## Monday, January 05, 2009

### Types of elementary particles

One of the main features of "progress" in theoretical physics is the unification of concepts and the emergence of tight links between previously unrelated concepts and assumptions. As this process continued (and continues), our theory were (and are) becoming more robust because they were (and are) built on a smaller number of independent assumptions.

This process is often misunderstood by the laymen who would often prefer the "progress" that invents or discovers completely new things that are unrelated to everything we have ever seen. Well, don't get me wrong, new effects and objects have to be sometimes discovered and science has to cover an ever larger set of phenomena. On the other hand, this is the "zoological" part of the progress in which new species are being constantly added. As soon as they are added, they are understood at a superficial level only.

The "philosophical" part of the progress that connects and unifies the old discoveries into an ever tighter network is more profound. This sort of development may be document by our perspective on the question what are the elementary particles in Nature. Modern quantum field theory describes pretty much all kinds of particles we know by the same formalism: the differences between types of particles that were previously viewed as "radically different" become either technicalities or different faces of the same underlying structure that inevitably follow as long as one understands this structure well.

In this text, I will describe different "iron curtains" that seem to separate "completely different" categories of particles in the minds of many people and explain the relationships between the individual "blocs". These relationships could be found by "pure thought" of sufficiently intelligent observers, at least in principle.

Particles vs waves

Since the early days of quantum mechanics, and even the old quantum theory, people knew that particles and waves were just two aspects of the same thing. Electromagnetic waves and similar objects that were historically identified as waves became a stream of particles such as photons. The energy carried by one quantum of energy always has to be proportional to "hf" where "h" is Planck's constant and "f" is the frequency.

(It's easier to type "h" than "hbar" and "f" than "omega" here.)

On the other hand, objects first identified as particles, for example electrons, were found to exhibit wave-like behavior, including interference. All of them are described by wave functions (probability waves) that became a prototype of a quantum field later, when multiparticle theories were studied by the methods of the second quantization.

Why were the photons initially known as waves (and classical forces) while electrons were known as particles (that can be counted)? Well, it's because the photons are bosons which means that they like to be grouped with their identical friends in the same state. That's why there are usually many photons in the same state, "N", and this large number effectively becomes continuous which is why the collective probability wave describing the state becomes a classical wave, in this case an electromagnetic one.

On the other hand, electrons are fermions that have to obey Pauli's exclusion principle. That's why they can never be found in the same state as another particle of the same type and they can never give rise to coherent fields and long-range forces. One always has to observe them individually, and because the waves in their typical wave functions were much shorter than the resolution of the 19th century physicists (roughly than the atomic radius), people initially didn't know about their quantum/wave properties.

Bosons vs fermions

This discussion leads us to bosons and fermions. It may look like they are completely different types of animals. Bosons prefer to look like classical waves while fermions prefer to look like classical particles. However, when you describe them in the language of quantum field theory, this seemingly qualitative difference boils down to a single sign in which they differ.

Wave functions of bosons are symmetric under the exchange of the coordinates (and other quantum numbers) of pairs of identical particles,
psi(x,y) = +psi(y,x),
while wave functions of fermions are antisymmetric: they flip the sign:
psi(x,y) = -psi(y,x).
In terms of quantum fields, bosons are described by commuting quantum fields while fermions by the anticommuting ones. Anticommuting (Grassmannian) numbers are a bit counter-intuitive for the newbies but if you learn how to deal with all aspects of quantum theory properly, you will see that the Grassmannian numbers work as well as the ordinary, commuting numbers for all the purposes and all the differences arise from the single sign that differs.

A realistic theory must be able to deal with bosons as well as fermions. Moreover, it makes no sense to imagine that a theory should only treat bosons as fundamental particles or it should only treat fermions as fundamental particles. Why?

If you have fermions, it is always possible to construct their bound states with an even number of fermions. These bound states inevitably behave as bosons and their properties are pretty much identical to the properties of elementary bosons: elementary and composite bosons (and other particles) should really be studied together, as we will argue. Moreover, real bosonic particles in an interacting theory (for example glueballs in QCD) also contain a mixture of fermionic particle-antiparticle pairs (for example quarks and antiquarks): one can never "remove all traces of fermions" from the real particles.

On the other hand, fermionic excitations can also emerge from a purely bosonic starting point but one needs more sophisticated methods than ordinary bound states, namely solitonic solutions and their excitations. However, a more important fact is that fermions are observed to exist and a remotely realistic theory simply has to agree with their existence.

To summarize, you should get used to the fact that a good enough theory has to contain both bosons and fermions, pretty much on equal footing, even though their physical manifestations in real-world, complex situations may look very different.

Particles of different spins in general

Bosons that like to be grouped with others can be shown to have an integer intrinsic angular momentum i.e. spin while the spin of fermions is half-integral such as 1/2 or 3/2. Particles, especially the nuclei, can have very high spins comparable to 10 or more.

However, the particles that you would ever consider "elementary" tend to have spins that never exceed 2. In fact, gravitons are the only elementary particles whose spin is 2: their polarizations with mixed signature (one time-like coordinate) would behave as "bad ghosts" that lead to negative probabilities. This potential catastrophe is prevented by gauge invariance, and the only possible gauge invariance for spin-2 fields is general covariance (diffeomorphism symmetry) of GR.

In this setup, the tensor field (the metric tensor) has to couple to the conserved stress-energy tensor. There can only be one such tensor in a sensible interacting theory of one Universe, which is why there can only be one kind of a graviton. However, the graviton may be higher-dimensional and its Fourier decomposition into four-dimensional particles can lead to new types of particles (Kaluza-Klein modes of the graviton; graviphotons; new scalar fields).

Analogously, the removal of unphysical, ghostly modes of spin 3/2 particles requires a conserved spin-3/2 current, corresponding to a spin-1/2 conserved quantity. That's inevitably a "supercharge". Such a supercharge is inevitably fermionic, by the spin-statistics relationships, and the anticommutator of two copies of it inevitably includes translations.

That's why we inevitably end up with diffeomorphisms as a part of the gauge invariance whenever there are spin-3/2 fields: we have a theory of general relativity with local supercharges, also known as supergravity, and the spin-3/2 fields are the gravitino fields. There can be a couple of them but not too many because the interactions become increasingly constrained as you add new supercharges.

The only elementary spin-1 fields are gauge fields such as the electromagnetic fields creating photons or their Yang-Mills counterpart connected with gluons or W bosons or Z bosons. The unphysical, ghostly, time-like component must be removed by a standard gauge invariance, Abelian or non-Abelian one. This gauge invariance can be unconfined and unbroken (like in electromagnetism), confined (like in the strong force) or spontaneously broken (like in the electroweak force) but it is still a gauge invariance, despite the very different behavior of these three types of gauge fields. The underlying mathematics is virtually identical in the three cases and all the qualitative differences in the everyday life are "emergent".

Spin-1/2 and spin-0 fields have no modes with negative probabilities. That's why they require no additional gauge invariances. These ordinary fields are often called "matter fields" (in the narrow sense) in particle physics. It is natural for the elementary spin-0 fields not to be easily observed because they can easily become very massive. On the other hand, the masses of spin-1/2 particles are often protected to be low.

As you can see, elementary fields can only have spin up to 2, and as you approach 2, they require an increasingly specific structure underlying them. That doesn't mean that particles with spins above 2 don't exist: the nuclei or highly excited closed strings surely do exist. But there's no way to write a sensible Lagrangian with a finite number of fields where they would be treated as elementary fields. Such a Lagrangian would need new, high-spin gauge symmetries to get rid of the negative probabilities (from the creation of timelike modes) and such complicated symmetries would require the interactions to essentially vanish for the corresponding Noether charges to be conserved.

Elementary vs composite particles

In the previous paragraphs, I discussed elementary particles as something very different from the composite ones. However, such a difference is only "qualitative" if one intends to describe physics by a particular classical Lagrangian (that can later be quantized). That means that certain fields are chosen to be fundamental - and the particles that they create are usually close to some real particles we can observe - while all interactions are added as small corrections.

In general, interactions are not that weak and the actual observed particles are not identical to the quanta of the fields in a Lagrangian. That also means you can't "qualitatively" distinguish which particles are fundamental and which particles are composite. In this sense, the difference is only useful for the people who write Lagrangians on a sheet of paper, not for people who only want to observe the reality.

If you go to the opposite extreme limit where the interactions become "infinitely strong", the question which fields are elementary and which fields are composites (or solitons, to be explained below) gets mixed up dramatically - in some sense, the answer is turned upside down. We will discuss this issue later.

Stable vs unstable particles

Some particles such as electrons and photons (and maybe even protons?) are stable, others such as the W boson decay after some time. Quite generally, particles are stable if there doesn't exist anything lighter with the same value of conserved charges that they can decay into. By the rules of quantum mechanics, the mass of unstable particles is complex, with the imaginary part being dictated by the width (essentially the inverse lifetime).

Stable particles have real masses, i.e. their width equals to zero.

You could think that this difference between stable and unstable particles is qualitative. But it is only as qualitative as the difference between 0 and another real number. Whether a particle is stable or not depends on dynamics, not on some predetermined categorization of the particles. If you learn the mathematics, it naturally treats stable and unstable particles in the same way - just the width is zero in the stable case. There's no "real" iron curtain between the two groups. More concretely, if a particle is unstable, it doesn't mean that it must be composite.

The neutron is unstable and composite but it is not really "made out of" the decay products, i.e. of a proton, an electron, and an antineutrino (such a bound state would be more similar to a much larger Hydrogen atom). Instead, it is made out of three quarks (and some QCD mushed potatoes in it).

The W boson and the Higgs boson are also unstable but they are completely elementary fields of the Standard Model - a statement that is really uncontroversial in the case of the W boson and the controversy in the Higgs case is largely speculative in character. How can they decay if they are elementary? Well, they can. Particles in quantum field theory can be destroyed and created as long as the conservation laws are obeyed. If no law prevents a process from occurring, it will always occur with a nonzero probability or rate.

Real vs virtual particles, particles vs resonances

There is a related way to look at the same question. Unstable particles appear as resonances. For example, if you collide an electron and a positron, they can annihilate into "pure energy" and a Z boson may be born from this energy (because it doesn't carry any charges, anyway). If the total energy of the two initial particles is close to the Z boson mass, you will produce the Z boson quite often. But because the Z boson mass is complex - the Z boson has a nonzero width because it is unstable - you will produce it even if the total initial energy is slightly off. You will observe the Z boson as a resonance in the scattering of your electron and your positron.

The concept of a resonance is a method to see an unstable particle. It is really another aspect of the very same thing. Whenever you observe a resonance, you can never be certain that the resonance is connected with an elementary or a composite particle. It can be both: except for extremely weakly coupled theories (where all the interactions are weak), there is no God-given qualitative difference between elementary and composite particles.

Color neutral vs confined particles

Quantum Chromodynamics, QCD, offers us lots of new classes of particles. Generally, all particles that interact via the strong force and that can actually be observed in isolation (in reality) are called hadrons. The most important subclasses of hadrons are baryons (with 3 quarks plus mushed potatoes: e.g. protons and neutrons) and mesons (with a quark and an antiquark plus mushed potatoes: e.g. pions and kaons).

But there can exist other types of similar particles such as tetraquarks and pentaquarks (with four or five quarks, plus potatoes) or glueballs (with several gluons only, plus potatoes). These increasingly composite particles become increasingly less fundamental although one must often be careful about such statements because in a different description, with a different Lagrangian, they can be differently composite. For example, glueballs may be dual to some gravitons according to the AdS/CFT correspondence.

These hadrons are very different from the particles that are inserted as elementary fields to the QCD Lagrangian: gluons and quarks. We never observe gluons and quarks in isolation because they carry color and the color (strong charge) is so strongly interacting that it always forces all colorful particles to get neutralized and form color-neutral combinations. In reality, we only observe these elementary colorful particles as "jets": the elementary particle with color tries to escape but because of the strong interaction, other particles are being glued to it and as a result, you will create a stream of color-neutral particles going in the same direction as the original quark or gluon.

But at some deeper level, partons (quarks and gluons) are particles in the very same sense as hadrons. If you study particle physics at distances much shorter than the proton radius, the confinement won't influence you and you will see many colorful particles running inside a hadron. They will be described by the same kind of quantum fields that you can also effectively use at very long distances to describe hadrons. The description in terms of quarks and gluons will be more accurate and well-defined (the theory with these fields is renormalizable) but it will be further from the observational reality because it is the hadrons, and not the quarks and gluons, that we directly observe.

Elementary excitations vs solitons

I have explained that the difference between elementary and composite particles depends on a particular Lagrangian. In fact, more dramatic effects of this kind often occur. There exist particles that are more composite than the normal composite particles, the so-called solitons. Solitons may be identified with classical solutions of some classical field equations: the magnetic monopoles are among the most famous examples. They usually exist because they carry some topologically nontrivial subtlety, a topological charge, or a similarly qualitative feature.

When you quantize your field theory, you will find out that the classical solution behaves as an object that becomes just another species of a particle. It interferes with itself and it does all the things that you expect from other types of particles. If your original field theory was weakly coupled, the solitons usually end up being very heavy, with masses going like "1/g^2" where "g" is the coupling constant. Note that "g" goes to zero so "1/g^2" goes to infinity.

In string theory, there are several possible counterparts of the gauge coupling constant "g". It can be the closed-string coupling constant "g_{closed}" which is why string theory contains ordinary solitons (like NS5-branes and magnetic monopoles) whose mass goes like "1/g_{closed}^2".

However, you may also identify "g", the gauge coupling from field theory, with "g_{open}", the interaction strength of the open strings that goes like "sqrt(g_{closed}). That's why string theory also contains a new, lighter kind of solitons, the D-branes, whose mass (or tension, if they have additional spatial dimensions) goes like "1/g_{open}^2 = 1/g_{closed}". If "g_{closed}" is small, this tension goes to infinity but it is smaller than "1/g_{closed}^2", the parametric dependence of the tension of the ordinary solitons such as NS5-branes.

However, when you send any of these "g" constants to infinity, these particles naturally become light. That's why you shouldn't be shocked that there often exists an equivalent, "S-dual" description of your theory where the role of "g" and "1/g" gets interchanged, much like the elementary particles and solitons. What used to be light small waves on some quantum fields become complicated extended solitons, and vice versa. This S-dual description in terms of the initially heavy objects is more likely to exist in supersymmetric theories where supersymmetry guarantees that the "1/g^2" or "1/g" formula for the mass (or tension) is correct even for large "g", and the object indeed becomes light when "g" is large.

Besides S-duality, modern quantum field theory and string theory offers other examples showing that whether or not a particle is elementary or whether it has an internal structure depends on the description - or the Lagrangian - you choose. Whenever possible, you should naturally choose a description in which all coupling constants are small (interactions are weak). However, such a choice doesn't exist in general. You must live with the fact that quantum field theories have to describe elementary and composite particles together which sometimes makes it very difficult to determine their properties.

When you know your starting point, the elementary particles and their interactions, the problem may look straightforward. In general, such a choice either doesn't exist or it is not unique. There may exist quantum field theories that have no classical Lagrangians, i.e. no allowed choice of elementary fields, but they still predict everything about the particle species and forces that should exist in the world described by this theory.

The (2,0) theory in 6 dimensions is believed to be an example. However, there may be a fivebrane minirevolution in the future that will show that this exotic theory actually has a universal Lagrangian, much like it recently happened with the 3-dimensional M2-brane theories.

Elementary particles vs black holes

The comments above should have convinced you that many (overlapping) types of particles belong to the "core" of the standard quantum field theory or they are linked by insights that have been well understood and "logically" follow from a purely theoretical analysis of quantum field theories: bosons, fermions, leptons, quarks, gauge bosons, gravitons, gravitinos, superpartners, atoms, molecules, animals, planets, stars, other complicated & composite bound states of known particles, solitons, hadrons, mesons, baryons, tetraquarks, pentaquarks, glueballs, neutrinos, Higgs bosons, magnetic monopoles, other solitons, wrapped D-branes, and many others.

All of them may carry an energy/momentum vector, all of them may be associated with some fields, all of them interfere with themselves etc.

However, there exists an additional type of objects that seem different: black holes. Are they composed out of electrons or other known particles? It doesn't seem to be the case. In general relativity, black holes seem to be solitons, classical solutions of some classical field equations. However, quantum theory guarantees that black holes must come as well-defined species, the black hole microstates. They are macroscopically indistinguishable and their number is huge, comparable to "exp(S)", where "S" is the black hole entropy which is very huge because it is the event horizon area in Planck units, and the maximum entropy you can ever squeeze to the same volume.

Despite their very different origin, black hole microstates behave just like other types of particles. They will appear as resonances in a scattering. If you could isolate them in a situation where they are stable, they would interfere with themselves, and so on. Previously, we have seen that the number of possible particles can clearly be infinite because you can create many types of composite objects (such as stars) and they can be excited in very many ways.

But the black holes take this set of possibilities into the extreme because the number of black hole microstates exceeds any number we have previously discussed. They should have a well-defined spectrum with well-defined widths but because they are not really made out of electrons or other known elementary particles, it seems that we don't know any straightforward method to precisely calculate the spectrum of the black hole microstates. Nevertheless, string theory shows that the answer to this question is completely unique even if you can't say that the objects are made out of specific elementary building blocks.

Quantum field theory vs string theory: what remains to be answered

String theory may be thought of as the most conservative extension of quantum field theory that adds gravity - with spin-2 gravitons - to the other forces and matter fields. It is a theory of quantum gravity and the black holes become the newest, most original type of particles that such an upgraded quantum field theory predicts.

On the other hand, string theory may also be viewed to be exactly as complicated an animal as a quantum field theory, due to the AdS/CFT correspondence. A string theory on a curved AdS-like space - which seems to be equally complicated as a quantum gravitational theory on a flat space - is exactly equivalent to a lower-dimensional non-gravitational quantum field theory on a flat space. So if you don't care about the spacetime dimensionality, gravitational and non-gravitational theories (QFTs or string theory vacua) seem to be equally complex.

Quantum field theory and string theory have been understood well enough for the people to qualitatively follow the nature of interference, interactions, scattering, poles in the interactions, confinement, spin, the role of gauge symmetries, and all other features of physics that were mentioned in the text above, besides other features that have not been mentioned.

However, these methods still don't exactly tell you what is the spectrum of particles and their masses. There are many options - there are many string-theoretical vacua and there are even more quantum field theories.  Despite the AdS/CFT-like equivalences, we usually want to talk about a differently filtered set of vacua when we talk about string theory vacua, and a differently filtered set of quantum field theories when we talk about quantum field theories. So the two sets are not really equally large.

In quantum field theory with a Lagrangian - that you can put on a lattice, among other approaches - you can completely calculate all physical phenomena, at least in principle. You will find out that if the theory is well-defined at very short distances, it must be completely specified by its qualitative spectrum at long distances and a few parameters (masses and coupling constants: the marginal and relevant deformations). So QFTs with a Lagrangian seem to form a set that is more or less understood.

QFTs without a Lagrangian are slightly more difficult. This class must include exotic theories such as the (2,0) theory in six dimensions. Nevertheless, it is natural to assume that this broader class, where the Lagrangian can be absent but where the other features of quantum field theory exactly hold, is comparably large to the class of QFTs with a Lagrangian.

In some sense, this class is similarly understood or misunderstood as the set of the string-theoretical vacua, also referred to as the landscape. It is important to note that even though we can't fully construct particles in a generic stringy vacuum out of a specific finite selection of elementary particles, the properties of all particles and interactions (including arbitrarily heavy black hole microstates) seems to be completely determined by the dynamics. For example, the black hole microstates in 11-dimensional M-theory can be calculated from the BFSS matrix model, at least in principle.

In some sense, this returns us to the framework of "bootstrap" - the idea that quantum field theories (and their extensions) are able to determine themselves, by obeying the basic consistency criteria, but without having any explicit methodology based on some preferred starting points (such as a set of elementary particles). This meme, originally promoted by Werner Heisenberg and heavily followed by the S-matrix theorists in the late 1960s, was largely unsuccessful (except for two-dimensional CFTs where it can be almost fully followed).

The history of physics has chosen a different path: all truly successful theories in the 20th century have always been constructed out of some specific starting points and elementary particles, such as quarks & gluons or relativistic strings. However, the recent duality revolution has shown that the choice of the elementary particles is not unique: in fact, there are many equivalent ways to approach a particular strongly-coupled quantum theory. In some sense, the number of classical Lagrangians is even higher than the number we need, i.e. than the number of physically distinct quantum theories.

On the other hand, the duality revolution also suggests that some theories or points on the landscape should exist even if we have no weakly-coupled description or a Lagrangian method based on elementary particles to approach the point. It is very tempting to ask what are the principles that can determine all properties of such theories without constructing them out of some specific elementary building blocks. However, we must realize that such a program is not guaranteed to be on the right path. It is a research project and its success may be a matter of a wishful thinking only.

While question marks remain, physics has already achieved an amazing degree of unification of its basic concepts that are enough to understand the observable world around us.

And that's the memo.