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How are point-dependent concepts such as correlation functions, renormalization, cluster decomposition...etc interpreted in axiomatic QFT?

In traditional quantum field theory we often speak about things happening at a point. For example, the correlation function $$\langle 0 | \phi(x) \phi(y)\ | 0\rangle \tag{1}$$ can be thought of the ...
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What is the field map in 0+1-dimensional QFT?

I'm a beginner trying to learn something about the Wightman axioms. I got the idea that from an abstract Hilbert space and an abstract Hamiltonian operator, I should be able to produce a trivial ...
Upasker's user avatar
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Existence of Schwinger Functions for QCD?

It seems to me the 'naive' approach to proving the existence of Yang-Mills in a rigorous context (via Osterwalder-Schrader $\to$ Wightman axioms), would be: Study gauge invariant lattice QCD ...
QCD_IS_GOOD's user avatar
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How are Schwinger and Wightman functions used in practice?

In Reed & Simon's Methods of Mathematical Physics Volume II, they define a (Hermitian scalar) quantum field theory to be the quadruple $\langle \mathcal{H}, U, \varphi, D\rangle$ that satisfies ...
CBBAM's user avatar
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Can quantum fields be smeared in space (rather than spacetime)?

I am interested in what is known about the possibility of smearing interacting quantum fields on a Cauchy slice. This is easy to do for free fields and their conjugate momentum, and indeed this is ...
Pranav Pulakkat's user avatar
5 votes
2 answers
1k views

What is an operator-valued distribution?

I am a mathematician who is trying to understand Wightman axioms. I do not understand what an operator-valued distribution is, because in this context people say that operators can be unbounded, and ...
Nicolò Cavalleri's user avatar
1 vote
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120 views

Help with Green functions of Wightman fields

I am trying to prove equation 97 in Chapter I of Zavialov's "Renormalized Quantum Field Theory" book, the statement claims that the Green functions of Wightman fields $\check{\phi}$, defined ...
Gaussian97's user avatar
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Is it known whether Wightman's axiomatic QFT is logically equivalent to Osterwalder–Schrader's axiomatic QFT?

Constructive QFT has provided some interesting models for dimension $d < 4$ of space-time, satisfying specific axiomatic versions of QFT. On the other hand, it is a well known fact that an ...
Davius's user avatar
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Fock space and the GNS construction of a free field

Let's consider a scalar field obeying Wightman's axioms. In the Wightman reconstruction theorem, we can reconstruct the Hilbert space by starting with the algebra $\mathcal{A} = \bigoplus_{n=1}^\infty ...
Todor Markov's user avatar
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0 answers
141 views

How to analytically continue Schwinger functions?

To get Wightman functions $W(t_1, \dots, t_{k-1})$ from Schwinger functions $S(\tau_1 = i t_1, \dots)$, we use analytical continuation. But I don't think this is simply an issue of plugging $it_a$ for ...
Prof. Legolasov's user avatar
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Wightman functions in quantum field theory

(Source of the 'theorem': click) Given a field $\Phi(x)$ with spin $s$ and its adjoint $\Phi^*(x)$, define the expectation values \begin{align} f(x-y)&:=\langle v,\Phi(x)\Phi^*(y)v\rangle, \\ ...
Balter 90s's user avatar
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Are propagators in QFT Wightman Functions?

I am studying relativistic quantum mechanics and I can't really understand how propagators arise from the theory. They are generally defined as the Wightman function $$ W_F (t',\vec{x}', t,\vec{x}) \...
Summoned Egar's user avatar
5 votes
1 answer
118 views

Existence of the S-matrix in AQFT

I am reading the book "An introduction to Symmetry and Supersymmetry in Quantum field theory" by Lopuszanski, and I have some problems understanding his argumentation about the existence of ...
S.Farr's user avatar
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How to show time evolution operator obeys causality?

If we are given a time evolution function $K_t(\phi,\phi')$ which give the amplitude for a field starting in confiruation $\phi$ to go to configuration $\phi'$ after time t. What is the condition that ...
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1 answer
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Continuity of $\phi(f)\vert\Omega\rangle$ in Wightman QFT

I have to prove the following statement: $f\mapsto \phi(f)\vert\Omega\rangle$ is continuous, where $\phi$ is a scalar Wightman quantum field, $\Omega$ the vacuum state of the theory and $f$ a ...
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8 votes
2 answers
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Wilson loop operator in electrodynamics

I'm trying to prove that the Wilson loop operator is well-defined in non-interacting quantum electrodynamics without matter, that is, $\hat{W}(\gamma)$ is a bounded operator on the Hilbert space. ...
Prof. Legolasov's user avatar
4 votes
3 answers
385 views

Wightman quantum field - Interpretation

I have a question regarding the interpretation of the Wightman quantum field in mathematical quantum field theory. A quantum field $\phi$ is a operator-valued distribution. This means that $\phi$ is ...
B.Hueber's user avatar
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How is a half-odd integer spin field defined in the Wightman axioms?

Can anyone give me a proper definition of 'half-odd integer spin fields' in terms of the Wightman axioms? I'm trying to prove the anti-unitarity of the PCT operator in 'PCT, Spin and Statistics, and ...
user353840's user avatar
4 votes
1 answer
271 views

Wightman distributions

According to the Wightman axioms, for Wightman fields $\phi_1,\dots,\phi_n$, the vacuum expectation value $$\langle\Omega, \phi_1(f_1)\dots\phi_n(f_n)\Omega\rangle$$ is a multilinear continuous map ...
user353840's user avatar
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Glimm and Jaffe's paper on the construction of 2D QFTs

I am presently interested in the construction of low-dimensional (2D/3D) QFTs where all the Wightman axioms have been proved and to this end, I started reading this article, which is recommended in ...
IchKenneDeinenNamen's user avatar
4 votes
1 answer
642 views

Why should the vacuum be unique?

Among the Wightman axioms is the requirement that there is a unique Poincare-invariant state called the vacuum. We know that QFT vacuua are not necessarily unique, for example in situations where ...
Nanashi No Gombe's user avatar
6 votes
2 answers
215 views

Extending Wightman axioms to gauge theories

I understand that Wightman axioms were defined for scalar quantum field theories. However, what prevents axiomatization of non-scalar quantum field theories, such as gauge theories, even for free ...
michelav's user avatar
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Classical field theory: marriage of differential geometry to functional analysis?

I have always wondered (and thoroughly - but perhaps not enough - searched for literature) how the purely geometrical formulation of classical field theory (take for example a hardcore text on this ...
DanielC's user avatar
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4 votes
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Wightman's theorem, in terms of time-ordered functions$.$

According to Wightman's theorem, given a set of distributions $\{W_n\}$ satisfying a set of axioms, we may conclude the existence of a set of operators $\{\phi\}$ which satisfy some properties, and ...
AccidentalFourierTransform's user avatar
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1 answer
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What is a precise mathematical statement of the Yang-Mills and mass gap Clay problem?

I am a mathematician writing a statement of each of the Clay Millennium Prize problems in a formal proof assistant.  For the other problems, it seems quite routine to write the conjectures formally, ...
Thales's user avatar
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0 answers
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Axiomatic QFT: Time-slice Axiom vs Transformation Properties

I am studying Wightman axioms and Haag–Kastler axioms for QFT from Haag's book "Local Quantum Physics". In both axiomatic frameworks, he introduces the "Time-slice Axiom" (axiom G) as "There should ...
user avatar
1 vote
1 answer
162 views

Error in Kac's "Vertex algebra for beginners" proof that a Wightman QFT gives rise to a vertex algebra?

Given $$ i[Q_k,\Phi_a(x)]=((x_0^2-x_1^2)\partial_{x_k}-2\eta_k x_k E - 2\Delta_a \eta_k x_k) \Phi_a(x), \quad (1.1.8) $$ applying a coordinate change $t= x_0-x_1$, $\bar{t}= x_0+x_1$ and defining $$ ...
Gytis's user avatar
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3 votes
1 answer
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partition function for Wightman and Haag-Kastler QFT

From what I hear, some modern mathematical approach quantum field theory uses the following definition "A $d$-dimensional $S$-structured quantum field theory $Q$ is a mathematical object, ...
Yul Otani's user avatar
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3 votes
3 answers
315 views

QED as a Wightman theory of observable fields? With a collision theory?

[Note: I'm using QED as a simple example, despite having heard that it is unlikely to exist. I'm happy to confine the question to perturbation theory.] The quantized Aᵘ and ψ fields are non-unique ...
user avatar
8 votes
1 answer
446 views

Asymptotic Completeness, generalized free fields, and the relationship of thermodynamics with infinity

Asymptotic completeness is a strong constraint on quantum field theories that rules out generalized free fields, which otherwise satisfy the Wightman axioms. If we were to take a limit of a list of ...
Peter Morgan's user avatar
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