As meta, I suggest most everyone is vastly underestimating the magnitude of potentially transformative improvement being left on the table by current science education. But that's a discussion for another time.
Kindergarten Quantum Mechanics
11–20 of 34 posts
Re: Kindergarten Quantum Mechanics
#12Earlier quoted context omitted.
Normally I'd go with the second, but I'm trying to learn physics as a hobby and, you know, technologically save the world with my messiah complex apparently, and I want to keep tabs on things that may be insightful to know about
Just in case you are serious about having a messiah complex, I am prone to having a messiah complex, thanks to schizoaffective disorder. It took me a while to figure out which medications actually help me, without too many unpleasant side effects, but I am very grateful to have discovered the cocktail that works for me now.
i'm glad to hear you've found something that helps you. I was joking about the messiah complex, though it is something my therapist has made the occasional joke about too when i get over-ambitious about things i wanna do (he did specify i don't seem to actually have it, though - just good old-fashioned depression).
Re: Kindergarten Quantum Mechanics
#13"Of course we do not expect you to know about category theory, nor do we want to encourage you here to do so." Ugh. Well, I can encourage you to do so, with some Baez [0][1]. [0] http://math.ucr.edu/home/baez/rosetta.pdf [1] https://arxiv.org/abs/quant-ph/0404040
Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
Re: Kindergarten Quantum Mechanics
#14Kindergarten? They just took bra/kets, Feynman diagrams and substituted block-looking shapes. The underlying logic is hardly changed to kindergarten level. The kid better have a masters in physics to follow this.
"...i.e. we need to conjugate the first Hilbert space (although this gives an isomorphic copy). In the above argument establishing the bijection we used the matrix representation of Hilbert spaces and hence essentially the whole vector space structure, but in fact we need none of this, and we will show that in any picture calculus we always have..."
Re: Kindergarten Quantum Mechanics
#15I have only skimmed this paper, but the diagrams look somewhat similar to Penrose graphical notation. Is there any connection there?
Re: Kindergarten Quantum Mechanics
#16Kindergarten Quantum Mechanics: if you ask mom for something, she will say "yes" or "no" (even if she wasn't sure before you asked). Asking her again won't change her answer. At this point, to try to get a different answer to the same question, you have to ask dad.
Re: Kindergarten Quantum Mechanics
#17Earlier quoted context omitted.
Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
This is rather uncharitable.
Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language. Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic).
There's certainly a community trying to bring a category-theoretic approach into other fields such as statistics, economics, or other areas of computer science, but it's too early to say it's been "quite a few years".
Re: Kindergarten Quantum Mechanics
#18"Of course we do not expect you to know about category theory, nor do we want to encourage you here to do so." Ugh. Well, I can encourage you to do so, with some Baez [0][1]. [0] http://math.ucr.edu/home/baez/rosetta.pdf [1] https://arxiv.org/abs/quant-ph/0404040
Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.
As grand-nibling comment suggests, it's primarily about vocabulary and definitions, but also how objects can be put together to form new objects. In combination, we end up with "universal properties", behaviors of objects which are the same in many different contexts.
If you want to get some surface-level mind-blown experiences, then there are some cool pages out there, in addition to the papers I already linked, which have great diagrams. [0] shows a periodic table which relates categories to truth values and sets, which is helpful for understanding what folks mean by "category theory is just as good as set theory for foundational work".
More mind-blowingly, category theory is "formally formal", which means that we can use category theory to formalize its own concepts. This leads to n-category theory. Because of this, we have tables like [1] which relate category theory's own building blocks to type theory and logic.
Category theory was originally the study of natural transformations. Any time that you find enough data to define a natural transformation, then that's interesting; the component categories are then also interesting objects of study. That's still what motivates its presence in physics. There are at least two other perspectives though, a logical perspective (a category is a deductive system for a formal logic) and a spatial perspective (a category is a topological space), and they're both rich as well. But the richest perspective is going to come after you've explored all three.
[0] https://ncatlab.org/nlab/show/periodic+table
[1] https://ncatlab.org/nlab/show/computational+trinitarianism
Re: Kindergarten Quantum Mechanics
#19Earlier quoted context omitted.
Man everyone seems to be talking about category theory lately and I still have no idea what it is except some general sense in which different types of mathematical objects can operate on each other (which is probably wrong). Looks like I should probably actually put the effort in and learn about it.
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
I'm obliged, I suppose, to list off a few big wins. From the 50s and 60s, we have classic theorems which use abstract nonsense to generalize big statements about entire classes of objects, like Freyd's adjoint functor theorem [0], Yoneda's lemma [1], and Lawvere's fixed-point theorem [2]. Yoneda's lemma is the slogan that "an object is equivalent to the arrows coming/leaving it", but formal. Lawvere's theorem is a deep and permanent generalization of Gödel, Tarski, Turing, et al. on incompleteness and undecideability.
Starting in the 60s and continuing to the present, there's been a theme of exploring category theory as a prime foundation for maths. There's been complete foundations like the Elementary Theory of the Category of Categories (ETCC) [3], and also much smaller but pointed presentations that focus on just simplifying set theory. I like [4] in particular.
On a deeper level, in terms of structure and philosophy, the entire existence of homotopy type theory (HoTT) relies on categorical presentations and memes. HoTT is, more than any other type theory, an investigation into what equivalence means, and it dovetails wonderfully with 2-category theory and the understanding that algebraic laws can be transformed into transformations.
[0] https://en.wikipedia.org/wiki/Formal_criteria_for_adjoint_fu...
[1] https://en.wikipedia.org/wiki/Yoneda_lemma
[2] http://tac.mta.ca/tac/reprints/articles/15/tr15.pdf
Re: Kindergarten Quantum Mechanics
#20Earlier quoted context omitted.
Just a cautionary note - category theory has seemingly been at "There's tremendous potential here! We just need to find a big win to make that clear to everyone else..." for quite a few years now.
This is rather uncharitable. Category theory is an essential part of the vocabulary of 20th-century mathematics. Large swaths of algebra, topology, geometry, and logic are fairly inextricably formulated in this language. Similarly, it seems irreplaceable for certain parts of programming language theory (arguably due to its connection to mathematical logic). There's certainly a community trying to bring a category-the…
I think parent meant big wins in physics, where the search started well over a decade ago. See e.g. the dates of the references here:
https://en.wikipedia.org/wiki/Categorical_quantum_mechanics#...