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## Main Question or Discussion Point

Hi. I am trying to understand a statement from Peskin and Schroeder at page 59 they write;

"The one particle states

[tex] |\vec p ,s \rangle \equiv \sqrt{2E_{\vec p}}a_{\vec p}^{s \dagger} |0\rangle[/tex]

are defined so that their inner product

[tex]\langle \vec p, r| \vec q,s\rangle = 2 \vec E_\vec{p} (2\pi)^3 \delta^{(3)}(\vec p - \vec q) \delta^{rs}[/tex]

is Lorentz invariant. This implies that the operator [itex]U(\Lambda)[/itex] that implements Lorentz transformations on hte states of the Hilbert space is unitary, even tough for boosts [itex]\Lambda_{1/2}[/itex] is not unitary."

Then they draw the conclusion from the above equations that

[tex]U(\Lambda)a_\vec{p}^s U^{-1}(\Lambda) = \sqrt{ \frac{ E_{\Lambda \vec{p} } }{E_{\vec p} }} a_{\Lambda \vec p}^s.[/tex]

So my question is; how do they see that [itex]U(\Lambda)[/itex] must be unitary? And how do they conclude with the last equation? :)

"The one particle states

[tex] |\vec p ,s \rangle \equiv \sqrt{2E_{\vec p}}a_{\vec p}^{s \dagger} |0\rangle[/tex]

are defined so that their inner product

[tex]\langle \vec p, r| \vec q,s\rangle = 2 \vec E_\vec{p} (2\pi)^3 \delta^{(3)}(\vec p - \vec q) \delta^{rs}[/tex]

is Lorentz invariant. This implies that the operator [itex]U(\Lambda)[/itex] that implements Lorentz transformations on hte states of the Hilbert space is unitary, even tough for boosts [itex]\Lambda_{1/2}[/itex] is not unitary."

Then they draw the conclusion from the above equations that

[tex]U(\Lambda)a_\vec{p}^s U^{-1}(\Lambda) = \sqrt{ \frac{ E_{\Lambda \vec{p} } }{E_{\vec p} }} a_{\Lambda \vec p}^s.[/tex]

So my question is; how do they see that [itex]U(\Lambda)[/itex] must be unitary? And how do they conclude with the last equation? :)