Time Evolution - Time Evolution Operators

Time Evolution Operators

Consider a system with state space X for which evolution is deterministic and reversible. For concreteness let us also suppose time is a parameter that ranges over the set of real numbers R. Then time evolution is given by a family of bijective state transformations

F_{t, s}(x) is the state of the system at time t, whose state at time s is x. The following identity holds

To see why this is true, suppose x ∈ X is the state at time s. Then by the definition of F, F_{t, s}(x) is the state of the system at time t and consequently applying the definition once more, F_{u, t}(F_{t, s}(x)) is the state at time u. But this is also F_{u, s}(x).

In some contexts in mathematical physics, the mappings F_{t, s} are called propagation operators or simply propagators. In classical mechanics, the propagators are functions that operate on the phase space of a physical system. In quantum mechanics, the propagators are usually unitary operators on a Hilbert space. The propagators can be expressed as time-ordered exponentials of the integrated Hamiltonian. The asymptotic properties of time evolution are given by the scattering matrix.

A state space with a distinguished propagator is also called a dynamical system.

To say time evolution is homogeneous means that

In the case of a homogeneous system, the mappings G_t = F_t,0 form a one-parameter group of transformations of X, that is

Non-reversibility. For non-reversible systems, the propagation operators F_{t, s} are defined whenever t ≥ s and satisfy the propagation identity

In the homogeneous case the propagators are exponentials of the Hamiltonian.