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3 Time Evolution of Observables

[209.2.1] Time is commonly considered as the set of Aristotelian time instants t˙. [209.2.2] The set of all time instants is represented mathematically by the set of real numbers R. [209.2.3] Time is “measured” by observing clocks. [209.2.4] Clocks are physical systems. [209.2.5] Let a be an observable quantity (e.g. the position of the sun, the moon or some hand on a watch), and let A be the set of observables of such a physical system. [209.2.6] A dynamical system is a triple A,R,T where A is the set of observables of a physical system, R represents time, and the mapping

 T:A×R → A a,t˙ ↦ T⁢a,t˙ (1)

is its dynamical rule [25]. [209.2.7] It describes the change of observable quantities with time. [209.2.8] For the dynamical rule T the following properties are postulated:

1. [209.2.9] For all time instants s˙,t˙R the dynamical rule obeys

 T⁢T⁢a,s˙,t˙=T⁢a,s˙+t˙ (2)

for all aA.

2. [209.2.10] There exists a time instant t˙*R, called beginning, such that

 T⁢a,t˙*=a (3)

holds for all aA.

3. [209.2.11] The map T is continuous in time in a suitable topology.

[209.2.12] The set of observables reflects the kinematical structure of the physical system. [209.2.13] The dynamical rule prescribes the time evolution of the system. [page 210, §0]    [210.0.1] Setting s˙=t˙* in eq. (2) and using eq. (3) shows, that either t˙*=0 must hold, or else the observable must be time independent. [210.0.2] The time evolution of observables is the one-parameter family ATt˙t˙R of maps ATt˙:AA defined by

 A⁢Tt˙⁢a:=T⁢a,t˙ (4)

for t˙R. [210.0.3] The time evolution obeys the group law

 A⁢Tt˙⁢A⁢Ts˙=A⁢Ts˙+t˙ (5)

for all t˙,s˙R, and the identity law

 A⁢T⁢0=1 (6)

where 1 is the identity on A. [210.0.4] The continuity law requires a topology. [210.0.5] It is usually assumed, that A is a Banach space with norm , and that

 limt˙→0+⁡A⁢Tt˙⁢a-a=0 (7)

holds for all aA. [210.0.6] Equations (5),(6) and (7) define a strongly continuous one parameter group of operators ATt˙t˙R on A, called a flow [26, 27]. [210.0.7] For bounded linear operators strong and weak continuity are equivalent [28].

[210.1.1] Identifying a=a0 and writing Ta,t˙=at˙ the time evolution becomes time translation to the left, i.e.

 A⁢Tt˙⁢a⁢s˙=a⁢s˙+t˙ (8)

for all t˙,s˙R. [210.1.2] If the arrow of time is taken into account, then the flow of time is directed, and only the time instants t˙0 after the beginning can occur. [210.1.3] In that case, inverse elements do not exist, and the family ATt˙t˙0 of operators forms only a semigroup [29, 28] instead of a group.