I'm just reading the schroeder/peskin introduction to quantum field theory. On Page 21 there is the equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\phi(x)=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{ \sqrt{2\omega_{\vec{p}}} }

(a_{\vec{p}} e^{i \vec{p} \cdot \vec{x}}

+a^{+}_{\vec{p}} e^{-i \vec{p} \cdot \vec{x}}

)[/tex]

and in the next step:

[tex]\phi(x)=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{ \sqrt{2\omega_{\vec{p}}} }

(a_{\vec{p}}

+a^{+}_{\vec{-p}}

)e^{i \vec{p} \cdot \vec{x}}[/tex]

with [tex]\omega_{\vec{p}}=\sqrt{|\vec{p}|^2+m^2}[/tex]

I don't understand that. When I substitute [tex]\vec{p}[/tex] for [tex]-\vec{p}[/tex] shouldn't the Jacobi-determinant then put a minus sign such that:

[tex]\phi(x)=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{ \sqrt{2\omega_{\vec{p}}} }

(a_{\vec{p}}

-a^{+}_{\vec{-p}}

)e^{i \vec{p} \cdot \vec{x}}[/tex]

What's wrong with me?

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# Understanding Klein-Gordon

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