What does boson-type realization mean?

Question

I have seen several different contexts the expression "boson-type realization", for instance in the study of algebras growth and realization of affine algebras.

To be or not be a boson-type realization is a property which deserve to be checked doing what?

What is a good basic reference for the nomenclatures boson, fermion and so on?

Answer 1

Based on a very cursory examination of a paper that uses this term, here's my guess.

Let $V$ be a vector space. The symmetric algebra $S(V)$ of $V$ is a simple version of what in physics is called bosonic Fock space, and the exterior algebra $\Lambda(V)$ of $V$ is a simple version of what in physics is called fermionic Fock space. The names come from the fact that if $V$ is the Hilbert space of states of some particle, then (a suitable completion of) $S(V)$ is the Hilbert space of states of an arbitrary number of copies of that particle if that particle is a boson, and (a suitable completion of) $\Lambda(V)$ is the Hilbert space of states of an arbitrary number of copies of that particle if that particle is a fermion.

Let $V$ be $n$-dimensional with basis $x_1, ... x_n$. Then bosonic Fock space can be identified with the polynomial algebra $\mathbb{C}[x_1, ... x_n]$, and then the Weyl algebra $\mathbb{C}[x_1, ... x_n, \frac{\partial}{\partial x_1}, ... \frac{\partial}{\partial x_n}]$ of polynomial differential operators in $n$ variables naturally acts on it. This action is closely related to the creation and annihilation operators which create or annihilate copies of particles.

The Weyl algebra is in particular a Lie algebra under the commutator bracket, and $\mathbb{C}[x_1, .. x_n]$ is a representation of this Lie algebra. So, my guess as to what a "boson type realization" of a Lie algebra is is that it's a representation analogous to this one. This interpretation is suggested by a few lines in this paper, e.g. "Fock space realization," "boson realization of $\hat{\mathfrak{sl}}(2)$ on the Fock space," "an embedding of $\hat{\mathfrak{g}}$ into the Weyl algebra in infinitely many variables and hence a realization"...

Answer 2

In particle physics, bosons and fermions are classes of elementary particles. Every elementary particle is either a boson or a fermion - for example, electrons are fermions and photons are bosons.

Fermions are characterized by the property that only one fermion may occupy a given quantum state, whereas many bosons may occupy the same quantum state. Mathematically, this means that fermions are described using Fermi-Dirac statistics and bosons are described using Bose–Einstein statistics. Many physical properties of these particles are derived from this essential difference.

I would guess that a definition of "boson-type realization" would be a reference to this mathematical difference. When looking around for a definition, I noticed that this term occurs most often in papers by Ben Cox, a professor at CofC. I imagine the most reliable way to get an answer would be to email him.

Answer 3

I include this example merely to illustrate. The usage of the terms bosonic and fermionic are far more broad.

In the study of $N=1$ supersymmetry one finds that the super Poincare group has a natural action on what is known as $\mathbb{R}^{4|4}$ superspace. This space supports 4 commuting coordinates and 4 anticommuting coordinates. To keep manifest the interaction with the super Poincare group it is customary to take the fermionic coordinates (that is the anticommuting ones) as dotted and undotted Weyl spinors or a Majorana spinor. I use Weyl spinor notation in what follows. A function $F:\mathbb{R}^{4|4} \rightarrow \mathbb{C}_c$ which takes values in commuting complex supernumbers is called a superfield. It has a component field expansion which is expressed as a polynomial in the anticommuting coordinates (modulo any other restrictions). In particular, you can find $f,m,n,v^m,d:\mathbb{R}^{4|4} \rightarrow \mathbb{C}_c$ and $\phi^{\alpha}, \bar{\chi}_{\dot{\alpha}},\bar{\lambda}_{\dot{\alpha}}, \psi^{\alpha}:\mathbb{R}^{4|4} \rightarrow \mathbb{C}_a$ where $m$ takes on $4$ values whereas $\alpha,\dot{\alpha}$ take on 2 values. These 16 component functions build the superfield $F$ below: $$ F = f+ \theta \phi + \bar{\theta}\bar{\chi}+ \theta \theta m + \bar{\theta}\bar{\theta} n + \theta \sigma^{m} \bar{\theta} v^m + \theta \theta \bar{\theta}\bar{\lambda}+ \bar{\theta}\bar{\theta}\bar{\theta}\bar{\lambda}+\bar{\theta}\bar{\theta}\bar{\theta}\bar{\lambda}$$ The notation $\mathbb{C}_a$ denotes anticommuting complex supernumbers. The neat thing about this superfield is it balances 8 bosonic fields (take their values in the grade zero commuting supernumbers) with 8 fermionic fields (take their values in the grade 1 anticommuting supernumbers). Further restrictions on this superfield such as $D_{\dot{\alpha}} \Phi = 0$ (Chiral) or $\bar{D}_{\alpha}\bar{\Phi}$ (anti Chiral) or $F = \bar{F}$ (vector superfield) are made such that the balance between fermions and bosons is maintained. This is the basic content of supersymmetry, to each boson a fermion is paired.

The approach I outline above was initiated by Wess and Zumino in 1974. Superfields simplified the construction of supersymmetric Lagrangians. If you work out what this all means at the level of the component fields it's really quite a beautiful construction. The text by Wess and Bagger or the review by Lykken were popular when I studied these things.

(there are other ways of understanding superfields, but, whatever construction one chooses there will be a $\mathbb{Z}_2$ grading and this is where the terms boson and fermion are obtained)