Dirac algebra and Foldy-Wouthuysen transform (Q2708576)

The author treats the Dirac equation \(\partial\psi/dt+ iH\psi= 0\) with \(H= h(x,D)\), \(D_j= (1/i)\partial x_j\). Here \(h(x,\xi)= \sum_{j=1}^3\alpha_j(\xi_j- A_j(x))+\beta+ V(x)\) is the selfadjoint \(4\times 4\)-symbol matrix. Let \(A= a(x,D)\in Op\psi c_m\), \(m= (m_1,m_2)\), be the pseudodifferential operator ``\(\psi\)do'' such that the symbol \(a(x,\xi)\in \psi c_m\) is \(O(\langle x\rangle^{m_2}\langle\xi\rangle^{m_1})\), \(\langle p\rangle= (1+|p|)^{1/2}\). The author defines a class \(P_m\subset Op\psi c_m\) of \(\psi\)do's \(A\) with \(4\times 4\)-symbols \(a(x,\xi)\) such that NEWLINE\[NEWLINE\partial^k_t A_t= \partial^k_t(e^{iHt}a(x, D)e^{-iHt})\in \mathbb{C}^\infty(\mathbb{R}, Op\psi c_{m-ke^2}).NEWLINE\]NEWLINE He shows the possibility of the decomposition \(a= q+ z\in\psi c_m\), with \(z{\i}\psi c_{m-e}\), \([h,q]= 0\), \(x,\xi\in\mathbb{R}^3\). F-W transform is the unitary transform \(U^*AU\) of \(\psi\)do's \(A\) by \(U= u(x,D)\in Op\psi c_0\). Under the condition \(\lim_{|x|\to\infty} (V(x),A_j(x))= 0\), the class \(P_m\) precisely consists of all operators \(A\in Op\psi c_m\) with \(U^*AU\) given by \(2\times 2\) matrices \(B,E\in Op\psi c_m\) in diagonal position and \(C,D\in O(-\infty)\). If \((V(x),A_j(x))= 0\), \(P= \bigcup_m P_m\) is invariant under a proper Lorentz transform.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].