Time evolution

Adapted from Wikipedia page.

Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called stateful systems) and in particular to dynamical systems. Time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies is governed by the principles of Classical Mechanics.

The concept of time evolution may be applicable to other stateful systems as well. For instance, the operation of a Turing machine can be regarded as the time evolution of the machine's control state together with the state of the tape (or possibly multiple tapes) including the position of the machine's read-write head (or heads). In this case, time is considered to be discrete steps.

Stateful systems often have dual descriptions in terms of states and observables. In such systems, time evolution can also refer to the change in observable values. This is particularly relevant in quantum mechanics where the Schrödinger picture and Heisenberg picture are equivalent descriptions of time evolution.

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 \to X)_{s, t \in R} .

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

F_{u, t} (F_{t, s} (x)) = F_{u, s} (x) .

In some contexts in mathematical physics, the mappings $F_{t, s}$ are related to propagators.
A state space with a distinguished propagator is also called a dynamical system.