture remains consistent as far as this single reaction goes.

However, we know that this picture can't be true globally. First of all, the theory predicts that the total number of electrons is equal to the total number of positrons (plus minus one). That can't be the case because the electrons dominate over the positrons in the visible Universe.

Moreover, we know why they are allowed to dominate. It's because there are also reactions that allow you to change the "charged part" of the lepton number, i.e. the difference between the number of electrons and the number of positrons: because Wheeler could only create or destroy the electron-positron pairs, the "charged" lepton number was always preserved. In many real reactions, however, you can change e.g. a neutron to a proton, electron, and an antineutrino.

So the "charged" lepton number is not preserved: an electron was created in the neutron decay but there was no positron. However, you could also claim that the total lepton number was preserved because the electron was created together with the antineutrino - that carries the same lepton number as the positron. So the antineutrino is replacing the positron.

However, there are good reasons to think that even the total lepton number isn't exactly preserved. Something like baryogenesis or leptogenesis is needed after the Big Bang to create the matter/antimatter symmetry we observe today. Both leptogenesis and baryogenesis contradict Wheeler's naive picture, even a generalized one.

Moreover, we know that it's not needed for the electrons of the world to "be the same electron" if we want to explain why they're identical. While Wheeler's picture would "qualitatively" explain why all the electrons were equal, it wouldn't tell us how they actually behave in particular situations. To know how they behave, you need the full quantum field theory (or its extension, i.e. string theory). And once you have a quantum field theory, you also know that all electrons are created by the same anticommuting quantum field - which also guarantees that they have identical properties; and that their wave functions must be antisymmetric with respect to their exchange.

At any rate, the discovery of the antimatter doubled the number of particle species in Nature - according to our best theories. Some people could have claimed that it was uneconomic for Nature to have many more species and Nature would be ugly. Except that it's actually more beautiful: the existence of antiparticles follows from the Lorentz symmetry of special relativity combined with some kind of symmetry of the microscopic laws of physics that exchanges the past and the future. The total picture makes much more sense than any picture without antiparticles could ever make.

Needless to say, I am saying these comments because of supersymmetry. It also doubles the number of elementary particle species. And some people also say that it is uneconomic if not ugly. But at the level of theory, they're wrong: supersymmetry is much more beautiful and nontrivial a symmetry than some simple past-future reversal. And it's even plausible that they will be proved wrong experimentally in the near future, too.

And that's the memo.


There are many interesting hep-th preprints today. Hartle and Srednicki posted another paper explaining that the "xerographic distribution" - the assumption about the likely location(s) of "ours" - is a part of the hypotheses that get tested in science. Unlike many anthropic folks, I completely agree with that but I guess there's no really new stuff in the new paper.

Adam Brown and Alex Dahlen posted a new fascinating paper about misanthropic giant leaps in the landscape. They argue that because of a new mechanism, the tunneling to very distant vacua actually becomes more frequent than the small steps to the neighboring vacua, especially if the vacuum energy goes to zero. That means many things - for example, if they're right, it means that the eternal inflation tries to avoid the "hospitable" vacua and Weinberg's scenario that we should end up in the favored states becomes less likely, to say the least.

Chiang-Mei Chen and Jia-Rui Sun have a paper deriving the entropy of a charged rotating black hole, extending the recent paper of Strominger, Castro, and a collaborator to the case of a nonzero electric charge. If you assume that a CFT2 dual actually exists, you get the right Bekenstein-Hawking entropy even for the most general black hole in the 3+1 dimensions. That's pretty cool. In some sense, it must be true that "all" such symmetries (similar to the conformal symmetry) are always present and they can only be spontaneously broken - and it is always a good idea to try to describe the system as a spontaneous breaking of the symmetry.

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