# Questions tagged [wightman-fields]

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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 ...
74 views

### 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 ...
92 views

### 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. ...
282 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 ...
29 views

### 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 ...
132 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 ...
73 views

### 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 ...
303 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 ...
138 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 ...
201 views

### 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 ...
199 views

### 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 ...
718 views

### 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, ...
364 views

### 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 ...
132 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  ...
213 views

### 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, ...