blue2script
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Hi all!
I worked for hours on this simple commutator of real scalar fields in qft:
\left[\Phi\left(x\right),\Phi\left(y\right) \right] = i\Delta\left( x-y \right)
where
\Delta\left(x\right) = \frac{1}{i}\int {\frac{{d^4 p}}<br /> {{\left( {2\pi } \right)^3 }}\delta \left( {p^2 - m^2 } \right)\operatorname{sgn} \left( {p^0 } \right)e^{ - ip \cdot x} }
The task is to solve this integral and to look at the cases x^2 = 0 and m\rightarrow 0.
But, what I get in my calculations is that for space like x the commutator is zero whereas the integral diverges for time-like x. Normally there should be a Bessel function involved in the end and I am just totally confused now. But maybe I should show the steps I took:
i) Spacelike x
The expression is Lorentz-invariant. For space-like x we go into a reference frame where x_0 is zero, then
\Delta \left( x \right) = \frac{1}<br /> {i}\int {\frac{{d^3 \vec p}}<br /> {{\left( {2\pi } \right)^3 }}\frac{1}<br /> {{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - i\vec p \cdot \vec x} - e^{ - i\vec p \cdot \vec x} } \right)} <br />
and this is zero. What argument fails here? Calculating just one term of the bracket leads to a modified Bessel function. I followed Weinberg, p. 202 here - its strange...
ii) Timelike x
Then we go into a reference frame where \vex x = 0 and we get
\begin{gathered}<br /> \Delta \left( x \right) = \frac{1}<br /> {i}\int {\frac{{d^3 \vec p}}<br /> {{\left( {2\pi } \right)^3 }}\frac{1}<br /> {{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\<br /> = 4\pi \frac{1}<br /> {i}\int {\frac{{dp}}<br /> {{\left( {2\pi } \right)^3 }}\frac{{p^2 }}<br /> {{2\sqrt {p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\ <br /> \end{gathered}<br />
This integral diverges quadratically.
So there has to be something wrong, otherwise this task wouldn't make any sense... I hope somebody can help me here, I am totally lost with this task. I worked hours on that problem and I don't know what to do any more.
A big thanks in advance to everybody!
Blue2script
I worked for hours on this simple commutator of real scalar fields in qft:
\left[\Phi\left(x\right),\Phi\left(y\right) \right] = i\Delta\left( x-y \right)
where
\Delta\left(x\right) = \frac{1}{i}\int {\frac{{d^4 p}}<br /> {{\left( {2\pi } \right)^3 }}\delta \left( {p^2 - m^2 } \right)\operatorname{sgn} \left( {p^0 } \right)e^{ - ip \cdot x} }
The task is to solve this integral and to look at the cases x^2 = 0 and m\rightarrow 0.
But, what I get in my calculations is that for space like x the commutator is zero whereas the integral diverges for time-like x. Normally there should be a Bessel function involved in the end and I am just totally confused now. But maybe I should show the steps I took:
i) Spacelike x
The expression is Lorentz-invariant. For space-like x we go into a reference frame where x_0 is zero, then
\Delta \left( x \right) = \frac{1}<br /> {i}\int {\frac{{d^3 \vec p}}<br /> {{\left( {2\pi } \right)^3 }}\frac{1}<br /> {{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - i\vec p \cdot \vec x} - e^{ - i\vec p \cdot \vec x} } \right)} <br />
and this is zero. What argument fails here? Calculating just one term of the bracket leads to a modified Bessel function. I followed Weinberg, p. 202 here - its strange...
ii) Timelike x
Then we go into a reference frame where \vex x = 0 and we get
\begin{gathered}<br /> \Delta \left( x \right) = \frac{1}<br /> {i}\int {\frac{{d^3 \vec p}}<br /> {{\left( {2\pi } \right)^3 }}\frac{1}<br /> {{2\sqrt {\vec p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\<br /> = 4\pi \frac{1}<br /> {i}\int {\frac{{dp}}<br /> {{\left( {2\pi } \right)^3 }}\frac{{p^2 }}<br /> {{2\sqrt {p^2 + m^2 } }}\left( {e^{ - iEx_0 } - e^{iEx_0 } } \right)} \hfill \\ <br /> \end{gathered}<br />
This integral diverges quadratically.
So there has to be something wrong, otherwise this task wouldn't make any sense... I hope somebody can help me here, I am totally lost with this task. I worked hours on that problem and I don't know what to do any more.
A big thanks in advance to everybody!
Blue2script