An identity for T-ordered exponentials with applications to quantum mechanics

The solution of the Cauchy problem for Liouville's equation φ_xy = e^φ is shown to be equivalent to the calculation of a T-ordered exponential of a two-dimensional variable matrix. This gives rise to an identity for the T-ordered exponential involving two arbitrary functions. The formalism is applied to the time evolution of a magnetic dipole in a time dependent magnetic field and to the one-dimensional Dirac equation with external potential $\text{V(x) = (n +}\text{1}\text{2}\text{)λ}$ tanh λx which is relevant for field theoretic applications. A generalization to n-dimensional matrices is also given.