but the output of the tensor product lies in a different vector space. i think one would end up with something similar to a Fock space if one wants the space to be closed under tensor product. But not exactly a Fock space
it seems like this would define a chain complex...but not sure if it'd be interesting 0.o
@RyderRude hm i see yeah i was thinking schematically of something like this i suppose
oh the $C^n$ should be an $X$ in the bottom right
i guess it is probably not very interesting. since the cohomology classes will depend on the distinguished $\lambda$
well it could also be interesting for that reason :P
would we ever have in physics a chain complex structure induced by a sort of exotic ladder operator situation?
i.e. our creation operator acts on a given state exactly once before mapping to $0$
schematically something like this situation
also there is another typo in the first image...i meant to put $C^n \otimes ... \otimes C^n$ where there are $k$ tensor products for the $k$th chain in the chain complex
there's no problem :P just exploring. i was first trying to see if the set of pure states can be said to live naturally in an algebra (rather than textbook complex vector space). then, i was wondering about if there is a natural chain complex structure that you can work with if u consider the algebra of matrices containing density matrices--or something along these lines
it seemed interesting to consider if the tensor product could be used to make some sort of multiplication operation (in the sense of a multiplication of an algebra)
@SillyGoose but see: it seems to me you invented the nonexistent $\lambda\otimes\lambda = 0$ because you wanted to find a chain complex somewhere :P
it would make sense to me if you had found such a $\lambda$ first and then wondered about it, but this way it just looks like you recently learned about homological algebra and how you're trying to see it everywhere :P
@ACuriousMind right. well i wanted to see if there was a "natural" coboundary map. and the available structure is the algebraic structure. so i tried to see if the tensor product could induce such a natural coboundary map
@ACuriousMind this is true, but i was more generally thinking about what can be drawn from the basic structures provided in usual treatments of quantum mechanics. and stuff that is not just written ad nauseam in canonical textbooks
and i don't know of many things so there comes (co)homology :P
but also it is sort of natural to think about i feel like. we have this nice potentially graded vector space-like structure we look at and deal with all the time :P
it is also conceivable that if it did have a "use" then perhaps it would be an obscure use. or something more mathematically useful. so then conceivable that physicists writing textbooks on quantum mechanics would not necessary care about it or think about it
@RyderRude i'm not too sure i haven't done much of it. just looked at particular examples in a course i recently took
there might be more technical requirements but the gist is to have a sequence of modules connected by morphisms that square to $0$ indeed--to my understanding
@SillyGoose but we don't start with "I want to have a cohomology" and then go looking for stuff that might fit into a chain complex, that's not how applying (co)homology theory works. On the contrary, you start with some problem, then you notice that the quantities that interest you can be modeled in terms of quotients of images/kernels of a nilpotent map and then you pull out the (co)homology toolbox
Really, I think fixating on a particular mathematical structure and trying to make random physics fit into it is the completely wrong way to go about either mathematics or physics
@ACuriousMind is there a name for cohomology generalized to nilpotent (as opposed to square to zero, or 2-nilpotent) maps? is it just cohomology still :P
The wave equation for the electric or magnetic field contains a derivative with respect to time. In the Helmholtz equation the temporal component is omitted. In which cases is one used and in which cases the other?? For example, in my lecture we are talking about optical resonators, and the professor said that we will try to solve the Helmholtz equation for the field inside the resonator.
@Obliv I think my question was posed in bad faith since I wasn't looking for any answer in a specific framework but kind of fishing for something too big to handle (for my level anyway)
Like I understand reversibility in the thermodynamics framework "X process is reversible, Y process is not" type thing
but was curious about the deeper reasoning but I imagine that requires some complicated stuff
i think it's because this law is more like a mathematical truth, rather than a physical one. when the system is too large, the chances of reversibility r near zero
but there is also this thing called Poincaire recurrence theorem which says things r guaranteed to reverse back
I recall my mechanics professor last semester saying to me, in reference to that theorem, "It could be that we will have this conversation again at again ad infinitum at other points in time" to which I responded with laughter
I think we sometimes use humor to deflect/deal with troubling realizations :P but yeah I don't really have any reason to believe we live in a dynamical system that obeys such and such axioms or whatever.
It's like, being skeptical must be an automatic process. We do have to believe while doing physics but we also have to not believe so we can explore further. Not sure if that makes sense
look, you are making remarks like these because you didn't understand stat therm from the book you are using. If you truly understood stat therm, you will realise that there is no way you can get a deeper understanding from studying fundamental theories.
It is more of a mathematical derivation that is self-contained in stat therm. There would need to be a lot more in-depth understanding of quantum theory if we want to improve upon the current understanding. We do need it, because quantum stat therm is still a rudimentarily developed field.
@naturallyInconsistent The second law isn't a self containted truth of the universe though. It's not like a solid will sublime into a gas spontaneously because that would increase its entropy? I think there are some underying reasons for why systems evolve in the way they do, no?
I guess that example is poor because if you analyze the system properly, the system still evolves to largest number of microstate probably
Plus I don't know what the "true" system is. All the examples we did so far are only approximations/ideals anyway
Einstein solid, bose-einstein condensate, 2 state paramagnet, ideal gas, van der waal gas, fermion gases, etc are all approximate models to describe certain systems
miao miao doesn't seem to sleep. How do I see u on at 5am your time lol
Why can't any arbitrary tensor be written as tensor product of vectors? This seems to be a very well known fact which I have missed all these days...Can somebody give an easy counterexample or a proof of some kind?
@SillyGoose Exactly, what I was looking for: Thanks!
It is quite surprising that proving that such a decomposition into tensor product of vectors can be done for symmetric traceless tensors is a difficult task involving two component spinors, etc.
thought of as a linear functional over $\mathcal{H}_1 \otimes \mathcal{H}_2$
@Sanjana i wonder if Schmidt or singular value decomposition could be used to make a proof for this result?
would want to prove that all but one term in the sum of the Schmidt decomposition of an arbitrary covector in a tensor product space vanishes if...corresponding matrix to this functional is symmetric and traceless...or something
