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Hello
I recently stumbled upon an article about Newton Wigner states (cf 10.1103/RevModPhys.21.400). It is repeatedly mentioned in the literature that acting with a Lorentz transformation on a Newton Wigner state completely delocalizes the state. However I was not able to verify this, albeit I found an article by J. Mourad (arXiv:gr-qc/9310018v1), which deals with exactly this question.
The Newton Wigner eigenstates are given by
\psi (x) = \sqrt{\omega (\vec{p})} e^{-i\vec{p}\cdot\vec{q_0}
(a state localized at position q_0 at a time t).
Then the transformed state can be written as (in a basis {|\vec{q}\rangle} of Newton Wigner eigenstates)
\langle \vec{q} | \psi '\rangle = \int \frac{d^3p}{(2\pi)^{3/2}\omega} \langle \vec{q}|\vec{p}\rangle \langle{\vec{p}|\psi '\rangle
where \psi' denotes the Lorentz transformed state. The integral measure has a factor of \omega in the denominator in order to be Lorentz invariant and \langle \vec{q}|\vec{p}\rangle = (2\pi)^{-3/2} e^{i\vec{p}\cdot\vec{q}}.
Unfortunately I do not succeed in evaluating the last inner product.
The whole integral in Mourad reads
\langle \vec{q} | \psi '\rangle = \sqrt{\gamma} \int \frac{d^3p}{(2\pi)^{3}} \sqrt{\left(1-\frac{\vec{\beta}\cdot\vec{p}}{\omega}\right)}e^{i\vec{p}\cdot (\vec{q}-\vec{q_0'})}
where \gamma^2 = 1-\beta^2.
Any help would be highly appreciated :). Thanks
I recently stumbled upon an article about Newton Wigner states (cf 10.1103/RevModPhys.21.400). It is repeatedly mentioned in the literature that acting with a Lorentz transformation on a Newton Wigner state completely delocalizes the state. However I was not able to verify this, albeit I found an article by J. Mourad (arXiv:gr-qc/9310018v1), which deals with exactly this question.
The Newton Wigner eigenstates are given by
\psi (x) = \sqrt{\omega (\vec{p})} e^{-i\vec{p}\cdot\vec{q_0}
(a state localized at position q_0 at a time t).
Then the transformed state can be written as (in a basis {|\vec{q}\rangle} of Newton Wigner eigenstates)
\langle \vec{q} | \psi '\rangle = \int \frac{d^3p}{(2\pi)^{3/2}\omega} \langle \vec{q}|\vec{p}\rangle \langle{\vec{p}|\psi '\rangle
where \psi' denotes the Lorentz transformed state. The integral measure has a factor of \omega in the denominator in order to be Lorentz invariant and \langle \vec{q}|\vec{p}\rangle = (2\pi)^{-3/2} e^{i\vec{p}\cdot\vec{q}}.
Unfortunately I do not succeed in evaluating the last inner product.
The whole integral in Mourad reads
\langle \vec{q} | \psi '\rangle = \sqrt{\gamma} \int \frac{d^3p}{(2\pi)^{3}} \sqrt{\left(1-\frac{\vec{\beta}\cdot\vec{p}}{\omega}\right)}e^{i\vec{p}\cdot (\vec{q}-\vec{q_0'})}
where \gamma^2 = 1-\beta^2.
Any help would be highly appreciated :). Thanks
