Questions tagged [wightman-fields]
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33 questions
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Must all correlation functions come from a measure?
In the Wightman setting a quantum field is defined as an operator valued distribution satisfying the Wightman axioms. Using the vacuum state of this theory we can define correlation/Wightman functions ...
- 3,992
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109
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Renormalisation in rigorous algebraic formulation of QFT
Before I get to the point, let me quickly describe the context and what level of understanding I’m trying to achieve, if possible. I’d like to get some intuition on more rigorous approaches of QFTs, ...
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Wightman correlation, local Lorentzian correlations
They say local Lorentzian correlations are not real functions, but a tempered distribution for Schwarz functions. That is, they are only defined with some smearing. But I always see this only in the ...
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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 ...
- 3,992
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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 ...
- 120
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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 ...
- 7,030
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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 ...
- 3,992
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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 ...
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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 ...
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125
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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 ...
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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 ...
- 1,670
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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 ...
- 121
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160
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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 ...
- 16.3k
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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, \\ ...
- 21
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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}) \...
- 108
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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 ...
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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 ...
user84158
2
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103
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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 ...
- 884
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485
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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.
...
- 16.3k
4
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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 ...
- 884
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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 ...
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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 ...
- 327
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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 ...
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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 ...
- 2,646
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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 ...
- 143
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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 ...
- 4,421
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460
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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 ...
14
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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, ...
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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 ...
user108500
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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)
$$
applying a coordinate change $t= x_0-x_1$, $\bar{t}= x_0+x_1$ and defining
$$
...
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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, ...
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321
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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 ...
Greg Weeks
8
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482
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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 ...
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