Fractional Derivatives
This is meant to be a very quick exposition on fractional derivatives, with particular emphasis paid to functions in \(H^{1/2}\). When I introduce my area of research to colleagues, I inevitably discuss four-spinors \(\psi\in H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\). This usually leads to an impromptu discussion of \(H^{1/2}\) and why we care about it. I’ll attempt to give a brief overview of these ideas here.
The Sobolev Space \(H^s\)
Recall that Sobolev spaces comprise of functions who’s derivative(s) exist in some weak sense. One of the purposes of considering such functions is to solve a PDE over a larger class of functions than the PDE’s domain. One can then try to recover appropriate regularity of the “weak” solution of the PDE. In this sense, one defines functions with \(k\) weak derivatives for some positive integer \(k\). It turns out you can generalize this notation to define fractional derivatives of functions. We reference (Evans 2010) and (Lieb and Loss 2001) for this discussion.
The space \(H^s(\mathbb{R}^n)\) for any positive real number \(s\) consists of functions \(f\in L^2(\mathbb{R}^n)\) with the property that \((1+|x|^s)\widehat{f}\in L^2(\mathbb{R}^n)\). The associated norm is,
\[||f||_{H^s(\mathbb{R}^n)}:=||(1+|x|^s)\widehat{f}||_2\]
In the particular case of \(H^{1/2}\) this is sometimes equivalently characterized as,
\[||f||_{H^{1/2}(\mathbb{R}^n)}^2:=||(1+|x|^2)^{1/2}|\widehat{f}(x)|^2||_2^2\]
This Fourier characterization of Sobolev spaces for integer \(s\) is equivalent to the more traditional definition of weakly differentiable functions. It can be very useful for both analytical as well as technical reasons. First, let me give a simple example of the former. Using the fourier characterization, one easily shows that if \(f\in H^s(\mathbb{R}^n)\) where \(s>\frac{n}{2}\), then in fact \(f\in L^\infty(\mathbb{R}^n)\). This follows from the following computation,
\[|f(x)|=\left|\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^n}e^{-ip\cdot x}\widehat{f}(p)dx\right|\leq\frac{1}{(2\pi)^{n/2}}\int\frac{1+|p|^s}{1+|p|^s}|\widehat{f}(p)|dx \]
\[\leq\frac{1}{(2\pi)^{n/2}}||f||_{H^s(\mathbb{R}^n)}^2\left|\left|\frac{1}{1+|p|^s}\right|\right|_2^2\]
where the latter inequality follows by Holder’s inequality. Since \(s>\frac{n}{2}\), the right hand side is finite, as needed.
Fractional Differential Operators
The fractional Sobolev space \(H^{1/2}\) arises naturally in relativistic quantum mechanics (Thaller 2011). The Klein-Gordon operator arises from quantizing the classical relativistic energy-momentum equation. The operator is,
\[\sqrt{m^2c^4-c^2\Delta}\tag{1}\]
How do we apply such an operator? Such fractional operators are defined using the Fourier transform.
\[\sqrt{m^2c^4-c^2\Delta}f(x):=\left(\sqrt{m^2c^4+c^2p^2}\widehat{f}(p)\right)^{\vee}\tag{2}\]
When solving variational problems in quantum mathematics, we study the energy of quantum systems and try to find the ground state of a quantum system. Alternatively, we seek critical points of quantum energy functionals. The operator (1) isn’t really what we analyze when studying a variational problem. In fact, we study the energy,
\[(f(x),\sqrt{m^2c^4-c^2\Delta}f(x))\tag{3}\]
which makes sense when \(f\) is only in \(H^{1/2}\), whereas operating on \(f\) as in (2) requires \(f\in H^1\) (since (2) needs to be in \(L^2\)). This amounts to a weakening of the weak differentiability requirements of a function we input into the energy (3). We say \(f\) is in the “form domain” of the Klein-Gordon operator.
Application to Dirac
The purpose of this post is not to give a complete exposition of the free Dirac operator; this is just a quick application of fractional operators to the operator I study every day. Let \(D\) denote the free Dirac operator with normalized units,
\[D_c:=-i\pmb{\alpha}\cdot\nabla+\beta\]
where the three-vector \(\pmb{\alpha}\) has components \(\begin{pmatrix}0&\pmb{\sigma}_j\\\pmb{\sigma}_j&0\end{pmatrix}\), \(\pmb{\sigma}_j\) being the usual Pauli matrices. \(\beta:=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\). Thus, \(D\) operators on 4-spinors.
The Dirac operator, under the Foldy-Wouthuysen Transformation, takes the form,
\[\begin{pmatrix}\sqrt{1-\Delta}&&0\\0&&-\sqrt{1-\Delta}\end{pmatrix}\tag{4}\]
This makes it easy to see that the form domain of the Dirac operator is \(H^{1/2}(\mathbb{R}^3;\mathbb{C}^4)\). The Klein-Gordon equations arise in (4) from the fact that \(|D|=\sqrt{1-\Delta}\). In fact, if we consider the energy of a particle under the influence of Dirac, we may write,
\[(\psi,H\psi)_2=(\psi^+,H\psi^+)_2+(\psi^-,H\psi^-)_2\]
Here, \(\psi=\psi^++\psi^-\) where \(\psi^+\) lies in the positive spectral subspace induced by \(D\) and \(\psi^-\) lies in the negative spectral subspace. Since \((\psi^+,H\psi^+)=(\psi^+,|H|\psi^+)\) and \((\psi^-,H\psi^-)=-(\psi^-,|H|\psi^-)\), we have,
\[(\psi,H\psi)_2=||\psi^+||_{H^{1/2}}^2-||\psi^-||_{H^{1/2}}^2\]
so that the energy may be written entirely in terms of the \(H^{1/2}\) norm of the positive and negative spectral components of \(\psi\).
In the analysis of energy functionals based on Dirac (say, the Dirac-Fock energy), it turns out that the best way to prove the existence of critical points is to search for weak critical points; that is, the critical points are in the form domain \(H^{1/2}\) (Maria J. Esteban and Séré 1999). Afterwards, one may recover some regularity of the solutions and, in fact, the self-adjointness of the Dirac operator perturbed by the Coulomb may be used to prove that the solutions are in fact in \(H^1\) (M. J. Esteban and Sere 2001).