Tensor product and ultraproduct construction

[nomadreid]

I do not know if this is the proper rubric to ask this question, but I picked the one that seemed the most relevant.

I have noticed some superficial resemblance between the tensor product and the ultraproduct definitions. Does this resemblance go any further?

While I am on the subject of tensor products: when there is a physical interaction, is there necessarily entanglement involved, even for a short time?

 

[fresh_42]

I haven't heard of ultraproducts before and I am not sure whether you mean the same thing, that Wiki means by it.
If so I don't get the "superficial resemblance". Can you elaborate this in more detail?
And - still assuming coincidence - "Logic" might be the appropriate rubric.

 

[nomadreid]

Thanks, fresh_42. First, the very superficial resemblance goes as follows: tensor products are used for combining Hilbert spaces into a new Hilbert space. A Hilbert space can be viewed as (a theory of) a model. The Ultraproduct construction (yes, as in Wiki) is used to combine models into a new model. More along algebraic lines, the analogy is that both are completions, using a collection of sequences modulo an equivalence relation.

The reason I am keeping this question here is that I realize that my second question has a better chance of being answered than the first one, and is in fact of greater interest. I should have added that I meant physical interaction between two particles (yes, I know that "particle" is just an abbreviation for "a local excitation of a field"); in other words, if two particles interact, is the result at anyone point necessarily a state that cannot be reduced to the tensor product of two separate states? The reason I ask this is that I seem to recall (sorry, I do not have the reference -- I know that this is bad) a discussion about decoherence saying that the information from a state becomes entangled with its environment. But I'm not sure that I see an algebraic reason for the necessity of this entanglement. This question would not go over well in the logic rubric.