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Fermi-Dirac and Bose-Einstein statistics: amazingly different biological sisters

Enrico Fermi died exactly 59 years ago. (Click.)

This greatest Italian physicist after Galileo Galilei was also the ultimate example of a physicist who was both a great theorist and a great experimenter.

Fermi speaking in 1954

Fermi has done some amazing work on the nuclear bomb, nuclear energy. More theoretically, he's famous for the first theory of the beta-decay and for the Fermi-Dirac statistics as well as numerous other things.

His theory of the \(\beta\)-decay from the 1930s is rather simple to summarize. All the \(\beta\)-decays of the nuclei may ultimately be described as the decay of the neutron\[

n \to p+ e^- + \bar\nu_e

\] to a proton, an electron, and an electron antineutrino (using the modern terminology). Fermi was wise to insist on the energy (and angular momentum) conservation in nuclear reactions so he also posited the existence of a new, nearly invisible particle, the neutrino (which is translated as "a little neutral Italian thingy" to English).

Even great minds like Niels Bohr were ready to "sacrifice" the energy conservation law because of something as mundane as the \(\beta\)-decay.

It would be really bizarre if one (odd number) of fermions were transforming into two (even number) fermions. The initial state would change its sign under rotations by \(2\pi\) while the final state wouldn't. At any rate, he was sure that an extra fermionic particle had to be created.

At the quark level, we describe the transformation as\[

d \to u + e^- + \nu_e

\] instead of the protons and quarks. A similar interaction is also responsible for the dominant decay of the muon,\[

\mu^- \to \nu_\mu + e^- + \nu_e

\] and similar decays. Fermi's theory of the \(\beta\)-decay simply stated that the field-theoretical Lagrangian contains a term that is quartic (i.e. fourth-order) in the fermionic fields and that is directly able to change the initial fermion to the final three fermions at the same point (among related processes),\[

\LL_{\rm Fermi} = \frac{G_F}{\sqrt{2}} \cdot \bar \psi_d \Gamma_{\rm Fermi} \psi_u\cdot \bar\psi_{\nu,e}\Gamma_{\rm Fermi}\psi_e

\] When you expand the fields \(\psi\) and \(\bar\psi\) to creation and annihilation operators, this operator is able to destroy one fermion and create other three (or one of the similar processes). Fermi's constant \(G_F\) is chosen to predict the right neutron lifetime.

The object \(\Gamma_{\rm Fermi}\) may be the identity matrix or some other function of the Dirac gamma matrices. If needed, it carries free Lorentz indices and they're summed over because they appear twice in the Lagrangian above. The character of this \(\Gamma_{\rm Fermi}\) decides about the "tensor character" of the interaction. If \(\Gamma_{\rm Fermi}\) is

  • \(1\), then the interaction is of the type S, the scalar interaction
  • \(\gamma_5\), then it is a pseudoscalar interaction,
  • \(\gamma_\mu\), V for vector interaction
  • \(\gamma_\mu\gamma_5\), A for axial vector interaction,
  • \(\gamma_{[\mu}\gamma_{\nu]}\), T for tensor interaction
This pretty much lists all the possible tensor structures in \(d=4\); I don't want to explain and clarify all the details here. In the early 1960s, mostly thanks to Feynman and Gell-Mann (who insisted on their right theory even though some stellar experimenters were saying they had excluded it and proposed an S–T theory instead), it became clear that the right theory is V–A (vector minus axial vector) and\[

\Gamma_{\rm Fermi,\mu} = \gamma_\mu (1-\gamma_5)

\] which actually allows the parity violation to be "maximal". Only the left-handed spinors are coupled. In the early 1960s, a decade after Fermi's death, it became clear that Fermi's four-fermion interaction results from "integrating out massive W-bosons" which are coupled to charged currents.\[

\LL_{\text{charged currents}} = g A^{W,+}_\mu \cdot
\bar \psi_d \gamma^\mu(1-\gamma_5) \psi_u +\\
\bar \psi_\nu \gamma^\mu(1-\gamma_5) \psi_e +\dots
} + {\rm h.c.}

\] One must also add the massive yet electromagnetism-like kinetic terms for the gauge field \(A^{W,+}_\mu\). The electroweak theory (a key part of the Standard Model) was born.

But let me return to the Fermi-Dirac statistics.

In January 2012, I discussed the Bose-Einstein statistics, its derivation, and some properties. The derivation of the Fermi-Dirac statistics is analogous and arguably simpler.

If we deal with fermions, the allowed occupation numbers for a given one-particle state are \(N=0\) and \(N=1\). The value \(N=2\) is already too many because of Pauli's exclusion principle: there can't be two fermions in the same state. We will discuss this point later.

By Boltzmann's general rules, the probability of \(N=1\) obeys\[

\frac{P(N=1)}{P(N=0)} = \exp \zav{ - \frac{\epsilon-\mu}{kT} }

\] where \(\epsilon\) is the increase of energy caused by adding the fermion into the state i.e. by changing \(N=0\) to \(N=1\). The exponential decrease and the \(1/kT\) factor follow from the general Boltzmann's rules how the probability of a microstate decreases with energy. I have also shifted the energy \(\epsilon\) by the chemical potential \(\mu\) to encode the extra forces that may be trying to add or subtract particles even though it costs some energy.

If we demand\[

P(N=0) + P(N=1) = 1,

\] we may easily derive\[

P(N=0) = \frac{\exp[(\epsilon-\mu)/kT]}{ \exp[(\epsilon-\mu)/kT]+1 },\\
P(N=1) = \frac{ 1 }{ \exp[(\epsilon-\mu)/kT]+1 }.

\] It's easy to verify that the ratio of these probabilities and the sum are equal to the desired values. The expectation value of \(N\) is \[

\langle N \rangle = 0\times P(N=0) + 1\times P(N=1) = P(N=1)

\] the same as the probability that there is a particle in the state. It's so simple. Unlike the Bose-Einstein case, we didn't even have to sum any geometric series.

Bose-Einstein statistics and Fermi-Dirac statistics are the two new 20th century types of statistics that supersede the Maxwell-Boltzmann statistics of the 19th century. In both cases, the \(\pm 1\) term in the denominator may be pretty much neglected for \((\epsilon-\mu)/kT\gg 1\) and one gets the old-fashioned simple exponential formula for \(\langle N\rangle\).

But you shouldn't overlook the remarkable difference between Bose-Einstein statistics and Fermi-Dirac statistics. Even the names sound very different. Bose and Einstein would prefer to investigate fields, electromagnetic waves etc., and at least Einstein was skeptical about quantum mechanics. On the other hand, Fermi and Dirac loved particles and they were enthusiastic about quantum mechanics.

That's no coincidence. Bose-Einstein statistics has an unlimited \(N\) so you may have many particles in the same state – bosons actually love these parties more than the Boltzmann particles would. Because \(N\) is very large in a "classical limit", it is de facto a continuous quantity and one may define classical, continuous fields that are larger if \(N\) is larger.

On the other hand, Fermi-Dirac statistics prohibits \(N\gt 1\). Even in any classical limit, the number of particles in a given state is at most one. That makes it sensible to talk about the coordinates (or momenta) of these particles. For these reasons, the classical limit applied to bosons often leads us to fields while the classical limit applied to fermions leads us to particles. See How fields and particles emerge from quantum theory, for example, if you want some additional words.

In a viable world that should contain intelligent life and some games that the intelligent life may enjoy, you may need "something like classical particles" and "something like classical fields". These look like very different concepts. If Nature hired a government to prepare everything that is needed for intelligent life, the government would almost certainly establish at least two major independent "departments" that would have thousands of subdepartments. These departments for "particles" and "fields" would work qualitatively differently.

But Nature is much more unified, clever, and effective. As modern science has proven, Nature abhors bureaucracy and leftwingers. Nature only has "one department" for all these "seemingly vastly different concepts". It contains bosonic and fermionic fields, bosonic and fermionic creation and annihilation operators, and they only differ by one sign in the supercommutator:\[

a_i a^\dagger_j \mp a^\dagger_j a_i = \delta_{ij}

\] where the minus sign is relevant for the bosons and the plus sign is relevant for the fermions. Some fields "nearly commute" with each other while others "nearly anticommute" with each other. One single switch is enough to make a field behave very differently – and the qualitative implications seem profound.

Bosons lead to "classical fields" in a classical limit; fermions primarily allow "classical particles" in a classical limit (although we of course may oberve isolated bosonic particles, too). That's not the only difference. The set of allowed values of \(N\) for fermions (but not bosons), \(\{0,1\}\), is invariant under the operation \(N\to 1-N\) which turns the spectrum of \(N\) upside down i.e. which interchanges \(N=0\) with \(N=1\).

That's the reason why we may rename the "occupied and unoccupied state" as "unoccupied and occupied state" (don't overlook the order of the words) and why the creation operator may be interchanged with the annihilation operator if the "particle we may create" is identified with the "hole" i.e. "absence of particle" in the previous picture. That's behind the Dirac sea treatment of the quantum fields that may only be "literally" applied to the fermions. For bosons, the allowed spectrum of \(N\) is allowed to be arbitrarily positive but it is not allowed to be negative so there's no "upside down" symmetry.

Sometimes, the same DNA code is behind two very different beings, in this case, a handsome, efficient human being vs a mindless pile of protein.

Nature is extremely efficient in producing vastly different outcomes from a deeply unified mathematical perspective. Even if two objects or phenomena or concepts or paradigms look as different as you may get, these two objects or phenomena or concepts or paradigms may be related by a tiny switch, a subtle and seemingly innocent minus sign, and they may ultimately be solutions of the very same underlying equations.

Seemingly very different objects, phenomena, concepts, and paradigms in physics often turn out to be biological siblings that are related by the DNA of physics. The process of discovering these superficially hidden examples of kinship is something that I find incredible about modern physics.

And that's the memo.

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