The semiclassical limit of the Dirac equation
数学专题报告
报告题目(Title)：The semiclassical limit of the Dirac equation
报告人(Speaker)：Jakob Möller 博士（维也纳大学）
地点(Place)：后主楼1220
时间(Time)：2026年10月29日（周四）15:30-16:30
邀请人(Inviter)：袁迪凡
报告摘要
The Dirac equation was discovered by Paul Dirac in 1928 in order to find a first-order in time Lorentz-invariant Schrödinger equation for particles with half-integer spin. It replaces the scalar Schrödinger equation and the 2-spinor-valued Pauli equation (both Galilei invariant) by a 4-spinor-valued equation which includes the two spin states of a charged fermion and predicts its antiparticle. The vector-valued nature implies that the Dirac operator is matrix-valued. In order to treat the semiclassical limit, we rewrite the Dirac equation in the Wigner formalism, where the density matrix is transformed into its matrix-valued Wigner function which converges to the positive definite matrix-valued Wigner measure. The latter obeys a matrix-valued Liouville equation where the oscillating part of the Wigner equation converges to a Lagrange multiplier that maintains the constraint that the Wigner matrix commutes with the Weyl symbol of the Dirac operator. The relativistic Vlasov equations for the electron and positron can then be recovered by projecting the Wigner matrix on the eigenspaces of the Dirac symbol and taking the trace over the spin components.
