Algebraic structure in time evolution

[tom.stoer]

In quantum mechanics time evolution is defined via a unitary operator

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

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 with restricted applicability 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 multiply any two group elements, but that we have a different structure S where only specific elements can 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 a name for 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.

 

[jambaugh]

You are actually describing a Category. The operators become the "morphisms" and the "objects" are the times or rather the Hilbert space at each time. However this structure is a bit more specific than the general idea of a Category and I'm not sure there will be many useful results in general Category Theory which you could apply to this structure.

You can also work with a fiber-bundle structure where the base is range of the t variable ([itex]\mathbb{R}[/itex]) and the fibers are distinct Hilbert spaces at each time point. Then the unitary operators become linear isomorphisms between these distinct Hilbert spaces.

 

[tom.stoer]

Thanks.

I am not sure whether I like the fiber-bundle structure b/c it still reminds me to quantum mechanics. I would like to focus on the U's and forget the objects they are acting on (the Hilbert spaces). It should be possible to identify a generic structure for time evolution applying to different frameworks like classical and relativistic mechanics, quantum mechanics, field theory etc.

I know nearly about category theory. Any idea how to start?

 

[jambaugh]

You can define them as actions on the fiber bundle without focusing on the fibers themselves. There is a reason they will still remind you of quantum mechanics.

But given each fiber in the bundle is isomorphic, you can (as is usually done) revert to the algebra of operators with an external t translation operator and work totally within a single operator algebra, in particular within a single Lie group. [itex]U(n)\otimes (\mathbb{R},+)[/itex]

The $64,000 question is of course "Why?" What is it specifically you are trying to accomplish here?

 

[tom.stoer]

I am trying to understand this algebraic structure, especially the multiplication, which can be achieved by using a Lie group, but which looks different at first sight.

Maybe I can rephrase my question as follows: given that we have the well-known axioms regarding the neutral and the inverse element, the slightly changed multiplication (it need not be defined for any two elements but only for specific ones) and the requirement that

$$U(t^\prime,t) = U(t^\prime,.) \cdot U(.,t)$$

is unique, i.e. the l.h.s. does not depend on the midpoint "." on the r.h.s.:
a) is it possible to prove that this is always equivalent to a group structure?
b) if not, what is the most general structure fulfilling these axioms and requirements?

 

[tom.stoer]

Considering the time evolutions U on a Hilbert space they belong to the group of unitary operators on that space, but they are not a subgroup; they define - as far as I can see - a Lie-grupoid. The reason is that the representation as time-ordered exponential

$$U(t^\prime,t) = \text{T}\, \text{exp}\left[-i\int_t^{t^\prime} ds \, H(s)\right]$$

prevents multiplication for U(t',a) and U(b,t) for b not equal a. Of course the product exists as a unitary operator, but for time-dependent H(t) the product fails to be a time-evolution represented by a time-ordered exponential.

As far as I can see this Lie-grupoid is connected b/c for any U and any V I can always find S and T such that

$$V = SUT$$

holds. Instead of dropping totality I could try to discard associativity. This seems not to be a good idea in quantum mechanics b/c as far as I can see associativity is always guaranteed by the spectral theorem for unitary operators.So I end up with a connected Lie-grupoid. Is this correct?