Gauge Theories2.5 Fermions, Bosons and Anyons First I apologize for not being able to write this series in most of the past month because of some unexpected assignments. I appreciate the readers, the great Scholar Shortriver, in particular, for their patience.
In the previous 2 postings, we have learnt that quantum field theories are the natural, necessary and indeed inevitable outcome if quantum mechanics has to be incorporated with the special theory of relativity. In passing, we were also told that antiparticles are a must for quantum field theories. With antiparticles, the notion that “particles and fields are the same stuff” becomes more transparent: fields are the configuration of a physical system in the real space and particles are the fields in the momentum space, or, k-space. Fields and particles are just two different but equivalent descriptions of nature.
What I did not mention is that the inclusion of the special theory of relativity has another ‘deep’ consequence: the particles and antiparticles, or the excited states of the field, have internal degrees of freedom. For instance, a particle may carry spin, or magnetic moment. It is well known that the electron has a negative charge, but it is less well known that the electron also has a magnetic moment, i.e., an electron is a tiny bar magnet. The fact that an electron has a charge is the foundation of the mature electrical and electronic industries; the fact that an electron has a spin is the foundation of the emerging technology called spintronics. In the following, we will reveal the fundamental origin of this internal degree of freedom. With that, I hope you will have a better and deeper understanding of our spacetime and the fields/particles it permits.
Now we are in the position of ending the conceptual introduction of quantum field theories by presenting all the commutators related to quantum field theories. We were informed at the beginning of Section 2 that all computations in quantum field theories, such as in particle physics, nuclear physics and cosmology, boil down to the manipulations of commutators.
Now that we have antiparticles, we need to introduce the corresponding creation and annihilation operators, b’ and b. To include antiparticles in quantum mechanics, we need to stipulate the difference between doing measurements in different orders, i.e., we need to specify new commutators for b, b’ as well as between a’, a and b’, b. The idea is exactly the same, only more lines of formula are written.
Experience tells us that even for one type of particles, they may be different physically, namely they may take different values for a given physical property. That is of course obvious. An electron, for example, may have different kinetic energies, momenta or spin. It is convenient in practice to use extra subscripts to specify these properties and creation and annihilation operators now look like a’_i, b’_j, a_i, b_j etc. The most important gadgets in the entire toolbox of the quantum field theories are the following commutators: [a’_i, a_j], [a_i, a_j], [a’_i, b’_j], [a_i, b_j], b’_i, b_j], [b’_i, b’_j], [b_i, b_j]. What more happy is, most of them are simply zero. For instance, they are all zero if i is not equal to j. Similarly, any pair creation operators and any pair of annihilation operators are zero. These facts are easy to understand in perspective of viewing a commutator as doing measurements in different orders.
The only nonzero commutators are [a’_i, a_i] = h and [b’_j, b_j] = h (ignoring a constant). In words, the following two cases have different outcomes and the difference is characterized by the Planck’s constant h: (i) creation of a particle before its annihilation and (ii) annihilation of a particle before its creation. In case (i), the particle has not existed yet, but in case (ii), the particle has already been there.
Therefore, a quantum field theory is not that different from quantum mechanics. It is still about measurement albeit on systems with infinite number of degrees of freedom. The commutation rules are not much different from those in quantum mechanics. The only spectacular thing, as it seems, is the introduction of antiparticles imposed by the special theory of relativity.
Is it the end of the story?
Unfortunately, it is not. No. Fortunately, it is not.
It turns out that the above commutators only apply to one class of particles called bosons. The other class of particles, called fermions, satisfy a slightly different commutation rules, i.e., anti-commutation rules: {a’_i, a_i} == a’_i a_i + a_ia’_i = h. That is to say, the minus sign in a commutator has to be changed into the plus sign for fermions. The reason is that bosons and fermions have different internal degrees of freedom: bosons have an integer spin such as 0, 1, 2,…while fermions have a half-odd-integer spin such as 1/2, 3/2,...
Why is it so? Why do we have two classes of particles? Or why do we have two classes of excited states of the fields? The question can also be askedin this way: Why do we have ONLY two classes of particles? The root of cause is that our fields are “contained” in spacetime. We all know that a container necessarily imposes some constraints on its content. The oil in a capillary and the oil in a film on the surface of, say, water, may show very different properties. The properties of spacetime must affect the properties of the fields it contains.
From the Lorentz invariance we learnt in the previous posting, we know that our spacetime is isotropic or democratic, i.e., there are no noble positions and times nor autocratic directions. It seems a really plain, unremarkable, even boring, fact of nature, but it is one of the most important facts for physicists. Imagine that our spacetime does not have this symmetry. Then this would imply that every point of our spacetime has its own physical laws, possibly unrelated, or, related to each other in a very complicated manner. Billions of Newtons and Einsteins would not be sufficient to find a physical theory applicable to our universe!
Fortunately, our spacetime does have this beautiful, simple symmetry (at least a very good approximation). A completely democratic spacetime has the highest symmetry as you can see it immediately, denoted SO(3) or SO(3,1) if time is also included. That is the most important property of our spacetime. Understandably, we will come back to this group frequently.
It is this obvious property of spacetime that leads to the two classes of particles, or, two sets of commutation rules (commutation and anticommutation rules).
To pinpoint which property of spacetime determines the commutation rules, we take a closer look at the meaning of a commutator, the symbol denoting the difference of doing two measurements in different orders.
In the language of ‘particles’, for a system consisting of identical particles, exchanging any two (or more) particles should make no differences. A country full of identical individuals would not make any difference if any two of them exchange positions. In quantum mechanics, this means that the exchange of any two particles should not make any measurable effect, i.e., it is of no significance on any measurements. However, we must be aware of that no effect on measurement results does not require no effect on wavefunction or field configuration. As mentioned in Section 2.2, wavefunction is more a supporting tool to assist calculations than a physical object. In fact, as well known, if a wavefunction is multiplied by a phase factor ph=e^(i*theta), no physical difference is made. So is the case for field operators. Therefore, if two identical particles exchange positions, the wavefunction or field operators may be changed, at most, by a phase factor. Any two particles can exchange positions and bring a phase factor to the wavefucntion or field operators. Because of the equivalence between particles and fields, the effect on exchange in positions is exactly the same as that on exchange in momenta.
Therefore, the effect of exchanging particles on creation or annihilation operators brings about, at most, a phase factor, i.e., the general commutation rule, after taking this identical symmetry into account, is: [a’_i, a_i*ph]=h. It might be superfluous to mention that this identical symmetry is a consequence of our isotropic spacetime. The remaining job is to find what this phase factor ph is. As mentioned above, this ph must be a property of our spacetime.
[a slightly different way of argument: an identical particle system should possess exactly the same symmetry of the spacetime. the wavefunction of a particle under a full rotation (e.g., a 360-degree rotation around a fixed axis) must show no difference other than a phase factor. therefore, when two identical particles are exchanged, the effect is at most a phase factor. it is equivalent to say that change of the order of any two creation/annihilation operators, brings about at most a phase factor. ]
Well, exchange of two particles produces, at most, a phase factor. How many different values can ph take? The problem, therefore, turns into how many distinct ways we bring a system back. It seems we have an infinite number of ways. However, our spacetime has a high symmetry, specifically its symmetery group is SO(3,1). Among the seemingly infinite number of ways of bringing a system back, many of them may be identical. To see whether two ways are identical or not, we look at the parameter space of the symmetry group. A rotation is specified by the rotational axis and the angle of rotation. It can be expressed by a point in a space: its distance to the origin is the rotation angle and the direction of the vector linking the point and the origin is the rotational axis. The parameter space of a symmetry group is the collection of all such points. The parameter space (topological space of the symmetry group) of SO(3) is a ball with diameter pi. Considering the rotations of pi and –pi are identical, all antipodal points of the surface of the ball must be identified.
A loop (with its base point at the origin) in the group space corresponds to a series of transformations that return the system to its original configuration. If two loops can change from one to the other via a continuous deformation, they are equivalent. Two equivalent loops make no difference on the system. In particular, if a loop can shrink into a single point in the group space, the transformations are equivalent to a single rotation. If a loop can shrink into the origin, the transformations are equivalent to no rotation at all.
How many ways of shrinking a loop in the parameter space of the symmetry group of our spacetime? For the topological space of group SO(3,1), it turns out that there are only two classes of loop: A loop in the group space that can shrink into a single point (the origin) and a loop that cannot shrink into a single point but two of such loops can shrink into a single point. We find that the loops in the parameter space of the symmetry form a group, called the fundamental group. For SO(3) group, therefore, its fundamental group has only two elements: the class of loops that can shrink to the origin and the class of loops that need run two turns to shrink to the origin. The first class corresponds to ph=1 and the second to ph = -1. Taking into account the symmetry of spacetime, therefore, we find that there are only two possible distinct effects on a system by exchanging the particles.
In formal, concise language, the commutation of creation/annihilation operators is decided by the fundamental group of the topological space of the symmetry group of the particle system. It may sound bizarre for the people who hear this for the first time, but do not worry. Here we add a paragraph to dispel the mystery of the fundamental group. The reader who is already familiar with the group space may skip the following paragraph and simply jump to the last paragraphs of this posting.
Let us take another look at what is the topological space of a (symmetry) group. The symmetry group itself is a topological space, i.e., the space formed by exhausting the values of the parameters that characterize the group. For example, the symmetry group of a circle is SO(2), i.e., the circle shows no difference when it rotates around its symmetric axis by any angle. The parameter space of the symmetry group, therefore, is a real line. However, we know two more facts about rotation around a fixed axis: (i) that rotating by alpha is the same as by pi – alpha, (ii) rotating by one turn is the same as by any integer number of turns. Taking these two facts into account, we find the topological space of SO(2) to be something like this: a line but we have many, many equivalent points, such as (using pi as unit): 1 = 2 = 3 = 4 = …. n = -1 = -2 = -3 = …. = -n and a = 1 – a. Therefore, the parameter space of SO(2) is a segment [0, 1] with its two endpoints 0 and 1 identified (equalized). It is actually a circle. The group space of the group SO(2) (the symmetry group of a circle) is a circle. It is easy to find that the loops (of any turns) on a circle cannot shrink to a point. Any two loops of different turns belong to different classes. Therefore, the loops form a group with infinite number of classes (elements). The fundamental group of the topological space of SO(2) is all integers because any number of turns around the circle is different from other numbers of turns. This has a profound consequence: the exchange of any two particles in a 2+1 spacetime can bring about any phase factor ph. The commutation rule for the field operators, therefore, is the most general form: [a’_i, ph*a_i] = h. The particles satisfying this rule are called anyons.
In summary, the fundamental group can be simply described as the ways of contracting a loop in the space. Visually, the fundamental group identifies the ’holes’ in a space. The fundamental group of the topological space of SO(3) is a group with only two elements (denoted Z2) because there are only two different ways of contracting the loops in this topological space. For two-dimensional systems such as the electrons restricted in a film or bound in a membrane, the symmetry group is reduced to SO(2) whose fundamental group has infinite number of elements (Z). In our spacetime, i.e, 3+1 manifold, ph=+1 for bosons and ph=-1 for fermions. For 2+1 spacetime, ph can be any values and the corresponding particles are called anyons. Therefore, the commutation rules in quantum field theories have a deep origin that traces all the way back to the most fundamental properties of our spacetime, the topological structure of the symmetry group of our spacetime. Doing measurements in different orders must consider the symmetry of our spacetime.
Therefore, the topological property of the spacetime may provide an extraordinary restriction on the possible types of fields or particles. For 3+1 spacetime, there are only two types of particles. We may assign an internal degree of freedom, the spin, to the particles. In this picture, the commutation or anticommutation is decided by the spin of the particles, integer spins (bosons) obeying commutation rule, half-integer spins (fermions) obeying anticommutation rule. For 2+1 spacetime, however, an infinite number of types of particles may be allowed. Non-integer and non-half-integer spins (anyons) obey ‘any-commutation’ rule.
There is another way of describing above difference. According to the special theory of relativity, a particle can be described by its worldline—the “trajectory” of the particle in spacetime. The history of a particle can be represented by a curve in the spacetime. You may imagine that the worldlines of the particles form noodles or braids. Exchange of particles means the braids getting tangled. In 3+1 spacetime, the tangle by exchanging the worldlines of two particles can always be detangled. In 2+1 spacetime, however, the tangle cannot be detangled.
Now it is good time to upgrade our old picture of spacetime as a kind of passive, void background or box detached with the fields or particles it contains. In the previous posting, we learnt that the special theory relativity tells us that spacetime is not a passive framework, it has its unique property: it obeys Lorentz transformation rather than Galilean transformation, i.e, space and time are connected. In this posting, we see that the topological structure of spacetime provides a severe restriction on the possible types of particles.
At this stage, we have come to the end of quantum field theories, conceptually. The remainder is technical. We may ignore them completely without affecting our understanding of nature significantly. However, to expose the problematic aspects of modern quantum field theories, we would better go through some technical procedures. The good news is that we still do not need equations to accomplish the job. Words, logic and a fresh mind are all we need here.