The nonrelativistic limit problem for a relativistic spinless particle in an electromagnetic field
The nonrelativistic limit, i.e. the limit as the light velocity c tends to \(\infty\), is discussed of the solution \(\psi^ c(t,x)\) for the imaginary-time Schrödinger equation \[ \partial_ t\psi^ c(t,x)=- [H^ c-mc^ 2]\psi^ c(t,x),\quad t>0,\quad x\in {\mathbb{R}}^ d, \] on the basis of its path integral or probabilistic representation obtained by \textit{T. Ichinose} and \textit{H. Tamura} [Proc. Japan Acad. Ser. A 62, 91-93 (1986; Zbl 0597.58038), Commun. Math. Phys. 105, 239-257 (1986; Zbl 0606.60060)]. Here \(H^ c\) is the Weyl quantized Hamiltonian corresponding to the relativistic classical Hamiltonian \([c^ 2(p- eA(x))^ 2+m^ 2c^ 4]^{1/2}+e\Phi(x)\) of a spinless particle with mass \(m>0\) and charge e in an electromagnetic field. The vector and scalar potentials \(A(x)\) and \(\Phi(x)\) are assumed to be respectively \({\mathbb{R}}^ d\)- and \({\mathbb{R}}\)-valued \(C^{\infty}\) functions which together with all their derivatives are bounded in \({\mathbb{R}}^ d\). It is shown that \(\psi^ c(t,\cdot)\) converges, in both \(L^ 2({\mathbb{R}}^ d)\) and \(C_{\infty}({\mathbb{R}}^ d)\) uniformly on every finite interval in \(t\geq 0\), to the solution \(\psi\) (t,\(\cdot)\) for the heat equation \[ \partial_ t\psi (t,x)=-[(1/2m)(-i\partial-eA(x))^ 2+e\Phi(x)]\psi (t,x),\quad t>0,\quad x\in {\mathbb{R}}^ d, \] i.e. to its Feynman-Kac-Itô representation.
path integral or probabilistic representation
scientific article; zbMATH DE number 4002843
scientific article