All Questions
Tagged with fa.functional-analysis mp.mathematical-physics
221 questions
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
28
votes
6
answers
6k
views
Any real contribution of functional analysis to quantum theory as a branch of physics?
In the last paragraph of this last paper of Klaas Landsman, you can read:
Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
25
votes
6
answers
3k
views
Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
- 13.6k
23
votes
2
answers
3k
views
States in C*-algebras and their origin in physics?
in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$
Now, of course there is also in classical physics and quantum ...
- 243
21
votes
2
answers
2k
views
Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
- 24.9k
20
votes
6
answers
7k
views
Does the derivative of log have a Dirac delta term?
Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
- 16.6k
20
votes
2
answers
7k
views
Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
- 3,515
18
votes
6
answers
4k
views
What is the best place to learn about the mathematical foundations of quantum mechanics?
I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind ...
- 1,305
16
votes
3
answers
852
views
Quantum Hamiltonian for an Inverse Cube Force Law
If you have a nonrelativistic quantum particle in $\mathbb{R}^3$ in an attractive inverse cube force, its Hamiltonian is
$$ H = -\nabla^2 - \frac{c}{r^2} $$
where I'm keeping things simple by ...
- 22.3k
16
votes
2
answers
2k
views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
- 2,147
15
votes
2
answers
1k
views
Is zero a hydrogen eigenvalue?
This question has been bugging me for some time.
Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is ...
- 2,358
15
votes
6
answers
3k
views
Spectral theorem for self-adjoint differential operator on Hilbert space
I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
13
votes
6
answers
2k
views
Interesting examples of non-locally compact topological groups
Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with ...
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
- 5,824
13
votes
2
answers
1k
views
Applications of non-separable Hilbert spaces
In applications, Hilbert spaces of interest are often assumed to be separable. In addition to being extremely convenient mathematically, this assumption can often be justified on computational or ...
13
votes
1
answer
729
views
Making sense of the formula $\operatorname{Det} (I+M )= e^{\operatorname{Tr} \ln (I+M)}$, especially in the infinite dimensional cases
$\DeclareMathOperator\Det{Det}\DeclareMathOperator\Tr{Tr}$In physics literature dealing with quantum field theory, the formula
\begin{equation}
\Det(I+M) = e^{\Tr \ln(I+M)}
\end{equation}
appears ...
- 3,477
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 3,323
12
votes
1
answer
1k
views
Is there a physical reason that fields in QFT are globally defined?
I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
- 7,952
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 1,990
12
votes
0
answers
922
views
What's the appropriate notion of a Unitary representation of a Lie algebra?
Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...
- 922
12
votes
0
answers
478
views
What is known about the Yang-Mills stratification over 3-manifolds?
Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...
- 8,803
11
votes
2
answers
1k
views
Harmonic oscillator in spherical coordinates
It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry.
More precisely, the operator
$$-\frac{d^2}{dx^2}+x^2$$
can be ...
- 113
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
11
votes
1
answer
2k
views
Spectral theory for self-adjoint field operators on a symmetric Fock space
Background
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \...
- 195
10
votes
2
answers
2k
views
Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
10
votes
1
answer
521
views
About Friedrichs historical contribution to QFT cited in Reed and Simon
In the Reed and Simon book, Appendix X.7, they mention that Friedrichs provided the first examples of inequivalent representations of the canonical commutation relations via the Weyl relations in the ...
- 424
10
votes
2
answers
2k
views
When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
- 8,512
10
votes
1
answer
337
views
What are the predictive implications of conditional non-commutative probability?
To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$.
In this context a state $S$ is a positive semi-definite ...
- 323
9
votes
2
answers
848
views
$\zeta$-function regularized determinants
In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ...
- 21.8k
9
votes
2
answers
2k
views
Mathematical equivalent to ladder operators?
A powerful method in theoretical physics are ladder operators. They are used in QM to solve problems like the harmonic oscillator and the hydrogen atom. The idea is to solve with their help the ...
user37929
9
votes
1
answer
596
views
Classical analogue of the Stone-von Neumann Theorem?
Let $U_s$, $V_t$ be a pair of continuous $n$-parameter groups ($n < \infty$) of unitary operators on a complex Hilbert space $\mathcal{H}$. The Stone-von Neumann Theorem establishes that any such ...
- 657
8
votes
2
answers
1k
views
Rigorous definition of the commutator $[a(k_1), a^\ast(k_2)]$ of creation and annihilation operators in boson quantum field models
In their lecture notes "Boson Quantum Field Models" (in "Mathematics of Contemporary Physics", R.Streater (ed.)), Glimm and Jaffe define an annihilation operator $a(k), k \in \mathbb{R}$ on a certain ...
- 111
8
votes
1
answer
2k
views
Validity of functional derivative using the Dirac delta function
In physics, it's customary to compute the functional derivative as $$\frac{\delta F[\rho(x)]}{\delta \rho(y)}=\lim_{\varepsilon\to 0}\frac{F[\rho(x)+\varepsilon\delta(x-y)]-F[\rho(x)]}{\varepsilon}.$$ ...
- 81
8
votes
2
answers
496
views
Which complete orthomodular lattices arise from von Neumann algebras?
Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice.
Question 1: Is the construction $A \mapsto \Pi(A)$ a ...
- 63.9k
8
votes
1
answer
584
views
Tensor products of unitary irreducible representations of $SU(2,2)$
What is known about irreducible decomposition of tensor products of (infinite-dimensional) unitary irreducible representations of $SU(2,2)$ (or, more generally, simple groups of split rank greater ...
- 1,488
8
votes
1
answer
420
views
What is the general form of the duality transform for the Fock space?
I am interested in properties of the symmetric Fock space, looked at via the associated Wiener space. It is well known that for a Hilbert space $k$, the symmetric Fock space $$\mathcal{F}(L^2(\mathbb{...
- 81
8
votes
0
answers
221
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
7
votes
2
answers
1k
views
Energy levels of double well potential
Consider the (quantum) Hamiltonian on the real line
$$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$
Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
- 21.8k
7
votes
2
answers
1k
views
C*-algebraic representation of observables vs self-adjoint operators one
I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...
- 585
7
votes
3
answers
1k
views
Simultaneous diagonalization of self-adjoint operators on Hilbert space
Suppose I'm given a finite set of possibly unbounded commuting self-adjoint operators $T_i : \mathfrak H \supset \mathscr D(T_i)\to \mathfrak H, i = 1 , \dots , N$ on a Hilbert space (in the sense of ...
- 203
7
votes
2
answers
992
views
Is there a nice "synthetic" way for doing differential geometry on infinite dimensional vector spaces?
If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
- 922
7
votes
2
answers
592
views
Prove that the following function is positive
Consider the following function:
$$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right)
$$
This is Mehler's kernel, and can be ...
- 90
7
votes
3
answers
546
views
Do non-normal states exist in the Solovay model?
Let H be an infinite dimensional Hilbert space. Then there exist non-normal states on B(H) in ZFC (i.e. states that are not represented by a density operator).
Is this also true in the Solovay model ?...
- 2,753
7
votes
1
answer
1k
views
dependence of eigenvalues on parameters
Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$,
with $\phi = 0$ on the boundary. There exists a sequence of ...
- 1,233
7
votes
1
answer
4k
views
Functional/variational derivative and the Leibniz rule
I am currently trying to understand the BV-formalism, which makes heavy use of the functional derivative.
Let us consider the functional derivative, as defined in for example its Wikipedia article.
...
- 515
7
votes
2
answers
1k
views
Yang Mills gradient/heat flow on 4-torus
The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...
- 362
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 109
7
votes
0
answers
501
views
intuitive connection between The KdV equations and the Virasoro bott group
I posted this on stack exchange but had no joy, perhaps someone here can answer : The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group....
- 979
6
votes
2
answers
644
views
Explicit form of this unitary transformation
Disclaimer: This question has its motivation from physics. It is probably not entirely rigorous at the moment. I just want to clarify some steps and try to make the arguments rigorous afterwards, if ...
- 3,515