In quantum mechanics time evolution is defined via a unitary operator(adsbygoogle = window.adsbygoogle || []).push({});

$$U(t^\prime,t) = e^{-iH(t^\prime-t)}$$

Now let's forget about the fact that we know this exponential representation and that we know that the U's fulfill the group axioms, i.e. that we can multiply any two U's, regardless on which times they depend.

Let's focus on the basic structure defined by 1) multiplication, 2) the existence of an inverse element, 3) the existence of a neutral element and 4) associativity, but withrestrictedapplicability multiplication (1)

$$U(t^\prime,t_a) \cdot U(t_b,t)$$

and (2), (3), (4) to the case where the two times coincide

$$t_b = t_a$$

That means we do no longer insist on having a Lie group, where we can multiplyany two group elements, but that we have a different structure S where onlyspecific elementscan be multiplied. Of course this multiplication has to satisfy

$$U(t^\prime,t_a) \cdot U(t_b,t)|_{t_b = t_a} = U(t^\prime,t)$$

My question is if there is anamefor this structure S with this restricted multiplication law, if this structure has already been discussed, or if one can prove that this structure must always follow from a group structure.

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# A Algebraic structure in time evolution

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