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12:01 AM
@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
i see
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
not sure if you should but you could ;)
what is the alternative :0
just...don't? :D
no one is forcing you to expand operators in the basis of simple tensors
(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
12:10 AM
ah, sure, statements like "maximally mixed" rely on the chosen factorization
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
Or to go the other way around is fine too
not sure what exactly the question is
$\mathbb{I} = \mathbb{I}_\text{sys} \otimes \mathbb{I}_\text{env}$
wait let me write up what i mean i think it will be clearer than latex
also: $\frac{1}{d_\text{env}}\mathbb{I}$ is not a density matrix on $H_\text{sys} \otimes H_\text{env}$, the trace is not unity
and so my question is if the last line is true
if that is how I write the maximally mixed state of the environment as defined in this new tensor basis
12:24 AM
hol' up
how is a pure state $\lvert \phi\rangle\in H_\text{env}$ supposed to be maximally mixed
oh uhhh ignore that part oops XD
i should have written an arbitrary density matrix symbol there
I'm not really sure I understand the question
by construction of the partial trace, you have that $\mathrm{tr}_\text{sys}(\rho_\text{sys}\otimes \rho_\text{env}) = \rho_\text{env}$
for $\rho_\text{env}$ to be maximally mixed it does not matter what state you choose for $\rho_\text{sys}$
12:52 AM
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 $\mathcal{H}_{env} \cong \otimes_i \mathbb{C}^2_i$
1:13 AM
are tensors a physics concept or would a typical undergrad math program have that covered?
just learned transition matrices and change of basis in my linear algebra class and i vaguely remember tensors having something to do with that
1:24 AM
an undergraduate linear algebra course might cover basics of tensors
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).
Interesting, maybe it's also covered in abstract algebra? (I never finished that so idk)
My course did not but i wonder if others do
@AudenYoung It's nice that there's a choice at least :P some people don't want to go through all the rigor that an upper div course has
1:30 AM
@AudenYoung Do you use gradient descent where your cost function is defined in terms of quantum operations (tracing out, etc.) at all?
I should probably get a head start on the harder courses that I'm going to inevitably take.. since I got this free time between exam dates
but I'm going to finish up linear/abstract alg so I can understand notation better
@SillyGoose ...what?
where did this question come from lol
@SillyGoose ah see my upper div lin alg class used Linear Algebra Done Right as a text. Didn't really go beyond inner product spaces.
because you are a mod on quantum computing :)
ah sorry that question just came rather out of the blue haha
no, i do not - i have heard of its use but i am still an undergrad
who let this goose out of the barn
1:40 AM
who let the goose out who who
hello @ACuriousMind - long time no see
how've you been?
pretty well, can't complain :)
how about you?
fairly well
very busy with classes - but it's spring break now so a bit of a breather
my linear algebra class was strange :P i think it had a nice selection of topics but a lack in every other regard. at least it was good exposure
1:43 AM
it's very fun coming on p.se now and seeing how many more questions i understand/know the answers to each time!
how long of a break did you take
heh, progress!
I took a long break from 2016/17 to now so the regulars have changed
but I do remember you were on here at that time, @Audenyoung
ah yeah i've been on and off
1:45 AM
We have a goose in here as well as many new faces
I have yet to find as great of a resource as p.se/m.se online
(maybe chatgpt some day :P)
oh god chatgpt
not for a hot minute
yes it pales in comparison to our chatbot ACM
i was messing around with it and it completely made up the existence of several papers
lol, I got good general info out of it but I didn't test it much. I tried asking it physics questions and it didn't get very deep
1:47 AM
yukawa potential was a fail
okay okay is this question more clear heh
was a very surreal experience because at first i thought it might be helpful doing literature reviews but then uh...it was actively unhelpful
@ACuriousMind indeed! it's very satisfying :)
@Obliv it is an insult to me to call these pathetic language models "intelligent"
though i do occasionally shudder to look back on my old questions and answers
would you feel better if i compared you to jarvis/hal9000? @Acuriousmind
@AudenYoung as do I, but it is part of the process :P got to take it in stride sometimes
1:48 AM
I think the answer should be that the identity of $\mathcal{H}_{env}$ is just the tensor product of identities of the subsystems?
@SillyGoose yes
okay thank god it is as simple as this XD
2 hours ago, by ACuriousMind
$\mathbb{I} = \mathbb{I}_\text{sys} \otimes \mathbb{I}_\text{env}$
ah I see
also your $\rho$s look like they're dying, please water/feed them
1:51 AM
what is the worst greek letter to use as a variable
@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 i look forward to achieving that haha
@AudenYoung $\Xi$
there's that really scribbly looking one i forget the name
1:55 AM
Nov 7, 2020 at 14:40, by ACuriousMind
@NiharKarve part of me dies every time some physicist or mathematician just draws a random squiggle and calls it xi :P
what's the most memorable ACM avatar
I forgot what the first one I saw was, but the dnd ones were cool
i feel like it was like space or something..
I think the only space-related avatar I've had was HK-47 from KOTOR
but there wasn't any space in the picture
zeta and xi look quite similar to me :P
zeta is a ballon with one loop in the string and xi has two loops :)
2:31 AM
@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
3 hours later…
5:48 AM
just say let a = xi at the start ;)
then swap it back at the end - ofc that way you never get any practice
1 hour later…
7:14 AM
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
Inventing science is a very Bacon-related field
7:36 AM
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?
Depends what you mean by this
overall i am trying to understand why $I^+(P) \cup \partial I^+(P) \neq J^+(P)$
If you remove the point from the spacetime, it will not be in any subset of it, since it does not exist
shouldnt this mean that its not in the causal future or the chronological future 0.o
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 $I^+(p)$
7:42 AM
what do you refer to when you say "that region"
The deleted future pointing null curves
I think that's the issue, anyway, it's been a while
also it may be a better example to remove a whole region of the future light cone than a single point
Not sure that's necessarily true for a single point
Check out fig. 34 p. 192
It does work properly in 1+1 dimensions anyway :p
is it crucial that the segment we remove be on the boundary or it can be anywhere
i think it must be on the boundary
if so, i think i see
If it's not on the boundary you can have squiggly lines all around to fill the space around that point
But if it's on the boundary, no null curve can do that
since a null curve cannot bend inward in the light cone to avoid it
otherwise it would become timelike at that point
ok i think i see
thank you
7:59 AM
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
8:51 AM
Though otoh two null-related points can always be connected by a null geodesic, IIRC, so that may be a moot point
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?
@user36093 Yes
@JohnRennie Why are you D. B. Cooper now
Google, Google, D. B. Cooper the hijacker?
He is also known in the movie D. B. Cooper versus Big Foot, but I am likely talking about this yes
Apart from being a black and white sketch I struggle to see any similarity ...
9:47 AM
He is just the most famous police sketch of a person I guess
Outside of the Zodiac Killer
10:04 AM
@John Rennie So does it apply in General relativity also that if the rate of time passing was faster the weight of an object would increase?
2 hours later…
12:10 PM
@Slereah Studying keeps me off the streets, lol :)
12:40 PM
I'm wondering how consciousness gives birth to the notions of collection, elements and logic @Slereah @Amit @ACuriousMind
Is there any philosophical progress regarding this
There probably is if you consider the root cause to be neurology
But the concept of cause and effect already assumes sets
Cuz its like an ordered pair (cause, effect)
I'll just wait for you to list all things which you consider don't need an explanation first
I'll take consciousness for granted in this discussion
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?
We have established yesterday that chairs dont exist in nature. Becuz its hard to define that existence
So in this discussion, im assuming that consciousness gives birth to collections, elements and logic @Amit
But HOW!
I don't understand why you're talking about "collection, elements and logic"
"how" does consciousness "give birth" to anything?
what does that even mean
If you consider any notion involving more than one thing to involve collections, I think you're gonna have a hard time proving anything
Becuz from our conscious experience we know that we can speak of particular things @ACuriousMind
And yesterday, we've established that these ideas dont exist in nature
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
12:49 PM
@Amit then we r again flipping on yesterday's conclusion :). We're talking about "all the things" out there creating consciousness
@RyderRude I don't see what my "conscious experience" has to do with that
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 :)
My consciousness was shot off in 'Nam
It's becuz we yesterday said that the concept of collections isnt in nature. Nature doesnt hav structure!
like, sure, on a technical level all notions require consciousness because without consciousness there's no one to discuss anything
@RyderRude I neither said that nor do I agree with it
12:52 PM
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
I was not aware we had reached a conclusion
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
@ACuriousMind no. Existence isnt defined without consciousness, in the scientific falsification sense. And theres no sensible other definition either
This is also the conclusion we reached yesterday
again, I wasn't aware we had reached any conclusion and I don't really agree with this phrasing
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?
12:57 PM
U cant! Then do we say that consciousness is collections? There is no mechanism of "birth" becuz the ideas r the same @ACuriousMind
@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 mean u cant perceive a world without collections. I agree @ACuriousMind
all I'm saying is that I think "existence" is an ill-defined notion for non-material things like concepts
So collections and consciousness r inseparable
12:58 PM
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
not sure what that means
A specific chair exists. U separate it from the rest of nature in ur conscious experience
Ok lemme ask one thing. Do u agree that multiple things exist? @ACuriousMind
I really don't know where this is going
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
Yeah. And is this notion of existence inseparable from consciousness
1:03 PM
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" :)
@ACuriousMind all this is still multiple things. In descartes's words : the devil cudnt fool u about the existence of multiple things
Becuz as soon as u can think of two things, u already have two things
In ur existence
@Amit are you of the opinion that the concept of multiple things may not b insaparable from consciousness
I mean, maybe some cultures cant think of multiple things
This is also a good idea
@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
I mean, certaintly, at some point of development of a baby, the baby exists but isnt complex enough to comprehend multiple things
So collections may not b insaparable from consciousness. I agree @Amit
I feel you're just trying to very clumsily re-invent the Kantian idea of the categories of reason
1:10 PM
This is supported by the idea that babies develop slowly
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
@ACuriousMind pls make it non-clumsy :P. Im struggling hard here
I mean, Kant wrote a whole book about this :P
@Amit but i think even the most basic concept of distinction doesnt come for free with consciousness. Cuz babies develop slowly
@RyderRude The concept of distinction is not distinction :)
1:12 PM
@ACuriousMind i will check it out then. I got lots of philosophy to catch up to
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
Even the wiki is seeming undigestable to me :P
It's too sophisticated philosophy than wut i do
but, honestly, this is the level of specificity I would expect from such discussions
Yeah. Phisophers r trained for specificity
not that I necessarily agree with Kant about everything, but he's very careful developing all of his technical terms
1:17 PM
I think I'll lose patience with that stuff
I'll mostly stick to my pedestrian philosophy :P
But ill try reading some introductory books
Becuz i cant bear when math books try to b too rigorous
can you have punctured minkowski space?
like the punctured plane..
Then it wont b a manifold. It has to be locally $R^n$. It wud hav fewer dimensions near the puncture, i think
But i dont know GR much
But singularities do mean that GR doesnt describe a manifold for some portions of spacetime
So maybe punctures r allowed too
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
Puncture is not some technical term, right? Im just thinking of a hole on a paper
You can have a punctured Minkowski space
1:29 PM
yes you're correct - puncture is just deletion of a point on the (manifold)
Every open set of a manifold is a manifold
And every open set of a spacetime is a spacetime, similarly
@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
Then it disappears
@geocalc33 You meant $\mathbb{R^{3,1}}$ right?
1:33 PM
It's probably not a good idea but as far as GR goes it's not forbidden
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
Cant roam outside the map
Game restarts
When writing down the equations for gyroscopic precession, we generally don't consider the inertia of the rod that holds the disc right?
no idea really if it would be cauchy
or you mean you want to foliate into submanifolds that look like $\mathbb{R}^{2,1}$?
1:39 PM
basically foliate $\Bbb R^{2,1}$ into submanifolds that are all diffeomorphic to $\Bbb R^{1,1}/\{0\}$
I don't need a construction, just to know if it's possible
I think that's possible as pointed out before, since taking out a point leaves all open sets open...
It is allowed
Although it is an extendible manifold
which are generally considered unphysical
it's unphysical because of that puncture I'm assuming..hmm
It can be isometrically embedded into a complete manifold uniquely
Therefore there isn't much reason to not do that
@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
1:53 PM
@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
@RyderRude Yes but this "going by" is just adopting a certain belief / axiom that can't be really proved or investigated by any means
Yeah I think Plantonist notion takes existence itself for granted. So they dont define existence
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