All Questions
21 questions
3
votes
1
answer
219
views
Moment problem, ergodicity and spectral gap on the space of tempered distributions
Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$.
Just below formula [5] in p.122 of the Fröhlich paper, ...
- 3,477
6
votes
0
answers
159
views
Identification of Fock space and the $L^2$ space of tempered distributions
Let $\mathcal{S}'(\mathbb{R}^d)$ be the set of tempered distributions over $\mathbb{R}^d$ and $d\phi_C$ a Gaussian measure over $\mathcal{S}'(\mathbb{R}^d)$ with covariance operator $C$. Consider the ...
- 721
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
2
votes
0
answers
172
views
AQFT from a Lagrangian
In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...
- 424
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
4
votes
1
answer
291
views
Structure of all Wightman QFTs
I have two related questions related to constructive/axiomatic QFT.
Is there a structure on the collection of all QFTs, as defined by the Wightman axioms? Do they form some type of category?
...
- 43
2
votes
0
answers
463
views
Segal's axioms for CFT
In Segal's papers about Conformal Field theory, https://www2.math.upenn.edu/~blockj/scfts/segal.pdf, in section $1$, he describes the evolution of a system (a string moving about in a manifold $M$) by ...
2
votes
0
answers
85
views
Pullbacks of LCS-valued distributions
Suppose $X$ is a locally convex space. Since the distributions $\mathcal{D}'\!(M)$ ($M$ a manifold) are a nuclear space, there is a canonical meaning to the topological tensor product $X\,\widehat{\...
- 439
6
votes
0
answers
290
views
Two questions about Fock spaces
Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
- 3,515
2
votes
1
answer
164
views
Vacuum state generating functional
In Theorem 1 of this paper Segal stablish a relation between states and generating functionals.
He assert that in order to $\mu$ be a generating functional must satisfy
$$
\sum_{j,k\in F} \mu (z_j-...
- 424
6
votes
1
answer
505
views
Fermions, their path integrals and effective actions
I just read the nice exposition Fermionic Path Integral on nLab and began to wonder about some details to which references appear to be lacking. Suppose we live on Euclidean space as in the ...
- 651
2
votes
0
answers
67
views
Reflection positivity on weighted $L^2$-spaces
Denote by $(t, x_{1}, \ldots, x_{d-1})$ the coordinates of $x \in \mathbb{R}^{d}$ and set
$$\mathbb{R}^{d}_{+}=\left\{t, x_{1}, \ldots, x_{d-1} \in \mathbb{R}^{d}|t > 0\right\}. $$
Write $\theta$ ...
- 505
12
votes
1
answer
1k
views
Is there a physical reason that fields in QFT are globally defined?
I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
- 7,952
6
votes
2
answers
621
views
Relativistic scattering theory vs non-relativistic one
In relativistic scattering theory (e.g. in quantum electrodynamics) the existence of the $S$-matrix as well as of Moller operators is postulated as far as I understand (although at some stage it has ...
- 21.8k
16
votes
2
answers
2k
views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
- 2,147
15
votes
1
answer
1k
views
Borel-Écalle re-summation and resurgence: criteria and results
This is about the theory of Borel-Écalle re-summation and resurgence, see Refs below.
This states that the perturbative series (say of the vacuum expectation value of an operator $\mathcal{O}$ in ...
- 10.5k
8
votes
2
answers
1k
views
Rigorous definition of the commutator $[a(k_1), a^\ast(k_2)]$ of creation and annihilation operators in boson quantum field models
In their lecture notes "Boson Quantum Field Models" (in "Mathematics of Contemporary Physics", R.Streater (ed.)), Glimm and Jaffe define an annihilation operator $a(k), k \in \mathbb{R}$ on a certain ...
- 111
3
votes
0
answers
57
views
Integration of Weyl operators multiplied by quasifree state over a symplectic space
I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
25
votes
6
answers
3k
views
Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
- 13.6k
4
votes
2
answers
974
views
Wightman fields vs local functionals vs operators
In QFT literature one wants to look at $n-$point correlation functions of "operators" inserted at $x$ say, $\cal{O}(x)$ and if $\phi_i$ are the fields then the quantity one has in mind is written as, $...
- 3,541
11
votes
1
answer
2k
views
Spectral theory for self-adjoint field operators on a symmetric Fock space
Background
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \...
- 195