Time-Evolution in Quantum Mechanics is a “Flow” in the (Abstract) Space of Automorphisms of the Algebra of Observables

Spiral of life

In quantum mechanics, time is not a geometrical flow. Time-evolution is characterized as a transformation that preserves the algebraic relations between physical observables. If at a time t = 0 an observable – say the angular momentum L(0) – is defined as a certain combination (product and sum) of some other observables – for instance positions X(0), Y (0) and momenta PX (0), PY (0), that is to say

L(0) = X (0)PY (0) − Y (0)PX (0) —– (1)

then one asks that the same relation be satisfied at any other instant t (preceding or following t = 0),

L(t) = X (t)PY (t) − Y (t)PX (t) —– (2)

The quantum time-evolution is thus a map from an observable at time 0 to an observable at time t that preserves the algebraic form of the relation between observables. Technically speaking, one talks of an automorphism of the algebra of observables.

At first sight, this time-evolution has nothing to do with a flow. However there is still “something flowing”, although in an abstract mathematical space. Indeed, to any value of t (here time is an absolute parameter, as in Newton mechanics) is associated an automorphism αt that allows to deduce the observables at time t from the knowledge of the observables at time 0. Mathematically, one writes

L(t) = αt(L(0)), X(t) = αt(X(0)) —– (3)

and so on for the other observables. The term “group” is important for it precisely explains why it still makes sense to talk about a flow. Group refers to the property of additivity of the evolution: going from t to t′ is equivalent to going from t to t1, then from t1 to t′. Considering small variations of time (t′−t)/n where n is an integer, in the limit of large n one finds that going from t to t′ consists in flowing through n small variations, exactly as the geometric flow consists in going from a point x to a point y through a great number of infinitesimal variations (x−y)/n. That is why the time-evolution in quantum mechanics can be seen as a “flow” in the (abstract) space of automorphisms of the algebra of observables. To summarize, in quantum mechanics time is still “something that flows”, although in a less intuitive manner than in relativity. The idea of “flow of time” makes sense, as a flow in an abstract space rather than a geometrical flow.

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