61
votes
Do all Noether theorems have a common mathematical structure?
The core of the Noether theorem in all contexts where it arises is surprisingly elementary!
From a very general point of view, one considers the following structure.
(i) A set of "states" $x\...
57
votes
Book covering differential geometry and topology for physics
The first thing that must be said is that the question is not really specific enough: Applications to what exactly are you looking for? To me, a book on algebraic geometry and mirror symmetry, and how ...
56
votes
Fourier transform of the Coulomb potential
I really appreciate the physical explanations made in other answers, but I want to add that Fourier transform of the Coulomb potential makes mathematical sense, too. This answer is meant to clarify ...
48
votes
Can a mathematical proof replace experimentation?
No. Physics remains an experimental science and so it is not possible to replace experiment by a proof. Descartes tried this when he proposed his theory of propagation of light - very elegant - but ...
44
votes
Mathematically rigorous QFT text
This question cannot be aswered as it is asked. There is no general mathematical rigorous definition of QFT in general, but rather different approaches with different goals and applications.
First of ...
Community wiki
41
votes
Accepted
What is the issue with interactions in QFT?
There are many different problems with interactions; or, rather, many manifestations of the same problem.
For example, interactions are always non-linear in the equations of motion, e.g.,
$$
(\partial^...
41
votes
Does it make sense to say that something is almost infinite? If yes, then why?
Almost infinite can make a lot of sense in physics. There isn't a precise definition but I would interpret it as the following: when something is 'almost infinite' the properties we are considering ...
39
votes
Accepted
Newton's law requires two initial conditions while the Taylor series requires infinite!
On the other hand, it requires only two initial conditions x(0) and
x˙(0), to obtain the function x(t) by solving Newton's equation
For notational simplicity, let
$$x_0 = x(0)$$
$$v_0 = \dot x(0)$$...
37
votes
Is there something similar to Gödel's incompleteness theorems in physics?
No, there is not nor can there be a similar statement in physics. That is because we can know all there is to know about the mathematical systems we construct; after all, we have set them up ourselves ...
34
votes
Accepted
What's the deal with momentum in the infinite square well?
Is the momentum operator $P=-i\hbar \frac{\mathrm d}{\mathrm dx}$ symmetric when restricted to the compact interval of the well? Are there any subtleties in its definition, via its domain or similar, ...
33
votes
Accepted
Non-unique zero function in the function space (Hilbert space)
This is precisely why the $L^2(\mathbb R)$ is not simply the space of square-integrable functions from $\mathbb R$ to $\mathbb C$ (which we might call $SI(\mathbb R)$).
$SI(\mathbb R)$ consists of ...
33
votes
Does it make sense to say that something is almost infinite? If yes, then why?
"Almost infinite" is a sloppy term that might be used to mean "effectively infinite", in a given context. For example, if a large value of $x$ in $y = 1/x$ produces a value ...
28
votes
Hilbert space of harmonic oscillator: Countable vs uncountable?
The previous answers are all correct, but I thought I'd give a more conceptual explanation for why the delta-function basis is the "wrong" basis in which to expand when counting degrees of freedom. ...
28
votes
Accepted
Is rigged Hilbert space generally considered the correct structure for QM?
I do not think there is a unique answer. It mostly depends on personal taste.
However, I think that almost all mathematical physicists agree on the fact that the QM approach based on the rigged ...
25
votes
A rigorous definition of the exponential of an operator in QM?
Even if there is already a good accepted answer I would like to say something further to completely fix some details.
Is this definition correct for unbounded operators, too?
No, it does not work ...
25
votes
Accepted
What does it mean for a field to be defined by a measure?
I believe this is a common issue in bridging the mathematically rigorous analysis of Glimm and Jaffe and the standard Physics discussion in QFT textbooks. But the basic point is this. Imagine you have ...
24
votes
Accepted
Not all self-adjoint operators are observables?
The appendix of Mackey talks about superselection rules, and indeed superselection is the phenomenon where there are self-adjoint operators that are not observables. Whether this is obvious or not ...
24
votes
Accepted
What is the physical importance of topological quantum field theory?
I have personally done original research in the field of TQFTs, so I can tell you the reasons I find TQFTs interesting. Some that come to mind are:
Some "real life" theories are accurately ...
23
votes
Is there any physical system with a non-separable Hilbert space?
The standard formulations of QM and QFT are such that the resulting Hilbert space is always separable, namely there exist a finite or infinite countable Hilbert basis (and thus every Hilbert bases are ...
23
votes
What is the relationship between different types of quantum field theories?
The three classes of QFTs you are referring to are distinguished by different symmetry assumptions (Poincare invariance, conformal invariance, and volume-preserving diffeomorphism invariance) and ...
23
votes
Not all self-adjoint operators are observables?
In general, for physical reasons not all self-adjoint operators are observables. I will discuss the problem at the level of von Neumann algebras that are (perhaps unawares) more familiar to physicists....
23
votes
Are there undecidable statements in classical mechanics?
Yes. One simple (trivial) way of seeing it is to consider a mechanical computer, and ask whether one can predict its end-state given the initial state without running it. Since this is exactly the ...
22
votes
Why is de Rham cohomology important in fundamental physics?
You could resumé the importance of cohomology (not only de Rham's) in Physics (and all areas of Applied Mathematics) in a very concise way: if your closed forms are not exact, then cohomology matters. ...
22
votes
Can a mathematical proof replace experimentation?
If you only make assumptions that have been experimentally verified (up to a high degree of precision) then a purely mathematical proof might be fine. However there are two problems with this:
1) ...
20
votes
Do continuous wavefunction form a Hilbert space?
Item 2 is simply false if wavefunction is understood as a generic state vector of the system.
This is compatible with the answer NO to your question.
With the above interpretation wavefunctions are ...
19
votes
Fourier transform of the Coulomb potential
I am personally fond of this short and sweet argument.
If you believe the relatively easy to prove fact that in 3 dimensions
$$
\nabla^2\frac{1}{r} = -4\pi \delta^{(3)}(\bf{r})
$$
then taking the ...
19
votes
Accepted
Is it possible to show a diffraction caustic as a home experiment / lecture demonstration?
This may be easier than you think. I took this photo (with my iPhone) of a cusp caustic which I generated by darkening the bathroom, wetting the mirror, and angling the laser pointer so it hit a water ...
19
votes
Non-unique zero function in the function space (Hilbert space)
This is an important point that is usualy swept under the rug in introductory classes.
The elements of the functional Hilbert space used in Quantum mechancs (called $L^2[{\mathbb R}]$ in the ...
18
votes
How exactly is the formalism of thermodynamics based on contact geometry?
Here is the upshot:
On one hand, a strict contact manifold $(M,\alpha)$ is a $(2n+1)$-dimensional manifold $M$ equipped with a globally defined one-form $\alpha\in \Gamma(T^{\ast}M)$ that is ...
17
votes
Confusion with Virtual Displacement
Here on SE, you may already find many answers to your question. Even if most of them are correct, I feel that a plain and correct answer is still missing. Where plain does not mean non-rigorous. But ...
Only top scored, non community-wiki answers of a minimum length are eligible
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