4 ms·

Introduction to Hilbert Space (2022) [pdf]

106 points by azeemba 3y ago

azeemba 3y ago
The author's website has a treasure trove of articles explaining different things in Physics: https://www.cphysics.org/ https://www.cphysics.org/
- anta40 3y ago
  Nicely formatted PDFs there. Bookmarked :D
- rhymer 3y ago
  Agree, this is great! I wonder if there're some ML equivalent sites that present topics in a modular way?
  - esafak 3y ago
    https://course.fast.ai/ https://course.fast.ai/ https://www.deeplearningbook.org/ https://www.deeplearningbook.org/
  - smokel 3y ago
    Perhaps not exactly what you are looking for, but MLU-Explain is nice: https://mlu-explain.github.io/ https://mlu-explain.github.io/
  - _a_a_a_ 3y ago
    Does https://mlbook.explained.ai/ https://mlbook.explained.ai/ help? Been on HN before, got very positive comments. From the author of ANTLR no less.
ezion 3y ago
Introduction to Hilbert Space (from a physics perspective)** I prefer Halmos' book.
- Paul-Craft 3y ago
  Do you mean Introduction to Hilbert space and the theory of spectral multiplicity? Here's a link: https://archive.org/details/introductiontohi0000halm https://archive.org/details/introductiontohi0000halm
mh-cx 3y ago
Hmm, he lost me on page nine where "complete" is explained. > Complete means that every sequence of vectors |a1>, |a2>, ... satisfying lim ... How are the elements of this sequence related? And why are we only interested in the elements where the index n/m goes to infinity? What does that even mean if the sequence is arbitrary? That's probably why I also can't make sense of this: > Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.
- stracer 3y ago
  This is standard mathematical analysis. Infinite sequence of elements may look like it converges to some target element, judging by mutual distances converging to zero. Such sequence is a Cauchy sequence. When the target element actually exists, then the sequence is also convergent. A space where every Cauchy sequence is convergent, is called complete. Example: if the space is all real numbers except 0, then any sequence of real numbers accumulating around 0 (for howsoever small a distance, there is always infinite number of points closer to 0), the sequence is a Cauchy sequence, but not convergent (because 0 is not present). So that space is not complete (has a hole). If the space is all real numbers, then the same sequence is also convergent, and the space is complete (no holes).
- steppi 3y ago
  Agreed that this is pretty terse. The sequences they're talking about are called Cauchy sequences [0]. A sequence a_i is Cauchy if for any epsilon, there exists an N such that if m and n are both greater than N, then |a_m - a_n| < epsilon. A classic example, suppose your space is the set of rational numbers, and consider the sequence a0 = 1, a_1 = 1.4, a_2 = 1.41, ... a_n = sqrt(2) up to n digits after the decimal place. You can verify that this is a Cauchy sequence, successive points get arbitrarily close to each other. This means the rational numbers are incomplete, because this Cauchy sequence of rationals doesn't converge to a rational number. It's the real numbers that forms a complete space. Completeness is required for nice results like the spectral theorem for self-adjoint operators [1] to hold, which is pretty essential for Quantum Mechanics. [0] https://en.wikipedia.org/wiki/Cauchy_sequence https://en.wikipedia.org/wiki/Cauchy_sequence. [1] https://en.wikipedia.org/wiki/Spectral_theorem https://en.wikipedia.org/wiki/Spectral_theorem
aquafox 3y ago
Interesting fact about Hilbert spaces: The inner product of a Hilbert space induces a norm and thus every Hilbert space is a Banach space. But what about the converse? Say we only have a normed vector space, can we decide if there is an inner-product space that actually induces this norm? The answer is yes! Simply check if the Parallellogram Law [1] holds. [1] https://en.wikipedia.org/wiki/Parallelogram_law https://en.wikipedia.org/wiki/Parallelogram_law
- 3abiton 3y ago
  This makes me want to lrarn more about hilbert space.
- sgdpk 3y ago
  Yet another fact discovered by von Neumann.
  - queuebert 3y ago
    https://imgflip.com/i/7zaxoy https://imgflip.com/i/7zaxoy
antoine-levitt 3y ago
> The name quantum in quantum theory is related to the fact that in a separable Hilbert space, any set of mutually orthonormal vectors is countable. Pretty sure the name quantum comes from the fact that some physical phenomena (eg absorption spectrum) get discrete allowed values. That in principle has nothing to do with separability (eg you can come up with non separable spaces which have operators with discrete spectrum). In fact presenting things this way is pretty confusing since separable Hilbert spaces do support operators with continuous spectrum (which is not obvious!) As far as I know separability is mostly technical, and often added to make life a bit simpler, since it's pretty hard to come up with useful non separable Hilbert spaces.
- hovden 3y ago
  A lot of non-quantum waves have discrete allowed values. EM cavities, guitar strings, etc. Quantum waves are described by a special wave equation, actually a complex diffusion equation (first order in time, second order spatially).
queuebert 3y ago
I've made several runs now at trying to understand Hilbert space from a physics perspective, but I still do not have a good intuition for it. Something about it just breaks my brain.
mikhailfranco 3y ago
Q: So what is the dimension of my Hilbert Space? A: Just enough to describe all your independent physical states, which is proportional to the number of your particles. Q: So if some interaction creates more particles I have to increase the dimension of my Hilbert Space? A: Yes, I'm afraid so. Q: But I am used to physics providing some constant background to goings-on here and now. It was absolute Newtonian space and time, then it was warped Einsteinian spacetime, now you say it depends on how many particles I create? A: Yes. Q: But that's insane? A: Yes. A: OK, I'll allow you to create or destroy as many particles as you like, especially if you have an really big Large Hardon Collider. Q: Are you sure I can keep my Hilbert Space the same? A: Yes, aha, I have thought of a special number that does not change when you add or remove finite integers... Q: There is no such number ... oh wait, you mean infinite dimensional? A: Yes, infinite dimensional, and complex, of course. Q: Of course. Q: Does not sound in the least bit plausible. How do you add particles? A: It's called the second quantization of Quantum Field Theory using Fock Spaces. Q: Makes perfect Focking sense.
srean 3y ago
Professor Hilbert was unaware that other mathematicians had started calling infinite dimensional inner-product spaces, Hilbert spaces. In fact some of the foundational results were proven by Von Neumann. The story goes that once Neumann was giving a lecture in Germany on such spaces with Hilbert in attendance. Hilbert had supposedly raised his hands to ask, Dr. von Neumann, I would very much like to know, what after all is a Hilbert space? Hilbert was quite underwhelmed by the definition, "is that all" he remarked.

106 points by azeemba 3y ago

The author's website has a treasure trove of articles explaining different things in Physics: https://www.cphysics.org/ https://www.cphysics.org/

anta40 3y ago
Nicely formatted PDFs there. Bookmarked :D
rhymer 3y ago
Agree, this is great! I wonder if there're some ML equivalent sites that present topics in a modular way?
- esafak 3y ago
  https://course.fast.ai/ https://course.fast.ai/ https://www.deeplearningbook.org/ https://www.deeplearningbook.org/
- smokel 3y ago
  Perhaps not exactly what you are looking for, but MLU-Explain is nice: https://mlu-explain.github.io/ https://mlu-explain.github.io/
- _a_a_a_ 3y ago
  Does https://mlbook.explained.ai/ https://mlbook.explained.ai/ help? Been on HN before, got very positive comments. From the author of ANTLR no less.

ezion 3y ago

Introduction to Hilbert Space (from a physics perspective)** I prefer Halmos' book.

Paul-Craft 3y ago
Do you mean Introduction to Hilbert space and the theory of spectral multiplicity? Here's a link: https://archive.org/details/introductiontohi0000halm https://archive.org/details/introductiontohi0000halm

mh-cx 3y ago

Hmm, he lost me on page nine where "complete" is explained. > Complete means that every sequence of vectors |a1>, |a2>, ... satisfying lim ... How are the elements of this sequence related? And why are we only interested in the elements where the index n/m goes to infinity? What does that even mean if the sequence is arbitrary? That's probably why I also can't make sense of this: > Loosely speaking, saying that a Hilbert space is complete means that it contains all of its limits.

stracer 3y ago
This is standard mathematical analysis. Infinite sequence of elements may look like it converges to some target element, judging by mutual distances converging to zero. Such sequence is a Cauchy sequence. When the target element actually exists, then the sequence is also convergent. A space where every Cauchy sequence is convergent, is called complete. Example: if the space is all real numbers except 0, then any sequence of real numbers accumulating around 0 (for howsoever small a distance, there is always infinite number of points closer to 0), the sequence is a Cauchy sequence, but not convergent (because 0 is not present). So that space is not complete (has a hole). If the space is all real numbers, then the same sequence is also convergent, and the space is complete (no holes).
steppi 3y ago
Agreed that this is pretty terse. The sequences they're talking about are called Cauchy sequences [0]. A sequence a_i is Cauchy if for any epsilon, there exists an N such that if m and n are both greater than N, then |a_m - a_n| < epsilon. A classic example, suppose your space is the set of rational numbers, and consider the sequence a0 = 1, a_1 = 1.4, a_2 = 1.41, ... a_n = sqrt(2) up to n digits after the decimal place. You can verify that this is a Cauchy sequence, successive points get arbitrarily close to each other. This means the rational numbers are incomplete, because this Cauchy sequence of rationals doesn't converge to a rational number. It's the real numbers that forms a complete space. Completeness is required for nice results like the spectral theorem for self-adjoint operators [1] to hold, which is pretty essential for Quantum Mechanics. [0] https://en.wikipedia.org/wiki/Cauchy_sequence https://en.wikipedia.org/wiki/Cauchy_sequence. [1] https://en.wikipedia.org/wiki/Spectral_theorem https://en.wikipedia.org/wiki/Spectral_theorem

aquafox 3y ago

Interesting fact about Hilbert spaces: The inner product of a Hilbert space induces a norm and thus every Hilbert space is a Banach space. But what about the converse? Say we only have a normed vector space, can we decide if there is an inner-product space that actually induces this norm? The answer is yes! Simply check if the Parallellogram Law [1] holds. [1] https://en.wikipedia.org/wiki/Parallelogram_law https://en.wikipedia.org/wiki/Parallelogram_law

3abiton 3y ago
This makes me want to lrarn more about hilbert space.
sgdpk 3y ago
Yet another fact discovered by von Neumann.
- queuebert 3y ago
  https://imgflip.com/i/7zaxoy https://imgflip.com/i/7zaxoy

antoine-levitt 3y ago

> The name quantum in quantum theory is related to the fact that in a separable Hilbert space, any set of mutually orthonormal vectors is countable. Pretty sure the name quantum comes from the fact that some physical phenomena (eg absorption spectrum) get discrete allowed values. That in principle has nothing to do with separability (eg you can come up with non separable spaces which have operators with discrete spectrum). In fact presenting things this way is pretty confusing since separable Hilbert spaces do support operators with continuous spectrum (which is not obvious!) As far as I know separability is mostly technical, and often added to make life a bit simpler, since it's pretty hard to come up with useful non separable Hilbert spaces.

hovden 3y ago
A lot of non-quantum waves have discrete allowed values. EM cavities, guitar strings, etc. Quantum waves are described by a special wave equation, actually a complex diffusion equation (first order in time, second order spatially).

queuebert 3y ago

I've made several runs now at trying to understand Hilbert space from a physics perspective, but I still do not have a good intuition for it. Something about it just breaks my brain.

mikhailfranco 3y ago

Q: So what is the dimension of my Hilbert Space? A: Just enough to describe all your independent physical states, which is proportional to the number of your particles. Q: So if some interaction creates more particles I have to increase the dimension of my Hilbert Space? A: Yes, I'm afraid so. Q: But I am used to physics providing some constant background to goings-on here and now. It was absolute Newtonian space and time, then it was warped Einsteinian spacetime, now you say it depends on how many particles I create? A: Yes. Q: But that's insane? A: Yes. A: OK, I'll allow you to create or destroy as many particles as you like, especially if you have an really big Large Hardon Collider. Q: Are you sure I can keep my Hilbert Space the same? A: Yes, aha, I have thought of a special number that does not change when you add or remove finite integers... Q: There is no such number ... oh wait, you mean infinite dimensional? A: Yes, infinite dimensional, and complex, of course. Q: Of course. Q: Does not sound in the least bit plausible. How do you add particles? A: It's called the second quantization of Quantum Field Theory using Fock Spaces. Q: Makes perfect Focking sense.

srean 3y ago

Professor Hilbert was unaware that other mathematicians had started calling infinite dimensional inner-product spaces, Hilbert spaces. In fact some of the foundational results were proven by Von Neumann. The story goes that once Neumann was giving a lecture in Germany on such spaces with Hilbert in attendance. Hilbert had supposedly raised his hands to ask, Dr. von Neumann, I would very much like to know, what after all is a Hilbert space? Hilbert was quite underwhelmed by the definition, "is that all" he remarked.