Quantum Logic - Non-relativistic Dynamics

Non-relativistic Dynamics

In non-relativistic physical systems, there is no ambiguity in referring to time evolution since there is a global time parameter. Moreover an isolated quantum system evolves in a deterministic way: if the system is in a state S at time t then at time s > t, the system is in a state F_s,t(S). Moreover, we assume

• The dependence is reversible: The operators F_s,t are bijective.

• The dependence is homogeneous: F_s,t = F_{s − t,0}.

• The dependence is convexity preserving: That is, each F_s,t(S) is convexity preserving.

• The dependence is weakly continuous: The mapping R→ R given by t → Tr(F_s,t(S) E) is continuous for every E in Q.

By Kadison's theorem, there is a 1-parameter family of unitary or anti-unitary operators {U_t}_t such that

In fact,

Theorem. Under the above assumptions, there is a strongly continuous 1-parameter group of unitary operators {U_t}_t such that the above equation holds.

Note that it follows easily from uniqueness from Kadison's theorem that

where σ(t,s) has modulus 1. Now the square of an anti-unitary is a unitary, so that all the U_t are unitary. The remainder of the argument shows that σ(t,s) can be chosen to be 1 (by modifying each U_t by a scalar of modulus 1.)