Quantum mechanics implies that the angular momentum is quantized: in particular, its component J_z, with respect to any axis of z, has to be a multiple of hbar/2. Why it is so?

Animation by Flip Tanedo who wrote a nice article about related topics.Well, if you rotate the object around the z-axis, its wave function is changing. If the object is found in a state that is an eigenstate of J_z, it has to be transforming to its multiple and only its phase has to be changing. And it has to change by the factor of exp(i.m.phi) if you rotate the object by the angle phi. Because you have to get to the same state when you rotate the object by 4.pi i.e. by 720 degrees, m has to be either integer or integer+1/2.

Now, I deliberately wrote the (true) previous sentence so that many readers are inevitably screaming: Why 720 degrees and not 360 degrees? WTF? Well, the answer is that only the rotation by 720 degrees is continuously connected to "doing nothing"; a rotation by 360 degrees is not.

**Why half-integral spins are allowed**

To see why the first statement is true, consider continuous paths inside the SO(3) group manifold. You literally want to remember how a particular rotation occurred, a moment after moment. Now, let's represent the process of rotation by the angle 720 degrees as a function from the interval (0,720) that has the following property:

R(x) is the rotation by x degrees where x is between 0 and 720.

Now, then there is another function S(x) that is equal to the identity (no rotation of the three-dimensional space) for every x. The question is:

Can you write R(x) as T(x,0) and S(x) as T(x,1) so that T(x,y) is continuous for all x between 0 and 720 and y between 0 and 1? Also, the initial state T(0,y) must be fixed i.e. y-independent, i.e. T(0,y) = T(0,0), and similarly for the final state T(720,y) = T(720,0).The answer is Yes. You may divide the rotation by 720 degrees around the z axis by two rotations by 360 degrees around the same z-axis. But the latter may be continuously changed to a rotation by 360 degrees around another axis w. And the axis w may be continuously changed from z through x (for example) to -z. So the rotation by 360 degrees may be continuously connected to a rotation by -360 degrees, which is why a rotation by 0 is equivalent to a rotation by 720 degrees, and the latter must therefore bring any state vector to itself.

If you try to do the same thing with the rotation by 360 degrees only, you won't be able to "continuously undo it". A rotation by 360 degrees may end up with the objects that "look the same" but the process of the rotation nontrivially differs from doing nothing!

This was just a topological argument underlying the fact that there exists a double cover of the SO(3) group manifold - the Spin(3) = SU(2) group - and the SU(2) group that is locally isomorphic to Spin(3) allows the half-integral spins, too. SO(3) may be represented as the quotient SU(2) / Z_2.

Just like SO(3) is locally isomorphic to SU(2), SO(3,1) is locally isomorphic to SL(2,C). Let me enumerate a couple of these (local) isomorphisms:

- SO(3) = SU(2) = USp(2)
- SO(2,1) = SL(2,R) = SU(1,1)
- SO(3,1) = SL(2,C)
- SO(2,2) = SL(2,R) x SL(2,R)
- SO(4) = SU(2) x SU(2)
- SO(5) = USp(4)
- SO(6) = SU(4)
- SO(4,2) = SU(2,2)
- SO(3,3) = SU(3,1)
- SO(5,1) = SL(2,H)
- SO(9,1) = "SL(2,O)"

If there are just 2 spatial dimensions, and perhaps one time that is added to have 2+1 dimensions in total, the rotation by 720 degrees can't be continuously deformed to the identity, either. In that case, m may be an arbitrary real number - at least under certain conditions. The resulting particles are neither bosons nor fermions - they're anyons. But there exist "good classes" of theories that require the spin to be integer or half-integer even if there are less than 4 spacetime dimensions.

**Listing possibilities**

The irreducible representations of Spin(3) are labeled by the maximum value of J_z in a given representation - which is called J. It's non-negative and can take the same (non-negative) values that J_z could - 0, 1/2, 1, 3/2, 2, 5/2, and so on.

Representations of larger (pseudo)orthogonal groups such as SO(10,1) are more complicated and require more labels than just the maximum J_z. You need something like a Young diagram to uniquely describe an irreducible representation. But the essential things don't change: you may embed SO(3) into your larger group and study how the representations of the larger group decompose under SO(3). The latter is therefore universally important.

From some viewpoint, the particles of different spins only differ by an integer or half-integer label. However, this simple "numerical" difference forces them to play rather different physical roles.

**Integer vs half-integer**

First of all, it is important to realize that the integral spins - J = 0, 1, 2, and so on - will be carried by bosons while the half-integral spins will be carried by fermions. Bosons are described by commuting fields so that the wave function doesn't change the sign if you exchange a pair of particles. Fermions change their wave function if you permute two of them.

This relationship between J and the behavior of the wave function under permutations is known as the "spin-statistics theory" and it was proven by Wolfgang Pauli. You couldn't really write down well-defined Lagrangians for integer-spin fermions and half-integer-spin bosons.

*Wolfgang Pauli has arguably made the greatest integrated contributions to our knowledge about the spin and its physical consequences.*

Fermions obey the Pauli exclusion principle. Electrons are the most famous examples. So they don't like to be found in the same state. That's because psi(state1,state1) = -psi(state1,state1) = 0. I used the fact that if you interchange the two arguments, the wave function changes the sign. But because I got the same wave function back - assuming that we insert the same states - it's inevitable that the wave function has to vanish if the states of at least two totally identical fermions coincide.

This Pauli exclusion principle gives the fermionic matter species the "impenetrability" that you know if you think about the chair or any piece of solid matter. That's why we say that the "normal matter" is primarily composed of fermions.

Bosons don't respect the Pauli exclusion principle. In fact, they actively disrespect it. Photons are examples of bosons. Instead of being repelled by other photons from the same state, they actively look for coherence - group think and group sex. Lasers are based on group think and masers are based on group sex. ;-)

Because bosons like to be found in the same state, they may create "macroscopic condensates" of many bosons which manifest themselves as classical fields. A trillion of photons in the same state - the tensor product of their wave functions - will look like a classical wave whose profile pretty much coincides with the wave function of each individual photon.

On the contrary, fermions can't produce any condensates (unless you first pair them into bosons). So there can't be any nonzero "classical fermionic fields" that you could actually measure. The fermionic or Grassmannian fields only make sense in quantum mechanics - if you're allowed to excite the states by operators to produce other states. Classically and operationally, you should always imagine that the values of all fermionic fields are zero. That's despite the fact that the algebraic operations with the fermionic fields are analogous to those with the bosonic fields.

**Spin 0 particles**

Spin 0 particles arise from scalar fields - those that have no Lorentz or spinor indices. Consider such a field, M(x,y,z,t), which is assigned to each point in spacetime. The energy density in a piece of spacetime will in general depend on the value of M(x,y,z,t).

If there is no dependence, the value of M doesn't matter: V(M) = const. In that case, we say that M is a "modulus field". More typically, we use the plural form of the word. These fields are the moduli. Any value of M is as good as any other value. There may be several components of "M" that parameterize a higher-dimensional (generally curved) space - the moduli space - which may be thought of as the space of some "solutions to some conditions". Nature tries to minimize the energy but in this case, the minimization tells us nothing.

The value of M would be different at different points of space and time. These values would affect the measured value of the fine-structure constant or the ratios of particle masses. This variability is not observed. That's one reason to know that there are no moduli in the real world. Another reason is that such moduli would induce new long-range forces which are not observed, either. Such new forces would arguably be universally attractive but they would still fail to be proportional to the normal masses of the objects: so they would violate the equivalence principle. Tests show that the equivalence principle holds extremely accurately.

So the real world contains no moduli with V(M) = const. However, such moduli are very important in simple enough toy model field theories and symmetric enough vacua of string theory, especially if there is a lot of supersymmetry (supersymmetry may guarantee that M is exactly massless because M is related by the supersymmetry to higher-spin fields that are massless for other reasons). Such fields M often have nice interpretations - geometric interpretation (shape and size of hidden dimensions, positions of branes, Wilson lines, and so on). None of the values is energetically preferred.

In the real world - and the realistic vacua of string theory - there may be scalar fields but they have to be "stabilized". This adjective means that V(M) = minimum is only achieved for a discrete set of values of M. Those are the values that will be picked by large regions of space. The second derivative of V(M) around the minimum is interpreted as the squared mass of the corresponding particle excitation.

If the second derivative is negative, the particle is a tachyon and the point is a maximum, not a minimum, of the energy which implies that you will only build unstable spacetimes around that point. That is not a good starting point for a sustainable lifestyle - despite the fact that the tachyon field may give you lots of renewable energy! :-) By the way, consistent theories of quantum gravity only allow tachyonic spin-0 particles and no nonzero-spin tachyonic particles. There are many ways to see it.

So in the configuration space of all the scalar fields, we look at the minima of the potential energy. These are the candidate places where the worlds may live.

**Goldstone bosons, Higgs bosons, inflatons**

I will mention three subgroups of spin-0 fields: Goldstone bosons, Higgs bosons, and inflatons.

First, the Goldstone bosons are massless bosons, much like the moduli. However, there is a big difference: for "proper" moduli, the different values of M give you inequivalent environments. The fine-structure constant and the mass ratios depend on the values of M. However, for Goldstone bosons, you will get equivalent environments for different values of the fields.

In fact, the different values of the Goldstone fields G are related by a symmetry which I will call also G. By definition, the Goldstone bosons arise from a spontaneously broken symmetry G. If the Lagrangian - the basic laws of physics - respect the symmetry G but the ground state of your theory (the vacuum) doesn't (so that the vacuum is mapped to a different state by elements of G), it follows that there have to exist massless spin-0 particles called the Goldstone bosons.

If the spontaneously symmetry G above is actually a gauge symmetry, the resulting Goldstone bosons may be proven to be yummy. And if something is yummy, it may be proven to be eaten by someone - and someone will get very fat or massive. In the case of the Goldstone bosons of a gauge symmetry, the eaters are the gauge bosons corresponding to the broken symmetry generators - such as the W-bosons and Z-bosons. When they eat the Goldstone bosons, their mass jumps from 0 to a large value - 80+ and 90+ GeV for the W- and Z-bosons, respectively.

It's necessary for W-bosons and Z-bosons - to be discussed later - to become massive because if they stayed massless, they would induce new long-range forces just like electromagnetism. Because they're massive, the force is a short-range one and only induces a couple of innocent radioactive decays of the nuclei (and the neutron).

Another reason why the W-bosons and the Z-bosons have to eat the Goldstone bosons is that massless spin-1 bosons such as photons have 2 physical polarizations. But massive spin-1 bosons are "vector particles" so they have to have 3 components if there are 3 spatial dimensions around these heavy static guys. The third polarization requires a new degree of freedom and it is nothing else than the eaten Goldstone bosons. The Goldstone bosons co-operates and combines with the original spin-1 fields to produce a massive vector particle.

The Higgs boson is related to the Goldstone boson but it is not the same thing. You usually start with some general "Higgs fields" but some of them end up being the Goldstone bosons which are eaten by spin-1 bosons. The proper "Higgs boson(s)" are those that are not eaten. So they do lead to new spin-0 particles.

In particular, the Standard model has a complex 2-component Higgs field which contains the equivalent of 4 real fields. Three of them are eaten by the three massive gauge bosons - W+, W-, and Z_0 (another way to see that there are three of the massive gauge bosons is to notice that the four dimensional group SU(2) x U(1)_{hypercharge} was broken to a one-dimensional U(1)_{elmg} group, so three generators were broken) - and 4-3=1 Higgs boson therefore remains uneaten. This field produces a single real spin-0 particle that is identical to its antiparticle, the God particle. It's identical exactly because the field is real. So if some Christian readers used to believe that the Devil is someone else than God, I just proved that they're wrong - they're the same beast. ;-)

In the minimal supersymmetric standard model (MSSM), there are two Higgs doublets which means 8 real components and 8-3=5, leaving us with five God particles.

Inflatons I(x,y,z,t) are hypothesized new scalar fields that are important for the cosmic inflation to proceed. It's important for them that V(I), the potential energy, may be pretty high for some values of I - and it must be able to stay pretty high for a long enough time. This large potential energy acts just like a temporary cosmological constant - but its numerical value is (was) approximately 100 orders of magnitude (!) larger than the currently observed cosmological constant. Correspondingly, the accelerated expansion produced by this vacuum energy density is (was) much faster and it has arguably produced the Universe around us.

So this was the lesson about spin-0 fields and particles. Even if I were not comprehensible, you should try to study the meaning of four special types of spin-0 fields - moduli, tachyons, Goldstone bosons, Higgs bosons, and inflatons. By the way, the Higgs fields lead to a spontaneous symmetry breaking exactly because they behave as tachyons at the point of the unbroken symmetry. This causes an instability, the Higgs fields run to a random direction, and they happily find a new stable place to live where the symmetry is broken.

Let me also add, at this point at the end, axions A as the fifth or sixth group. They're ultimately scalar fields as well - even though in some cases, it is useful to rewrite them via their gradient, dA, which is equal to the Hodge dual of something else, namely to *dB where B is a 2-form (an antisymmetric tensor field) and d is the exterior derivative acting on differential forms. There may exist lots of axions in the world around us. Some people believe that their existence is made pretty much necessary by string theory (axiverse etc.). Independently of string theory, axions are helpful to solve the strong CP-problem.

Let me continue with the bosons, and return to the fermions at the end.

**Spin 1 particles**

In relativistic quantum field theory, spin-1 particles are excitations of fields that have to have a vector index, "mu". So the quantum fields are "A_mu" - the electromagnetic 4-potential is the most common example.

However, one of the values of "mu" is the time-like direction "0", and it would create particle excitations with a negative norm. They would make some predicted probabilities negative and that isn't consistent with the basic rules of maths and probabilities. So they have to become unphysical.

It follows that whenever you include fields with vector indices to your action, there has to be a symmetry that "decouples" the inappropriate, negative-norm excitations. It's the gauge symmetry. If you have many time-like components, you need many generators of symmetry.

So photons have a single vector A_mu. You need one generator of U(1) at each point. Gluons have eight vectors A_mu^i. You need an 8-dimensional gauge group, SU(3), at each point. Consistency guarantees that the group really has to be a group - it has to be closed under multiplication (or under the commutator, at the level of the Lie algebras).

In the electroweak theory, the gauge group is SU(2) x U(1) which has 4 generators. So you need 4 electroweak fields A_mu^i. Three basis vectors in this 4-dimensional space produce the massive W+, W-, Z_0 bosons that eat the Goldstone bosons from the Higgs doublet, as I discussed in the Goldstone and Higgs section. One of them, the photon gamma, remains massless. So it still has just two physical polarizations - the helicity may be +1 or -1 but not zero (the zero is only unnecessary because the photons are moving by the speed of light and they have no rest frame) - and the photons mediate a long-range force, the electromagnetism.

While the Higgs mechanism is the canonical way to make the spin-1 bosons (also: vector bosons) massive, this is only necessary for non-Abelian gauge groups. If your gauge group has lots of U(1)'s, their gauge bosons may get masses in a slightly inequivalent way, by the Stueckelberg mechanism. (I encourage Facebook to pay some money so that it would be called the Zuckerberg mechanism which is easier to write down.)

In the case of spin-0 fields, I discussed moduli and non-moduli. Well, the potential energy has to be zero for vanishing values of spin-1 fields. If it were not, the vacuum would choose a nonzero value of the vector A_mu, which would imply that the Lorentz symmetry would be violated. That would be too bad. Quite generally, all "mass terms" for the spin-1 bosons should come from interactions with other fields such as scalar fields - e.g. by the Higgs mechanism - otherwise the theory is sick.

In the spin-0 section, I mentioned axions. Here, I find it useful to mention that in higher-dimensional theories, there are many more types of fields that lead to spin-1 particles. For example, in 11-dimensional gravity, you have the 3-form potential C_{mnp}. Some people could incorrectly think that it is a spin-3 field because it has 3 indices.

But it is a spin-1 field because the tensor is antisymmetric. Embed SO(3) into SO(9,1) and study the rotation around the z-axis, i.e. in the xy-plane. Then the spin of a component of C_{mnp} only depends on the number of x or y indices among m,n,p. If the number is 2 - e.g. in C_{xy8}, you get a scalar because the antisymmetric tensor in xy is invariant under xy-rotations. You also get spin 0 if there's no x and no y, e.g. in C_{567}. But if there is a single x or a single y, you get spin-1 particles.

As I mentioned, the higher-dimensional representations of the pseudoorthogonal groups may be more complicated. But only the "symmetric groups of indices" are able to increase the spin. I will discuss spin 2 fields momentarily.

To summarize, spin-1 fields have to be associated with a symmetry that "decouples" unwanted modes. The corresponding conserved quantities are Lorentz scalars, spin-0 objects, whose commutators promote them to generators of a Lie algebra (in general). Depending on the context, the spin-1 fields may remain massless and their symmetries unbroken, like in the electromagnetism and U(1) with photons, or they may get broken by the Higgs mechanism, or get mass by the Stuckelberg mechanism.

Also, for gluons and the SU(3) gauge group of QCD, the symmetry remains unbroken and the gluons remain formally massless but because the gluons interact with each other (they interact with everything that is colorful and they're colorful themselves), one gets confinement - colors can't be macroscopically separated from each other. Consequently, local objects are colorless and only "residual" forces remain at long distances. For this reason, the strong nuclear interaction effectively becomes a short-range force as well (keeping the protons, neutrons, and nuclei together), despite the massless gluons.

**Spin 2 particles**

Much like fields A_mu, symmetric tensor fields such as g_{mn} have some time-like components that would lead to negative probabilities. Well, in this more general case, it's really the components with an odd number of time-like indices (such as one) that would have the wrong form.

Just like in the spin-1 case, you need to have a symmetry. But in this case, you need to get rid of a whole "vector" of components, g_{0i}, at the same moment. Three of them in the case of 2+1 dimensions. So the number of conserved quantities must be correspondingly greater.

But because the conserved quantities look like a spacetime vector, they must be linked to some symmetries in spacetime - spacetime translations. By definition, these symmetries are spacetime translational symmetries. These symmetries have to be made local to locally eliminate the wrong (negative-norm) degrees of freedom. In other words, a consistent theory with a tensor field such as g_{mn} has to include a diffeomorphism symmetry that has to be treated as a gauge symmetry just like in the spin-1 case.

When you write down the possible gauge-invariant actions, you will start with the Ricci scalar, and consistent theories with spin-2 fields are therefore inevitably theories of Einstein's gravity. You can only have one such a conserved energy-momentum vector and one diffeomorphism.

There exist some limited contexts in which you may have "several independent diffeomorphisms" at each point and in which you may "break" one of them to get a massive graviton but you must be very careful about such things. Whenever you look at those unusual theories more carefully, you will find something wrong with them. It's fair to say that all effective theories extracted from consistent stringy vacua only have one graviton - and exactly one graviton.

**Higher spin fields**

If you tried spin-3 fields, you would still need gauge symmetries that would require the (gauged) conservations of objects even more complicated than spacetime vectors. But if a whole "tensor" of components would be required to be conserved, you would get theories whose interactions must pretty much vanish. This was shown in the full glory by the Coleman-Mandula theorem.

The theorem was only "complete" for bosonic fields. For fermions, one could find exactly one kind of a loophole - the supersymmetry.

So there are good reasons not to consider field theories with fundamental spin-3, spin-4, or higher spin fields. Even spin-5/2 is already too much; I won't discuss this possibility further.

That doesn't mean that "fundamental" particles with higher spins can't exist. After all, a string may produce excitations of an arbitrarily high spin. But there must be infinitely many of them and the interplay between them isn't just about field theory. It is about some string theory or quantum gravity.

In quantum gravity, generic "very high spin" elementary particles must be interpreted as black hole microstates. You may make "increasingly more complicated" excitations of strings or their strongly coupled extensions but in the limit of a "high complexity", there's always some simplified dual description. In quantum gravity, it's the description involving black holes.

Now, let's get to fermions.

**Spin 1/2 particles**

The fermions carry a half-integral spin. They use the loophole that only rotations by 720 degrees have to be trivial. The fermions are excitations of anticommuting fields that transform as spacetime spinors (or spintensors).

The Dirac spinor describes the electrons and positrons - particles that differ from their antiparticles and that are typically massive.

However, we may also consider reduced spinor representations - the Weyl spinors and the Majorana spinors. The Weyl spinors are chiral (left-handed degrees of freedom are taken while the right-handed ones are eliminated, or something that is equivalent); the Majorana fermions are real (they obey a reality condition). In 3+1 dimensions, the Weyl and Majorana conditions restrict the degrees of freedom in the same way - they lead to the same subset of the Dirac degrees of freedom.

It's still useful to treat them separately because the two concepts naturally lead to different interactions. The Majorana particles are those in which the interactions don't preserve the difference between particles and antiparticles. In particular, Majorana masses - which are almost certainly responsible for the observed neutrino masses - allow a neutrino to oscillate into an antineutrino and vice versa. "Weyl fermions" would only deserve this name if they banned such Majorana masses - so they would have to stay massless.

Charged leptons (electron, muon, tau) and their corresponding neutrinos are spin-1/2, much like quarks. These matter fields may carry charges under the spin-1 symmetries - which makes them interact.

**Spin 3/2 particles**

So the last value of spin that deserves a special treatment is J=3/2. You may imagine the corresponding field R_{m,a} to carry one Lorentz vector index "m" and one spinor index "a", giving it the spin 1+1/2=3/2.

Just like in the J=1,2... case, there will be components of this "spintensor" field (Rarita-Schwinger field) with a negative norm. So you need some symmetries - and corresponding conserved quantities - to get rid of these pathological polarizations.

In the spin-2 case, I showed you why the conserved quantity had to be a Lorentz vector (just like it was a Lorentz scalar in the Yang-Mills spin-1 case). The value J=3/2 is in between, so the corresponding conserved quantity has to be a spacetime spinor.

A conserved spacetime spinor is only possible with supersymmetry. The commutator of two such conserved spinor symmetry generators (two such "supercharges") must inevitably contain a spacetime vector. And as I said, the latter has to be the energy-momentum vector, gauged into local diffeomorphisms.

It follows that the only way to get rid of the negative-norm polarizations of spin-3/2 fermions is to have a local supersymmetry that anticommutes to the local diffeomorphisms: it is a theory of supergravity. The minimum number of such spin-3/2 fields gives you the minimal supergravity; the extended supergravity - with a higher number of independent spinor-valued supercharges - will produce a correspondingly higher number of gravitinos.

Just like the Goldstone bosons originated from a spontaneously broken bosonic symmetry, goldstinos (fermions) arise from a spontaneously broken fermionic symmetry, a supersymmetry. Just like those Goldstone bosons were eaten by gauge bosons if the broken symmetry were local, the goldstinos are eaten by the broken symmetry "Goldstone" fermions (in this case, they're fermions), namely by the gravitinos. So the goldstinos allow the gravitinos to become massive.

**Summary**

Even if you didn't understand all the details, I wanted to convey a broader point. In physics, one often has just some "irrelevant numerical differences" between different objects - for example, different signs coming from the rotation by 360 degrees or, equivalently, the permutation of two particles; or different values of the spin. But such differences propagate through the rich logic of physics and you often end up with totally different outcomes.

Particles with different spins and different statistics - even though they differ "just by some integer labels" - end up having qualitatively different roles, properties, and options in a physical theory. They're connected with totally different fundamental features of the world we know - such as the observed forces, impenetrability of matter, broken and unbroken symmetries, stability and instability, and so on. In particular, there is no "egalitarianism". All of physics is about learning how different assumptions (about the building blocks in a theory, or about the initial state) lead to different outcomes.

And despite the common ancestry, they're very different, indeed.

**LHC babe warned of 9/11 in advance**

This could have been too technical so let me wrap up with a soft story. USA Today promotes a comic about a female LHC physicist called Sandra. Her husband died on 9/11. She had a very good idea - you're surprised that no physicist has done it before.

She has simply constructed an efficient time machine, took her iPad, and went to 9/11 before the terrorist attacks. She showed them the detailed documentation. The deniers were unimpressed and ignored her evidence. In particular, her husband denied her because she was too old a harpy: 10 years makes a difference. The full comic will be released in September.

Via NewsBusters

## snail feedback (0) :

Post a Comment