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Suppose [itex]\Lambda[/itex] is a Lorentz transformation with the associated Hilbert space unitary operator denoted by [itex]U(\Lambda)[/itex]. We have

[tex]U(\Lambda)|p\rangle = |\Lambda p\rangle[/tex]

and

[tex]|p\rangle = \sqrt{2E_{p}}a_{p}^{\dagger}|0\rangle[/tex]

Equivalently,

[tex]U(\Lambda)|p\rangle = U(\lambda)\sqrt{2E_{p}}a_{p}^{\dagger}|0\rangle[/tex]

Now, by definition,

[tex]|\Lambda p\rangle = \sqrt{2E_{\Lambda p}}a_{\Lambda p}^{\dagger}|0\rangle[/tex]

Therefore it follows that

[tex]\sqrt{2E_{p}}U(\Lambda) a_{p}^{\dagger}|0\rangle = \sqrt{2E_{\Lambda p}} a_{\Lambda p}^{\dagger}|0\rangle[/tex]

or

[tex]U(\Lambda)a_{p}^{\dagger} = \sqrt{\frac{E_{\Lambda p}}{E_p}}a_{\Lambda p}^{\dagger}[/tex]

But apparently the correct expression is

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

Can someone please point out my mistake?

Thanks.

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# Lorentz Transformation and Creation Operators

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