Quantum Statistics

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QUANTUM STATISTICS

One of the most basic facts about the physical world is that matter is built up from a few fundamental building blocks (e.g., electrons, quarks, photons, gluons), each occurring in vast numbers of identical copies. Were this not true there could be no lawful chemistry, because every atom would have its own quirky properties. But in nature we find accurate uniformity of properties, even across cosmic scales. The patterns of spectral lines emitted by atoms in the atmospheres of stars in distant galaxies match those we observe in terrestrial laboratories.

From the perspective of classical physics, the indistinguishability of electrons (or other elementary building blocks) is both inessential and surprising. If electrons were nearly but not quite precisely identical—say, for example, their masses varied over a range of a few parts per billion—then according to the laws of classical physics different specimens would behave in nearly but not quite the same ways. And since the possible behavior is continuously graded, we could not preclude the possibility that future observations, attaining greater accuracy than is available today, might discover small differences among electrons. Indeed, it would seem reasonable to expect that differences would arise, since over a long lifetime each electron might wear down, or get bent, in a way dependent on its individual history.

The first evidence that the similarity of like particles is quite precise and goes deeper than mere resemblance emerged from a simple but profound reflection by Josiah Willard Gibbs (1839–1903) in his work on the foundations of statistical mechanics. It is known as "Gibbs's paradox," and it goes as follows. Suppose that we have a box separated into two equal compartments A and B, both filled with equal densities of hydrogen gas at the same temperature. Suppose further that there is a shutter separating the compartments, and consider what happens if we open the shutter and allow the gas to settle into equilibrium. The molecules originally confined to A (or B) might then be located anywhere in A + B. Thus, since there appear to be many more distinct possibilities for distributing the molecules, it would seem that the entropy of the gas, which measures the number of possible microstates, will increase. On the other hand, one might have the contrary intuition, based on everyday experience, that the properties of gases in equilibrium are fully characterized by their volume, temperature, and density. If that intuition is correct, then the act of opening the shutter in our thought experiment makes no change in the state of the gas, and so of course it generates no entropy. In fact, this result is what one finds in actual experiments.

The experimental verdict on Gibbs's paradox has profound implications. If we could keep track of every molecule, we would certainly have the extra entropy, the so-called entropy of mixing. Indeed, when gases of different types are mixed, say hydrogen and helium, entropy is generated. Since entropy of mixing is not observed for (superficially) similar gases, there can be no method, even in principle, to tell their molecules apart. Thus we cannot make a rigorous statement of the kind "Molecule 1 is in A, molecule 2 is in A, . . . , molecule n is in A," but only a much weaker statement, of the kind "There are n molecules in A." In this precise sense, hydrogen molecules are not only similar, nor even only identical, but beyond that indistinguishable.

In classical physics, particles have definite trajectories, and there is no limit to the accuracy with which we can follow their paths. Thus, in principle, we could always keep tab on who's who. Thus classical physics is inconsistent with the rigorous concept of indistinguishable particles. It comes out on the wrong side of Gibbs's paradox.

In quantum mechanics the situation is quite different. The possible positions of particles are described by waves (that is, their wave-functions). Waves can overlap and blur. Related to this, there is a limit to the precision with which their trajectories can be followed, according to Heisenberg's uncertainty principle.

When we calculate the quantum-mechanical amplitude for a physical process to take place, we must sum contributions from all ways in which it might have occurred. Thus, specifically, to calculate the amplitude that a state with two indistinguishable particles of a given sort—call them g-ons at positions x₁, x₂ at time t_I will evolve into a state with two q -ons at x₃, x₄ at time t_f , we must sum contributions from all possible trajectories for the q -ons at intermediate times. These trajectories fall into two distinct classes. In one class, the q -on initially at x₁ moves to x₃, and the q -on initially at x₂ moves to x₄. In the other class, the q -on initially at x₁ moves to x₄, and the q -on initially at x₂ moves to x₃. Because (by hypothesis) q -ons are indistinguishable, the final states are the same for both classes of trajectories. Thus, according to the general principles of quantum mechanics, we must add the amplitudes for these two classes. We say there are a "direct" and an "exchange" contribution to the process. Similarly, if we have more than two q -ons, we must add contributions involving arbitrary permutations of the original particles.

It is found by experiment that the particles of nature, from which matter is built, fall into two great classes. For bosons the direct and exchange contributions are simply added. For fermions they are added after supplying a relative sign change—or, to put it more simply, subtracted. We say these two types of particles, bosons and fermions, display different quantum statistics.

Deep understanding of the origin of quantum statistics is obtained in relativistic quantum field theory. Undoubtedly the single most profound fact about nature that quantum field theory uniquely explains is the existence of different, yet indistinguishable, copies of elementary particles. Two electrons anywhere in the universe, whatever their origin or history, are observed to have exactly the same properties. We understand this as a consequence of the fact that both are excitations of the same underlying ur-stuff, the electron field. The electron field is thus the primary reality. The existence of classes of indistinguishable particles is the necessary logical prerequisite to a second profound insight from quantum field theory: the assignment of unique quantum statistics, boson or fermion, to each class. Given the existence of indistinguishability of a class of elementary particles, and complete invariance of their interactions under interchange, the general principles of quantum mechanics require that solutions forming any representation of the permutation symmetry group retain that property in time. But these general principles in themselves do not put any constraint upon the representations that are realized. Quantum field theory not only explains the existence of indistinguishable particles and the invariance of their interactions under interchange but also goes a step further and constrains the symmetry of the solutions. For bosons, only the identity representation is physical (symmetric wave functions); for fermions, only the one-dimensional odd representation is physical (antisymmetric wave functions). Put another way, the wave function for a many-boson system is unchanged when two particles are interchanged, whereas the wave function for a many-fermion system changes sign when two particles are interchanged. This rule is, of course, closely connected with the rule for direct and exchange processes mentioned earlier. Finally, the detailed mathematics of quantum field theory determines the quantum statistics of a particle from what superficially appears to be an entirely unrelated property, namely the magnitude of its spin. The spin-statistics theorem states that objects whose spin is a whole number (measured in units of Planck's constant) are bosons, whereas objects whose spin is half an odd integer spin are fermions.

Among the particles appearing in the Standard Model, quarks and leptons (and their antiparticles) have spin ½ and are fermions; whereas color gluons, photons, W and Z bosons, with spin 1, and the spin-0 Higgs particle, are bosons.

It is straightforward to determine the quantum statistics of composite particles from that of their constituents. The rule, easily derived, is that a composite particle is a fermion if and only if it is built up from an odd number of fermions. Protons and neutrons, according to the naive quark model, are built from three quarks and are therefore predicted to be fermions. In the more sophisticated picture of protons supplied by Quantum Chromodynamics (QCD), they can contain any number of quark-antiquark pairs and gluons. Nevertheless the "naive" prediction remains valid, since adding any number of quark-antiquark pairs changes the number of fermions by an even number, and adding gluons changes it not at all. A slightly more complicated example, which has dramatic experimental consequences (see immediately below), concerns the isotopes of helium. Atoms based on ³He, the isotope of helium containing two protons and one neutron, are fermions. Indeed, these atoms are built up from two protons, one neutron, and two electrons, for five fermions altogether, an odd number. Atoms based on the more common ⁴He isotope, on the other hand, are bosons.

At the level of elementary processes, the influence of quantum statistics comes through the different rules for combining direct and exchange processes. Since dominant nonstatistical—that is, electromagnetic—interactions at low energy are essentially the same for both isotopes of helium, differences arising in the results of low-energy scattering experiments involving the three possible combinations ³He-³He, ³He-⁴He, and ⁴He-⁴He must be ascribed to the operation of quantum statistics. For example, the probabilities of scattering through 90° in the three cases are in the ratio 0:1:2, since in the first case the direct and exchange processes cancel, in the second they add as probabilities, and in the third they add as amplitudes. (For simplicity of exposition, I have assumed that the nuclear spins of the ³He are all aligned in the same direction.) More generally, the quantum statistics of highly unstable or even confined particles, such as quarks and gluons, plays an essential role in predicting and interpreting the results of scattering experiments, which are the bread-and-butter of experimental elementary particle physics.

At the level of macroscopic phenomena, the quantum statistics of particles determines major aspects of the behavior of the matter they form. Roughly speaking, identical bosons like to occupy the same quantum-mechanical state. Laser action, wherein many photons of the same kind—same spectral color, same direction, same spatial cross-section, same polarization—are emitted in a correlated beam, is a manifestation of this behavior. The (closely related) phenomena of superfluidity, super-conductivity, and Bose-Einstein condensation are all characteristic of systems of bosons. They tend to occur at low temperature, when the tendency of bosons to occupy a common state overcomes the disordering influence of thermal agitation.

Identical fermions are forbidden to occupy the same quantum-mechanical state. This is the precise formulation of Pauli's exclusion principle. When forced into a small volume, therefore, additional fermions will be forced into ever higher energy states. Thus a system containing many identical fermions resists compression. The fermionic character of electrons, in particular, underlies the stability of matter, the structure of the periodic table, and the properties of metals. White dwarf stars are supported against gravity by the quantum statistical incompressibility of the electrons they contain. Neutron stars are supported by the quantum statistical incompressibility of their neutrons.

Although ³He and ⁴He are chemically identical, and the gases based on their atoms behave very similarly at ordinary temperatures, their behavior at low temperatures is radically different, reflecting the difference in their quantum statistics. ⁴He is the original superfluid, with vanishing viscosity below about 4K. ³He, on the other hand, is still a normal liquid down to much lower temperatures, around 10^-3K, below which it too becomes super-fluid. The difference in temperatures, and in many more subtle properties, reflects the very different mechanisms at work in the two cases. In ⁴He the bosonic atoms readily organize themselves into a common quantum state. In ³He the atoms are fermions, and only after they form delicately bound quasi-molecular pairs (which are bosons) is super-fluidity possible.

In recent years two developments have brought fundamentally new perspectives to the subject of quantum statistics.

Using only the currently established symmetry principles of special relativity and quantum field theory, it is impossible to connect particles of different spin—or therefore, according to the spin-statistics theorem, particles of different quantum statistics. Supersymmetry is a new kind of symmetry that extends the Lorentz symmetry of special relativity. Supersymmetry postulates the existence of additional purely quantum dimensions. When a particle takes a step into one of the quantum directions, its position in ordinary space-time does not change, but it undergoes changes in its spin and quantum statistics. For example, a spin-0 boson will transform into a spin-½ fermion. Supersymmetry transformations mix ordinary and quantum dimensions. In order for such transforms to be symmetries of physical law, there must be particles of different spin and statistics with closely related physical properties. At present the evidence is far from conclusive, but there are serious reasons to believe that (spontaneously broken) supersymmetry is a feature of fundamental physical law.

It has been discovered that particlelike excitations arising in condensed matter systems, specifically in the state of matter known as the fractional quantum Hall effect, obey new forms of quantum statistics, intermediate between bosons and fermions. Particles obeying the new forms of quantum statistics are called anyons. The existence of these new possibilities for quantum statistics transcends, but does not contradict, the principles discussed earlier. When a material is in the fractional quantized Hall state, the presence of an anyon within it modifies the wave-functions of all its underlying electrons. Thus anyons do not correspond to a simple (that is, spatially localized) composite of any definite number of electrons, and the usual rule for determining the quantum statistics of composite particles cannot be applied to them.