Fermions and the Jordan-Wigner transform II: the Canonical Commutation Relations

Back to fermions today, with a post introducing the canonical commutation relations fermions, and discusses something of their mathematical and physical significance. The next post will get more into the meat, describing the consequences these commutation relations have.

Note: This post is one in a series describing fermi algebras, and a powerful tool known as the Jordan-Wigner transform, which allows one to move back and forth between describing a system as a collection of qubits, and as a collection of fermions. The posts assume familiarity with elementary quantum mechanics, comfort with elementary linear algebra (but not advanced techniques), and a little familiarity with the basic nomenclature of quantum information science (qubits, the Pauli matrices).

Fermions

The canonical commutation relations for Fermions

Suppose we have a set of operators [tex]a_1,\ldots,a_n[/tex] acting on some Hilbert space [tex]V[/tex]. Then we say that these operations satisfy the canonical commutation relations (CCRs) for Fermions if they satisfy the equations

[tex] \{ a_j, a_k^\dagger \} = \delta_{jk} I; \,\,\,\, \{ a_j,a_k \} = 0, [/tex]

where [tex]\{ A, B \} \equiv AB+BA[/tex] is the anticommutator. Note that when we take the conjugate of the second of these relations we obtain [tex]\{ a_j^\dagger, a_k^\dagger \} = 0[/tex], which is sometimes also referred to as one of the CCRs. It is also frequently useful to set [tex]j = k[/tex], giving [tex]a_j^2 = (a_j^\dagger)^2 = 0[/tex].

How should one understand the CCRs? One way of thinking about the CCRS is in an axiomatic mathematical sense. In this way of thinking they are purely mathematical conditions that can be imposed on a set of matrices: for a given set of matrices, we can simply check and verify whether those matrices satisfy or do not satisfy the CCRs. For example, when the state space [tex]V[/tex] is that of a single qubit, we can easily verify that the operator [tex]a = |0\rangle \langle 1|[/tex] satisfies the Fermionic CCRs. From this axiomatic point of view the question to ask is what consequences about the structure of [tex]V[/tex] and the operators [tex]a_j[/tex] can be deduced from the fact that the CCRs hold.

There’s also a more sophisticated (but still entirely mathematical) way of understanding the CCRs, as an instance of the relationship between abstract algebraic objects (such as groups, Lie algebras, or Hopf algebras), and their representations as linear maps on vector spaces. My own knowledge of representation theory is limited to a little representation theory of finite groups and of Lie algberas, and I certainly do not see the full context in the way an expert on representation theory would. However, even with that limited background, one can see that there are common themes and techniques: what may appear to be an isolated technique or trick is often really an instance of a much deeper idea or set of ideas that become obvious once once one has enough broad familiarity with representation theory. I’m not going to pursue this point of view in these notes, but thought it worth mentioning for the sake of giving context and motivation to the study of other topics.

Finally, there’s a physical way in which we can understand the CCRs. When we want to describe a system containing Fermions, one way to begin is to start by writing down a set of operators satisfying the CCRs, and then to try to guess what sort of Hamiltonian involving those operators could describe the interactions observed in the system, often motivated by classical considerations, or other rules of thumb. This is, for example, the sort of point of view pursued in the BCS theory of superconductivity, and which people are trying to pursue in understanding high temperature superconductors.

Of course, one can ask why physicists want to use operators satisfying the Fermionic CCRs to describe a system of Fermions, or why anyone, mathematician or physicist, would ever write down the CCRs in the first place. These are good questions, which I’m not going to try to answer here, although one or both questions might make a good subject for some future notes. (It is, of course, a lot easier to answer these questions once you understand the material I present here.)

Instead, I’m going to approach the Fermionic CCRs from a purely mathematical point of view, asking the question “What can we deduce from the fact that a set of operators satisfying the CCRs exists?” The surprising answer is that we can deduce quite a lot about the structure of [tex]V[/tex] and the operators [tex]a_j[/tex] simply from the fact that the [tex]a_j[/tex] satisfy the canonical commutation relations!

We’ll take this subject up in the next post, where we’ll show that the CCRs essentially uniquely determine the action of the [tex]a_j[/tex]s, up to a choice of basis.