78
votes

### The Planck constant for mathematicians

Let's give it a try. Of course, the precise mathematical meaning is perhaps absent, so the answers are sort of heuristic. But if I understand correctly, you want to gain intuition ;)
The first ...

49
votes

### Any real contribution of functional analysis to quantum theory as a branch of physics?

I'm can't claim to have studied the relevant history in a lot of detail, but count me a skeptic of Landsman's claim. Let's take this little paper and the companion that it cites as a test case, which ...

Community wiki

38
votes

### States in C*-algebras and their origin in physics?

I'd like to try to give a more comprehensive answer.
In the elementary formulation of quantum mechanics, pure states are represented by unit vectors in a complex Hilbert space $H$ and observables are ...

33
votes

### The Planck constant for mathematicians

To build intuition for the Planck constant $\hbar$, which I understand is the purpose of the OP, I would start by noting that $\hbar$ is not a dimensionless number: it has dimensions of energy $\times$...

30
votes

### Representation theory and elementary particles

You can understand this philosophy as a generalization of Noether's theorem. Let me only state Noether's theorem in the quantum case because it's actually easier to understand there than in the ...

30
votes

### Any real contribution of functional analysis to quantum theory as a branch of physics?

This reminds me the following anecdote. K. Friedrichs once met Heisenberg on a conference. He thanked Heisenberg for creation of quantum mechanics which benefited mathematics so much, and added:
"...

Community wiki

29
votes

### Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

So, in short, the perfect squares arise as the sums of the first $k$
odd numbers, and the invariant subspaces arrange themselves into
energy levels that way because... well, here I get stuck.
To get &...

27
votes

### What are the strongest arguments for a genuine quantum computing advantage?

The short answer is that if you are looking for theoretical evidence in the form of provable theorems, the situation is not very satisfactory. But I would argue that the heuristic evidence is pretty ...

25
votes

### On independence and large cardinal strength of physical statements

Sorry, not an answer, but too long for a comment.
I am the author of the "Pitowsky's Kolmogorovian models and Super-Determinism" paper. I would reject the claim that this paper is a "philosophical ...

24
votes

Accepted

### Why are quantum groups so called?

Typically in math "quantum X" means a deformation of "X" which is in some sense "less commutative." So quantum groups should be deformations of groups which are "less commutative." Interpreting this ...

23
votes

Accepted

### Is there any published physics article where $q$-mathematics is applied?

There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-...

22
votes

### On independence and large cardinal strength of physical statements

In a recent result with Shay Moran, Pavel Hrubes, Amir Shpilka and Amir Yehudayoff, we show that the answer to a basic question in statistical machine learning is determined by the value of the ...

21
votes

### What determines the maximal dimension of the irreps of a (finite) group?

A simple bound on the largest dimension of a complex irreducible representation (which is either equal to or half of the largest dimension of a real irreducible representation) is the following: we ...

20
votes

Accepted

### John von Neumann's remark on entropy

An alternative version of Von Neumann's quote says "no one understands entropy very well". At the intuitive level, this makes sense, it is much harder to explain the concept of entropy to a ...

19
votes

Accepted

### What determines the maximal dimension of the irreps of a (finite) group?

Your question touches on many issues in group representation theory, and I can only give a few general remarks which may point you in interesting directions for further reading.
As to your question ...

17
votes

### Any real contribution of functional analysis to quantum theory as a branch of physics?

As jjcale mentions in a comment, the index of a Fredholm operator is very important in physics. One way to define the Chern number of a topological insulator is in terms of the index of a Fredholm ...

Community wiki

17
votes

### Rigged Hilbert spaces and the spectral theory in quantum mechanics

Why are rigged Hilbert spaces a paper subject, not usually treated in rigorous textbooks: this is a good question. If you learn QM the way I did, you start by understanding that the physicists' Dirac ...

16
votes

### What are the strongest arguments for a genuine quantum computing advantage?

The question as posed assumes an "ideal" quantum computer, I presume meaning a fully fault tolerant device. (Which does not yet exist.) There is one other limitation that needs to be ...

15
votes

### The Planck constant for mathematicians

To supplement the excellent answers previously given, some more remarks situated in the canonical quantization formalism, which is equivalent (at physicist level) to the path integral formalism ...

15
votes

### What are the strongest arguments for a genuine quantum computing advantage?

One major piece of evidence is not in the context of problems which are decision problems (which return a yes/no answer), or function problems, but rather distribution problems, where one wants to ...

15
votes

### Meaning of a quantum field given by an operator-valued distribution

tl;dr: The reason for operator-valued distributions is because the physically meaningful "measurements" in QFT are things that preserve locality and that can be measured at any location.
In ...

14
votes

### Infinite dimensional symplectic geometry

Aspects of symplectic topology hold in the infinite dimensional setting. A notable difference is that there are different notions for what a symplectic form should be: there is the notion of a strong ...

14
votes

Accepted

### Matrix exponential, containing a thermal state

It is actually easier to compute the first column of $e^{tM}$ for each $t \in \mathbb{R}$ rather than computing only the first column of $e^M$.
Indeed, let $u(t) = e^{tM}e_1$ for each $t \in \mathbb{...

14
votes

Accepted

### Why does Riesz's Representation Theorem apply in quantum mechanics?

Okay, there is a lot of confusion in this question.
First, I'm not sure why you say ``it is common to begin the discussion with embedding $A$'' into $B(H)$. The point of the C${}^*$-algebra approach ...

14
votes

### Rigged Hilbert spaces and the spectral theory in quantum mechanics

@Nik Weaver: The answer one is looking for is some clearly important, basic topic in QM which can be treated more easily with rigged Hilbert spaces.
The foremost topic is the theory of resonances.
A ...

14
votes

### What is the best place to learn about the mathematical foundations of quantum mechanics?

The question is a little unclear --- you want something axiomatic but not rigorous? Anyway, if you don't care about rigor and you like Dirac deltas, I don't think there's any better place to start ...

14
votes

### Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

As Igor said, it's a bit late for that. If you just finished undergrad and you discovered a passion for QFT from a rigorous mathematical standpoint, what you should do, for example, is apply for the ...

14
votes

Accepted

### Spectrum of matrix involving quantum harmonic oscillator

The Hamiltonian
$$H=\begin{pmatrix}
\alpha^2+a^\ast a&\alpha a+\beta a^\ast\\
\alpha a^\ast+\beta a&\beta^2+a^\ast a
\end{pmatrix}
$$
is known in the physics literature as the anisotropic Rabi ...

13
votes

### Applications of Jordan algebras

I would like to elaborate on the link to "associative" problems (such as the Zelmanov's solution of the restricted Burnside problems mentioned by Tom De Medts) - mainly because by saying that Jordan ...

13
votes

### The Planck constant for mathematicians

About your Q1: I think that the simplestâ€”and most obviousâ€”way to think mathematically about the physical meaning of Planck's constant $h$ is that it is a kind of quantitative measure of the departure ...

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