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The Dyson formula is an expression for the solution of the Schrödinger equation in time dependent quantum mechanics.
It expresses the parallel transport of the Hamiltonian operator regarded as a Hermitian-operator valued 1-form on the time axis.
In time-dependent quantum mechanics dynamics is encoded in a Lie-algebra valued 1-form
on the real line (time) with values in the Lie algebra of the unitary group on a Hilbert space $V$.
For $t = Id : \mathbb{R} \to \mathbb{R}$ the canonical coordinate function and $d t \in \Omega^1(\mathbb{R})$ accordingly the corresponding canonical basis 1-form, the Lie-algebra valued coefficient
of $A$ is called the Hamiltonian operator. If $H$ is a constant function, one speaks of time-independent quantum mechanics.
A state of the system is a function
A physical state is a solution to the Schrödinger equation
or equivalently
This differential equation is that which defines the parallel transport of $A$. Its unique solution for given $\psi(0)$ is written
This is called the Dyson formula . In the special case of time-independent quantum mechanics this becomes an ordinary exponential of an ordinary integral.