@SillyGoose this is again the fact that not every tensor is a simple tensor; yes, every operator is a sum of operators of that form but the operators that really represent interaction of the individual subspaces will always have more than one summand
that's just "entanglement" on the level of operators rather than states
so if I want to fix my hilbert space tensor factorization (e.g. as above), then i should write every state and every operator acting over the hilbert space as a sum of corresponding objects tensored in the order determined by the factorization
(to contexualize) Let $\mathcal{H} \cong \mathcal{C}^{\otimes n}$ be fixed in this tensor product factorization. I would like to say that the first factor corresponds to my system and the rest correspond to the environment without changing the factorization of the space. I would like to write a maximally mixed state of the environment in terms of this factorization
my understanding at the moment tells me that fixing the tensor product factorization fixes how you expand your operators :0
If I have the factorization $\mathcal{H}_{sys} \otimes \mathcal{H}_{env}$ I can definitely obtain a maximally mixed state of the environment: $1/d_{env} \mathbb{I}$. The difficulty I am facing now is trying to translate the maximally mixed state written in terms of the above factorization into the same state, which I will call the maximally mixed state of the environment, written in terms of the factorization $\otimes_i \mathcal{H}_i$ if that makes sense
right right, but given that we have this maximally mixed state of the environment, I want to then write it in a new tensor factorization of the environment space
in particular $\mathcal{H}_{env} \cong \otimes_i \mathbb{C}^2_i$
tensors are certainly not solely present in physics, but i'll note my lower div physics-dept lin alg class covered them but the math-dept upper and lower div lin alg classes do not
(why there exists a version of lower-div lin alg in every department, i do not know)
cf. Paul Halmos Finite Dimensional Vector Spaces which is an undergraduate linear algebra text. It talks about vector spaces and dual spaces, then direct product of vector spaces, then tensor products of vector spaces
The idea is to start with vector spaces and then see what other vector spaces you can construct from that vector space (dual spaces, direct product spaces, and tensor spaces are all such vector spaces you can construct from starting with other vector spaces).
@AudenYoung there's certainly cringe there but there's also the occasional satisfaction of reading some answer and going "this user got it" and then realizing you're reading your own answer from 5 years ago :P
@ACuriousMind once i got a homework problem that used xi, so every time i had to write it i added another loop. by the end of the problem, xi extended past three or four lines of work
According to Roger Bacon, one of the inventor of science, we must look at the natural world instead of thinking really hard about it because of the fall of man
Have you thought about sinning less instead of studying
Also he was not related to Francis Bacon for some reason
hello - i have a question about why, if we have minkowski space time with a point in future light cone deleted, this causal future of the space lacks this point but the chronological future + boundary contains it?
IIRC the issue you bring up is that if you delete a point on the future light cone, not only does it remove that point from $J^+(p)$, but also some of the future pointing null curves that stem from it
But that region is still in the boundary of $I^+(p)$
not sure if this works with a single point for more than two dimensions since it could theoretically squiggle round that point on the light cone itself, but the 1+1 dimensional case is very simple to grasp for that
In the wikipedia article about the Gibb's Paradox, it's said that the reason why the paradox occurs is because the entropy, considered by Gibbs, wasn't an extensive quantity. Is it because we consider indistinguishability that entropy becomes an extensive quantity (sackur tetrode eq. of it)?
@John Rennie Hello John Rennie, thankyou for answering my questions previously they are always the best answers. In your diagram of how gravity works in relativity where the two travellers move North from the equator and their paths converge, is travelling north analagous to time passing and the speed of travelling north the rate of time passing?
But my favorite thing is questioning consciousness. Look look. When you look at a chair, is the concept of chair created within consciousness, or is the chair creating the concept of the chair. Who is pulling the strings? The light falls on the chair, reaches the retina, and in that process retrieves all the memories / associations related to "chair". So isn't the chair creating you at that instant?
I can't accept that. As far as I can see, "something" exists. All the things under the category of "something" are the ones that are creating the consciousness, not the other way around. The final point is: even that statement is questionable, because the word "consciousness" also creates the idea that there exists a "consciousness", it's just another thing we picked up in some textbook or somewhere else. So where do we go from here? lol
I don't remember what happened yesterday, I can only recall one small philosophical question at time... but anyway even if that's so, what's wrong with yesterday's conclusion? Let's stick to it :)
Yeah, but we r also saying that the concept of "things", " Groups" doesnt exist without consciousness @ACuriousMind . So consciousness is not merely required here. It's also giving birth to these ideas
@ACuriousMind how would u describe yesterday's conclusion in your words
my point about models/pattern-matching was that "nature" is inaccessible, what we're imposing structures upon are our sensory perceptions and in a radically skeptical viewpoint the relation between "nature" and sensory perceptions is not clear
@RyderRude Does anything "exist" without consciousness?
you're using that word as if it is obvious what you mean by it
but I really have no idea what it means for a non-material thing to exist
But then, how can we not say that consciousness gives birth to the stuff we know to exist, like collections? I mean... If existence can only b meaningfully defined using consciousness
I know what existence means for material things: A chair exists in the sense that I can sit on it, and in a world in which it wouldn't exist I couldn't sit on it - the chair's existence makes a difference to my perception
I don't know what existence means for "collections"
how do I perceive the difference between a world with collections and a world without collections?
@RyderRude instead of thinking in terms of conclusions I suggest you take a more "mathematical" approach: if certain axioms / assumptions are correct then certain things follow -- but they are not "conclusions" in the sense that you really can't ever prove the truth of any of these axioms. If you can, the field turns from metaphysics into physics, but that can't happen for most of the questions you are raising
I don't think you can meaningfully talk about whether or not more specific concepts like "perfect groups" exist, either, that doesn't mean perfect groups and consciousness are inseparable
@ACuriousMind ok but one can still point to "particular" material things.. This is the idea of "collections" and "elements" that exists in the form of material things, not just in the form of concepts
Like u can talk about a chair specifically
This notion of existence must come with consciousness
again, I know how to ascertain the existence of material things: When someone says 'There exists a chair over there' then I can look at that chair and I see something different from a world where there's no chair over there
of course there is a notion of existence for more than a single specific chair
the phrase "there exists a chair over there" is essentially just a phrase that encodes my expectations about my perceptions: "the chair exists -> I can see something that has the shape of a chair & I can walk over there and sit on it & all other aspects of sensory perception of the chair"
@RyderRude I think it may be worthwhile for you to consider and perhaps research how different cultures approached "existence". For you the materiality of something like a "chair" may be obvious, but in different cultures seeing the world as a dynamic process took precedence over perceiving it as a material collection of "stuff". @Slereah quoted Thales yesterday: "All is water" :)
@RyderRude I think it's a cultural input to a large degree yes, in particular the emphasis on the importance of dividing and subdividing the natural world into various categories
There is a level at which the organism apparently has to distinguish between certain things. As I said yesterday distinguish the lion from the deer. But anything beyond what is required for survival comes from culture and society. And even what we say about the "organism distinguishing" is how we western influenced, scientifically oriented, etc. interpret the actions of an organism
your particular attempt at claiming that "collections" are somehow intrinsic to "consciousness" reminds me of Kant's metaphysical deduction, in particular the claim that the categories of "quantity" exist a priori, i.e. in the mind prior to any experience
@RyderRude Kant is very famously a terrible read :P
Gotcha, my question related to that is is there a smooth nonsingular foliation of $\Bbb R^{2,1}$ whose leaves are all diffeomorphic to the punctured Minkowski plane
I guess I don't care much about the puncture in the sense of the physics there
@Slereah Is this in the same sense in which singularities r allowed too? But wut im worried is that a particle can reach the puncture in finite proper time
Thank you slereah.. My idea (although it's really a shot in the dark) is to take infinite copies of these punctured minkowski planes and collapse all the punctures to the origin of $\Bbb R^{2,1}$ and to get a foliation of this 3 dim. space
@Amit Suppose collections r separable from consciousness. Which way to do u think the relation goes : 1. Collections necessarily need consciousness but not the other way around 2. Consciousness necessarily needs collections but not the other way around. 3. Both can exist independently
I think 2. gets ruled out becuz collections dont exist out there in no meaningful definition of existence. 1 seems plausible becuz it may b that fetuses r primitive beings that exist without any notion of collections
@RyderRude I say the existence of both is questionable. But if one exists the other exists. It's like "I think therefore I am" -- but what about when you don't think? Can you know if there is ever a moment when you don't think? Please make a comment on that, difficult eh? :)
Trying to use a collection of concepts, ideas, words... to investigate whether collections exist independently of these things, seems to me an act of futility. Because if you conclude that they do, this is trivially apparent from the very fact that you are using these things. But on the other hand, it tells you nothing about whether they really exist independently from this investigating appartus
@Amit but i dont agree with "if collections exist, then consciousness exists". If we're going by the Platonist notion of existence, then collections can exist without consciousness
But i personally dont like the platonist notion. I go by the scientification falsification notion of existence. In this sense, collections hav to b dependent on consciousness. Maybe they r one and the same, or maybe consciousness is a deeper idea
