- #1
center o bass
- 560
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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? :)