Equation for the WKB approximation and Wigner-Kirkwood expansions for systems with magnetic field

We discuss semiclassical expansions for the off-diagonal thermal kernel of quantum systems with a nonuniform, time-independent, magnetic field. In the process, we derive a new equation for the WKB approximation for such systems. In particular, we provide recursion relations and a diagrammatic method for calculating explicitly the coefficients in these expansions. As an illustration we present the off-diagonal coefficients up to order ${\hslash }^3$, and the on-diagonal coefficients up to ${\hslash }^6$ for a uniform magnetic field. We also give the on-diagonal coefficients up to order ${\hslash }^4$ for a nonuniform field.