WightmanAxioms

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The Wightman axioms are a set of conditions proposed by Arthur Strong Wightman for a quantum field theory to satisfy in order to provide a consistent mathematical formulation. These conditions are:

1. **State Space**: The states of a quantum field theory are represented as vectors in a separable Hilbert space $\mathcal{H}$.

2. **State Vector**: There exists a unique, normalized vacuum state $|0\rangle \in \mathcal{H}$ such that $P|0\rangle = |0\rangle$ for every Poincaré transformation $P$.

3. **Poincaré Invariance**: The Poincaré group acts unitarily on $\mathcal{H}$. This means there's a strongly continuous unitary representation $U: \mathbb{R}^4 \times SO(1,3)^{\uparrow} \rightarrow B(\mathcal{H})$ where $B(\mathcal{H})$ is the space of bounded operators on $\mathcal{H}$. Here, $\mathbb{R}^4$ is the group of translations and $SO(1,3)^{\uparrow}$ is the restricted Lorentz group. This action also satisfies the spectrum condition, which says the energy-momentum 4-vector $P = (P^0, P^1, P^2, P^3)$ (defined from the generators of the Poincaré group) has a spectrum that lies in the closed forward light cone, i.e., if $p$ is in the spectrum of $P$, then $p^0 \geq 0$ and $(p^0)^2 - (p^1)^2 - (p^2)^2 - (p^3)^2 \geq 0$, with equality only for $p = 0$.

4. **Field Operators**: For each type of particle of spin $s$, there are field operators $\phi(x, \sigma)$ ($\sigma$ ranges from $-s$ to $s$) which are operator-valued tempered distributions. They are covariant under the Poincaré transformations, i.e., $U(a,\Lambda)\phi(x, \sigma)U(a,\Lambda)^{-1} = \Sigma_{\sigma'} D^{(s)}_{\sigma \sigma'}(\Lambda)\phi(\Lambda x + a, \sigma')$ where $D^{(s)}$ is the $(2s+1)$-dimensional unitary representation of $SO(1,3)^{\uparrow}$ and the sum is over $\sigma'$. They also transform under internal symmetries in a particular way if such symmetries exist.

5. **Locality**: The field operators at spacelike separated points either commute or anticommute, i.e., if $(x - y)^2 < 0$ then $[\phi(x, \sigma), \phi(y, \sigma')] = 0$ or $\{\phi(x, \sigma), \phi(y, \sigma')\} = 0$, depending on whether the fields are bosonic or fermionic.

6. **Positivity of the Energy**: The generator $P^0$ of time translations is a positive operator, meaning it has non-negative eigenvalues. This is also known as the spectrum condition and is already implied by the third axiom.

7. **Existence of a Hermitian scalar field**: There exists at least one Hermitian scalar field (a quantum field whose corresponding quantum mechanical operator is Hermitian) among the set of fields for the theory.

8. **Completeness (or "Reeh-Schlieder theorem")**: For any open set $O$ in Minkowski spacetime, the set of vectors that can be obtained by acting on the vacuum with a field operator with support in $O$ is dense in $\mathcal{H}$.

Each of these axioms can be expanded into further mathematical detail and has its own proofs, interpretations, and physical implications. These axioms are used as a foundation in rigorous mathematical physics to build up quantum field theories.

The Wightman axioms are a set of conditions proposed by Arthur Strong Wightman for a quantum field theory to satisfy in order to provide a consistent mathematical formulation. These conditions are:

1. State Space: The states of a quantum field theory are represented as vectors in a separable Hilbert space $\mathcal{H}$.

2. State Vector: There exists a unique, normalized vacuum state $|0⟩ \in \mathcal{H}$ such that $P|0⟩ = |0⟩$ for every Poincaré transformation P.

3. Poincaré Invariance: The Poincaré group acts unitarily on $\mathcal{H}$. This means there's a strongly continuous unitary representation

$$U: \mathbb{R}^4 \times SO(1,3)^{up} \to B(\mathcal{H})$$ where $B(\mathcal{H})$ is the space of bounded operators on $\mathcal{H}$. Here, $\mathbb{R}^4$ is the group of translations and $SO(1,3)^{up}$ is the restricted Lorentz group. This action also satisfies the spectrum condition, which says the energy-momentum 4-vector $P = (P^0, P^1, P^2, P^3)$ (defined from the generators of the Poincaré group) has a spectrum that lies in the closed forward light cone, i.e., if p is in the spectrum of P, then $p^0 \geq 0$ and

$$(p^0)^2 - (p^1)^2 - (p^2)^2 - (p^3)^2 \geq 0$$ with equality only for $p = 0$.

4. Field Operators: For each type of particle of spin s, there are field operators $\phi(x, \sigma)$ ($\sigma$ ranges from $-s$ to $s$) which are operator-valued tempered distributions. They are covariant under the Poincaré transformations, i.e.,

$$U(a,\Lambda)\phi(x, \sigma)U(a,\Lambda)^{-1} = \sum_{\sigma'} D^{(s)}_{\sigma \sigma'}(\Lambda)\phi(\Lambda x + a, \sigma')$$ where $D^{(s)}$ is the $(2s+1)$-dimensional unitary representation of $SO(1,3)^{up}$ and the sum is over $\sigma'$. They also transform under internal symmetries in a particular way if such symmetries exist.

5. Locality: The field operators at spacelike separated points either commute or anticommute, i.e., if $(x - y)^2 < 0$ then the commutator $[\phi(x, \sigma), \phi(y, \sigma')] = 0$ for bosonic fields or the anticommutator $\{\phi(x, \sigma), \phi(y, \sigma')\} = 0$ for fermionic fields is used.

6. Positivity of the Energy: The generator $P^0$ of time translations is a positive operator, meaning it has non-negative eigenvalues. This is also known as the spectrum condition and is already implied by the third axiom.

7. Existence of a Self-Adjoint Scalar Field: There exists at least one self-adjoint scalar field (a quantum field whose corresponding quantum mechanical operator is self-adjoint) among the set of fields for the theory.

8. Completeness (or "Reeh-Schlieder theorem"): For any open set $O$ in Minkowski spacetime, the set of vectors that can be obtained by acting on the vacuum with a field operator with support in $O$ is dense in $\mathcal{H}$.

Each of these axioms can be expanded into further mathematical detail and has its own proofs, interpretations, and physical implications. These axioms are used as a foundation in rigorous mathematical physics to build up quantum field theories.